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Special IssueWell-Posed and Ill-Posed Boundary Value Problems for PDE

Guest EditorsAllaberen Ashyralyev, Sergey Piskarev, Valery Covachev, Ravshan Ashurov, and Hasan Ali Yurtsever

Well-Posed and Ill-Posed BoundaryValue Problems for PDE

Abstract and Applied Analysis

Well-Posed and Ill-Posed BoundaryValue Problems for PDE

Guest Editors: Allaberen Ashyralyev, Sergey Piskarev,Valery Covachev, Ravshan Ashurov,and Hasan Ali Yurtsever

Copyright q 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Abstract and Applied Analysis.” All articles are open access articles distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium,provided the original work is properly cited.

Editorial BoardDirk Aeyels, BelgiumRavi P. Agarwal, USAM. O. Ahmedou, GermanyNicholas D. Alikakos, GreeceDebora Amadori, ItalyPablo Amster, ArgentinaDouglas R. Anderson, USAJan Andres, Czech RepublicGiovanni Anello, ItalyStanislav Antontsev, PortugalMohamed Kamal Aouf, EgyptNarcisa C. Apreutesei, RomaniaNatig Atakishiyev, MexicoFerhan M. Atici, USAIvan G. Avramidi, USASoohyun Bae, KoreaChuanzhi Bai, ChinaZhanbing Bai, ChinaDumitru Baleanu, TurkeyJozef Banas, PolandGerassimos Barbatis, GreeceMartino Bardi, ItalyRoberto Barrio, SpainFeyzi Basar, TurkeyA. Bellouquid, MoroccoDaniele Bertaccini, ItalyMichiel Bertsch, ItalyLucio Boccardo, ItalyIgor Boglaev, New ZealandMartin J. Bohner, USAJulian F. Bonder, ArgentinaGeraldo Botelho, BrazilElena Braverman, CanadaRomeo Brunetti, ItalyJanusz Brzdek, PolandDetlev Buchholz, GermanySun-Sig Byun, KoreaFabio M. Camilli, ItalyAntonio Canada, SpainJinde Cao, ChinaAnna Capietto, ItalyKwang-chih Chang, China

Jianqing Chen, ChinaWing-Sum Cheung, Hong KongMichel Chipot, SwitzerlandChangbum Chun, KoreaSoon Y. Chung, KoreaJaeyoung Chung, KoreaSilvia Cingolani, ItalyJean M. Combes, FranceMonica Conti, ItalyDiego Cordoba, SpainJ. Carlos Cortes Lopez, SpainGraziano Crasta, ItalyGuillermo P. Curbera, SpainB. Dacorogna, SwitzerlandVladimir Danilov, RussiaMohammad T. Darvishi, IranL. F. P. de Castro, PortugalToka Diagana, USAJesus I. Dıaz, SpainJosef Diblık, Czech RepublicFasma Diele, ItalyTomas Dominguez, SpainA. I. Domoshnitsky, IsraelMarco Donatelli, ItalyOndrej Dosly, Czech RepublicWei-Shih Du, TaiwanLuiz Duarte, BrazilRoman Dwilewicz, USAPaul W. Eloe, USAAhmed El-Sayed, EgyptLuca Esposito, ItalyJose A. Ezquerro, SpainKhalil Ezzinbi, MoroccoJesus G. Falset, SpainAngelo Favini, ItalyMarcia Federson, BrazilS. Filippas, Equatorial GuineaAlberto Fiorenza, ItalyTore Flatten, NorwayIlaria Fragala, ItalyBruno Franchi, ItalyXianlong Fu, China

Massimo Furi, ItalyGiovanni P. Galdi, USAIsaac Garcia, SpainJose A. Garcıa-Rodrıguez, SpainLeszek Gasinski, PolandGyorgy Gat, HungaryVladimir Georgiev, ItalyLorenzo Giacomelli, ItalyJaume Gin, SpainValery Y. Glizer, IsraelLaurent Gosse, ItalyJean P. Gossez, BelgiumDimitris Goussis, GreeceJose L. Gracia, SpainMaurizio Grasselli, ItalyYuxia Guo, ChinaQian Guo, ChinaChaitan P. Gupta, USAUno Hamarik, EstoniaFerenc Hartung, HungaryBehnam Hashemi, IranNorimichi Hirano, JapanJiaxin Hu, ChinaChengming Huang, ChinaZhongyi Huang, ChinaGennaro Infante, ItalyIvan G. Ivanov, BulgariaHossein Jafari, IranJaan Janno, EstoniaAref Jeribi, TunisiaUn C. Ji, KoreaZhongxiao Jia, ChinaLucas Jodar, SpainJong S. Jung, KoreaVarga K. Kalantarov, TurkeyHenrik Kalisch, NorwaySatyanad Kichenassamy, FranceTero Kilpelainen, FinlandSung G. Kim, KoreaLjubisa Kocinac, SerbiaAndrei Korobeinikov, SpainPekka Koskela, Finland

Victor Kovtunenko, AustriaPavel Kurasov, SwedenMiroslaw Lachowicz, PolandKunquan Lan, CanadaRuediger Landes, USAIrena Lasiecka, USAMatti Lassas, FinlandChun-Kong Law, TaiwanMing-Yi Lee, TaiwanGongbao Li, ChinaPedro M. Lima, PortugalElena Litsyn, IsraelShengqiang Liu, ChinaYansheng Liu, ChinaCarlos Lizama, ChileMilton C. Lopes Filho, BrazilJulian Lopez-Gomez, SpainJinhu Lu, ChinaGrzegorz Lukaszewicz, PolandShiwang Ma, ChinaWanbiao Ma, ChinaEberhard Malkowsky, TurkeySalvatore A. Marano, ItalyCristina Marcelli, ItalyPaolo Marcellini, ItalyJesus Marın-Solano, SpainJose M. Martell, SpainMieczysław S. Mastyło, PolandMing Mei, CanadaTaras Mel’nyk, UkraineAnna Mercaldo, ItalyChangxing Miao, ChinaStanislaw Migorski, PolandMihai Mihailescu, RomaniaFeliz Minhos, PortugalDumitru Motreanu, FranceRoberta Musina, ItalyMaria Grazia Naso, ItalyGaston M. N’Guerekata, USASylvia Novo, SpainMicah Osilike, NigeriaMitsuharu Otani, JapanTurgut Ozis, TurkeyFilomena Pacella, ItalyN. S. Papageorgiou, Greece

Sehie Park, KoreaAlberto Parmeggiani, ItalyKailash C. Patidar, South AfricaKevin R. Payne, ItalyJosip E. Pecaric, CroatiaShuangjie Peng, ChinaSergei V. Pereverzyev, AustriaMaria Eugenia Perez, SpainDavid Perez-Garcia, SpainAllan Peterson, USAAndrew Pickering, SpainCristina Pignotti, ItalySomyot Plubtieng, ThailandMilan Pokorny, Czech RepublicSergio Polidoro, ItalyZiemowit Popowicz, PolandMaria M. Porzio, ItalyEnrico Priola, ItalyVladimir S. Rabinovich, MexicoI. Rachunkova, Czech RepublicMaria A. Ragusa, ItalySimeon Reich, IsraelWeiqing Ren, USAAbdelaziz Rhandi, ItalyHassan Riahi, MalaysiaJuan P. Rincon-Zapatero, SpainLuigi Rodino, ItalyYuriy Rogovchenko, NorwayJulio D. Rossi, ArgentinaWolfgang Ruess, GermanyBernhard Ruf, ItalyMarco Sabatini, ItalySatit Saejung, ThailandStefan Samko, PortugalMartin Schechter, USAJavier Segura, SpainSigmund Selberg, NorwayValery Serov, FinlandNaseer Shahzad, Saudi ArabiaAndrey Shishkov, UkraineStefan Siegmund, GermanyA. A. Soliman, EgyptPierpaolo Soravia, ItalyMarco Squassina, ItalyS. Stanek, Czech Republic

Stevo Stevic, SerbiaAntonio Suarez, SpainWenchang Sun, ChinaRobert Szalai, UKSanyi Tang, ChinaChun-Lei Tang, ChinaYoushan Tao, ChinaGabriella Tarantello, ItalyN. Tatar, Saudi ArabiaRoger Temam, USASusanna Terracini, ItalyGerd Teschke, GermanyAlberto Tesei, ItalyBevan Thompson, AustraliaSergey Tikhonov, SpainClaudia Timofte, RomaniaThanh Tran, AustraliaJuan J. Trujillo, SpainCiprian A. Tudor, FranceGabriel Turinici, FranceMehmet Unal, TurkeyS. A. van Gils, The NetherlandsCsaba Varga, RomaniaCarlos Vazquez, SpainGianmaria Verzini, ItalyJesus Vigo-Aguiar, SpainYushun Wang, ChinaXiaoming Wang, USAJing Ping Wang, UKShawn X. Wang, CanadaYouyu Wang, ChinaPeixuan Weng, ChinaNoemi Wolanski, ArgentinaNgai-Ching Wong, TaiwanPatricia J. Y. Wong, SingaporeRoderick Wong, Hong KongZili Wu, ChinaYong Hong Wu, AustraliaTiecheng Xia, ChinaXu Xian, ChinaYanni Xiao, ChinaFuding Xie, ChinaNaihua Xiu, ChinaDaoyi Xu, ChinaXiaodong Yan, USA

Zhenya Yan, ChinaNorio Yoshida, JapanBeong I. Yun, KoreaVjacheslav Yurko, RussiaA. Zafer, TurkeySergey V. Zelik, UK

Weinian Zhang, ChinaChengjian Zhang, ChinaMeirong Zhang, ChinaZengqin Zhao, ChinaSining Zheng, ChinaTianshou Zhou, China

Yong Zhou, ChinaChun-Gang Zhu, ChinaQiji J. Zhu, USAMalisa R. Zizovic, SerbiaWenming Zou, China

Contents

Well-Posed and Ill-Posed Boundary Value Problems for PDE, Allaberen Ashyralyev,Sergey Piskarev, Valery Covachev, Ravshan Ashurov, and Hasan Ali YurtseverVolume 2012, Article ID 492589, 2 pages

An Approximation of Semigroups Method for Stochastic Parabolic Equations,Allaberen Ashyralyev and Mehmet Emin SanVolume 2012, Article ID 684248, 24 pages

FDM for Elliptic Equations with Bitsadze-Samarskii-Dirichlet Conditions,Allaberen Ashyralyev and Fatma Songul Ozesenli TetikogluVolume 2012, Article ID 454831, 22 pages

Regularity for Variational Evolution Integrodifferential Inequalities, Yong Han Kang andJin-Mun JeongVolume 2012, Article ID 797516, 18 pages

Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-ParabolicEquations in Holder Spaces, Okan GercekVolume 2012, Article ID 237657, 12 pages

Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution,Asylzat Kopzhassarova and Abdizhakhan SarsenbiVolume 2012, Article ID 576843, 6 pages

A Note on the Inverse Problem for a Fractional Parabolic Equation, Abdullah Said Erdogan andHulya UygunVolume 2012, Article ID 276080, 26 pages

Exact Asymptotic Expansion of Singular Solutions for the (2 + 1)-D Protter Problem,Lubomir Dechevski, Nedyu Popivanov, and Todor PopovVolume 2012, Article ID 278542, 33 pages

A Note on the Second Order of Accuracy Stable Difference Schemes for the NonlocalBoundary Value Hyperbolic Problem, Allaberen Ashyralyev and Ozgur YildirimVolume 2012, Article ID 846582, 29 pages

Classification of Exact Solutions for Some Nonlinear Partial Differential Equations withGeneralized Evolution, Yusuf Pandir, Yusuf Gurefe, Ugur Kadak, and Emine MisirliVolume 2012, Article ID 478531, 16 pages

Numerical Solution of Stochastic Hyperbolic Equations, Necmettin Aggez andMaral AshyralyyewaVolume 2012, Article ID 824819, 20 pages

Spectral Properties of Non-Self-Adjoint Perturbations for a Spectral Problem with Involution,Asylzat A. Kopzhassarova, Alexey L. Lukashov, and Abdizhakhan M. SarsenbiVolume 2012, Article ID 590781, 5 pages

The Numerical Solution of the Bitsadze-Samarskii Nonlocal Boundary Value Problems withthe Dirichlet-Neumann Condition, Allaberen Ashyralyev and Elif OzturkVolume 2012, Article ID 730804, 13 pages

Existence and Nonexistence of Positive Solutions for Quasilinear Elliptic Problem, K. SaoudiVolume 2012, Article ID 275748, 9 pages

On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator forDistributions, Ravshan Ashurov, Almaz Butaev, and Biswajeet PradhanVolume 2012, Article ID 649848, 13 pages

A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a BanachSpace, Maksat AshyraliyevVolume 2012, Article ID 178084, 18 pages

Finite Difference Method for the Reverse Parabolic Problem, Charyyar Ashyralyyev,Ayfer Dural, and Yasar SozenVolume 2012, Article ID 294154, 17 pages

On the Regularized Solutions of Optimal Control Problem in a Hyperbolic System,Yesim Sarac and Murat SubasıVolume 2012, Article ID 156541, 12 pages

Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-NeumannConditions, Zafer CakirVolume 2012, Article ID 463746, 17 pages

Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of theLimiting Operator, Burkhan T. Kalimbetov, Marat A. Temirbekov,and Zhanibek O. KhabibullayevVolume 2012, Article ID 120192, 16 pages

Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order,Djumaklych Amanov and Allaberen AshyralyevVolume 2012, Article ID 973102, 16 pages

A Note on the Right-Hand Side Identification Problem Arising in Biofluid Mechanics,Abdullah Said ErdoganVolume 2012, Article ID 548508, 25 pages

On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-ParabolicEquations, Allaberen Ashyralyev and Okan GercekVolume 2012, Article ID 230190, 13 pages

On Global Solutions for the Cauchy Problem of a Boussinesq-Type Equation, Hatice Taskesen,Necat Polat, and Abdulkadir ErtasVolume 2012, Article ID 535031, 10 pages

Inverse Scattering from a Sound-Hard Crack via Two-Step Method, Kuo-Ming LeeVolume 2012, Article ID 810676, 13 pages

Existence and Uniqueness of Solutions for the System of Nonlinear Fractional DifferentialEquations with Nonlocal and Integral Boundary Conditions, Allaberen Ashyralyev andYagub A. SharifovVolume 2012, Article ID 594802, 14 pages

Efficient Variational Approaches for Deformable Registration of Images, Mehmet Ali Akinlar,Muhammet Kurulay, Aydin Secer, and Mustafa BayramVolume 2012, Article ID 704567, 8 pages

The Difference Problem of Obtaining the Parameter of a Parabolic Equation,Charyyar Ashyralyyev and Oznur DemirdagVolume 2012, Article ID 603018, 14 pages

Kamenev-Type Oscillation Criteria for the Second-Order Nonlinear Dynamic Equations withDamping on Time Scales, M. Tamer SenelVolume 2012, Article ID 253107, 18 pages

Generalizations of Wendroff Integral Inequalities and Their Discrete Analogues,Maksat AshyraliyevVolume 2012, Article ID 768062, 15 pages

A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrodinger Equations,Yildirim Ozdemir and Mehmet KucukunalVolume 2012, Article ID 687321, 12 pages

A New Approach for Linear Eigenvalue Problems and Nonlinear Euler Buckling Problem,Meltem Evrenosoglu Adiyaman and Sennur SomaliVolume 2012, Article ID 697013, 21 pages

An Approximation of Ultra-Parabolic Equations, Allaberen Ashyralyev and Serhat YılmazVolume 2012, Article ID 840621, 14 pages

Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition,Deniz AgirsevenVolume 2012, Article ID 682752, 31 pages

Existence Results for Solutions of Nonlinear Fractional Differential Equations, Ali Yakar andMehmet Emir KoksalVolume 2012, Article ID 267108, 12 pages

On the Global Well-Posedness of the Viscous Two-Component Camassa-Holm System,Xiuming LiVolume 2012, Article ID 327572, 15 pages

The Local and Global Existence of Solutions for a Generalized Camassa-Holm Equation,Nan Li, Shaoyong Lai, Shuang Li, and Meng WuVolume 2012, Article ID 532369, 26 pages

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 492589, 2 pagesdoi:10.1155/2012/492589

EditorialWell-Posed and Ill-Posed Boundary ValueProblems for PDE

Allaberen Ashyralyev,1 Sergey Piskarev,2 Valery Covachev,3Ravshan Ashurov,4 and Hasan Ali Yurtsever5

1 Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey2 Science Research Computer Center of Moscow State University, Vorobjevy Gory, Moscow 119899, Russia3 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University,123 Muscat, Oman

4 Universiti Putra Malaysia, Institute of Advanced Technology, 43400 Serdang, Selangor, Malaysia5 American Islamic College, Chicago, IL 60613, USA

Correspondence should be addressed to Allaberen Ashyralyev, [email protected]

Received 14 October 2012; Accepted 14 October 2012

Copyright q 2012 Allaberen Ashyralyev et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The studies of well-posed and ill-posed local and nonlocal boundary value problems forpartial differential equations are driven not only by a theoretical interest but also by the factthat several phenomena in engineering, various fields of physics and financial mathematicscan be modeled and investigated in this way.

The present special issue is devoted to the publication of high-quality researchpapers in the fields of the construction and investigation of analytic and numerical methodsfor solutions of well-posed and ill-posed boundary value problems for partial differentialequations.

The issue covers a wide variety of problems for different classes of ordinary andpartial differential equations, as well as dynamic equations on time scales. The topicsdiscussed in the contributed papers are traditional for qualitative theory of differentialequations. The issue contains papers on the global well-posedness of the viscous two-component Camassa-Holm system, local and global existence of solutions for a generalizedCamassa-Holm equation, global solutions for the Cauchy problem of a Boussinesq-typeequation, exact asymptotic expansion of singular solutions for the (2 + 1)-D Protter problem,on the regularity for variational evolution integrodifferential in equalities, right-hand sideidentification problem arising in biofluid mechanics, regularized solutions of optimal controlproblem in a hyperbolic system, generalized localization of Fourier inversion associated withan elliptic operator for distributions and Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on-time scales. Furthermore, classification

2 Abstract and Applied Analysis

of exact solutions for some nonlinear partial differential equations with generalized evolutionis presented.

A number of papers are concerned with well-posedness of partial differentialand difference equations. Interesting stability results are obtained for nonlocal boundaryvalue problems for hyperbolic Schrodinger equations, difference schemes of ultraparabolicequations and reverse parabolic problem, approximate solutions of delay parabolic equationswith the Dirichlet condition, difference problem of obtaining the parameter of a parabolicequation, integral-differential equation of the parabolic type in a Banach space, the first andsecond order of accuracy stable implicit different schemes for elliptic-parabolic equationsin Holder spaces, the second order of accuracy stable difference schemes for the nonlocalboundary value hyperbolic problem, and the numerical solution of the Bitsadze-Samarskiinonlocal boundary value problems with the Dirichlet-Neumann condition. Moreover,applications of generalizations of Wendroff integral inequalities and their discrete analoguesand operator approach to investigate for stability of hyperbolic equations are presented. Twopapers collected in this special issue address construction and investigation of differenceschemes for numerical solutions of stochastic parabolic and hyperbolic equations.

Several authors deal with different aspects of the theory of boundary value problemsfor fractional ordinary and partial differential equations. Interesting existence and uniquenessresults are obtained for a class of nonlinear fractional differential equations, system ofnonlinear fractional differential equations with nonlocal and integral boundary conditions,fractional partial differential equations of higher order. Moreover, applications of operatorapproach to investigate for stability of difference schemes for fractional parabolic partialdifferential equations with the Dirichlet-Neumann conditions and well-posedness of theinverse problem for a fractional parabolic equation are presented.

Two papers collected in this special issue address spectrum of differential operatorsand its applications for nonlinear Euler buckling problem and asymptotic solutions ofsingular perturbed problems with an unstable spectrum of the limiting operator. Moreover,basis properties of eigenfunctions of second-order differential operators with involution, andspectral properties of non-self-adjoint perturbations for a spectral problem with involutionare presented. Finally, some applied problems are also considered—an efficient variationalapproach for deformable registration of images, and an inverse scattering from a sound-hardcrack via two-step method.

This volume was a collection of 36 accepted manuscripts by 61 authors. The selectionof the papers included in this volume was based on an international peer-review procedure.The accepted manuscripts examine wide ranging and cutting edge developments in variousareas of well-posed and ill-posed local and nonlocal boundary value problems for partialdifferential equations. The papers give a taste of current research. We feel the variety of topicswill be of interest to both graduate students and researchers.

Further, we are very grateful to all authors for sending their new papers for thepublication in the present special issue.

Allaberen AshyralyevSergey PiskarevValery CovachevRavshan Ashurov

Hasan Ali Yurtsever


Research ArticleAn Approximation of Semigroups Method forStochastic Parabolic Equations

Allaberen Ashyralyev1, 2 and Mehmet Emin San3

1 Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey2 Department of Mathematics, ITTU, 74012 Ashgabat, Turkmenistan3 Certified Dental Supply LLC 43 River Road, Nutley, NJ 07031, USA

Correspondence should be addressed to Mehmet Emin San, [email protected]

Received 25 May 2012; Accepted 8 June 2012

Academic Editor: Ravshan Ashurov

Copyright q 2012 A. Ashyralyev and M. E. San. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A single-step difference scheme for the numerical solution of the nonlocal-boundary valueproblem for stochastic parabolic equations is presented. The convergence estimate for thesolution of the difference scheme is established. In application, the convergence estimates forthe solution of the difference scheme are obtained for two nonlocal-boundary value problems.The theoretical statements for the solution of this difference scheme are supported by numericalexamples.

1. Introduction

It is known that most problems in heat flow, fusion process, model financial instrumentslike options, bonds, and interest rates, and other areas which are involved with uncertaintylead to stochastic differential equation with parabolic type. These equations can be derivedas models of indeterministic systems and considered as methods for solving boundary valueproblems.

The method of operators as a tool for investigation of the solution to stochastic partialdifferential equations in Hilbert and Banach spaces has been systematically developed byseveral authors (see [1–4] and the references therein). Finite difference method for thesolution of initial boundary value problem for stochastic differential equations has beenstudied extensively by many researchers (see [5–15] and the references therein). However,multipoint nonlocal-boundary value problems were not well investigated.


In the present paper the multipoint nonlocal-boundary value problem

dv(t) = −Av(t)dt + f(t)dwt, 0 < t < T

v(0) =J∑

j=1

αjv(λj)+ ϕ(wλ1 , . . . , wλJ

),

J∑

j=1

∣∣αj∣∣ ≤ 1, 0 < λ1 < · · · < λJ ≤ T, 0 ≤ t ≤ T

(1.1)

for stochastic parabolic differential equations in a Hilbert spaceH with a self-adjoint positivedefinite operator A is considered. Here

(i) wt is a standard Wiener process given on the probability space (Ω, F, P).

(ii) For any z ∈ [0, T], f(z) is an element of spaceM2w([0, T],H1), whereH1 is subspace

ofH.

(iii) ϕ(wλ1 , . . . , wλJ ) is element of space M2w([0, T],H2) of H2-valued measurable

processes, whereH2 is a subspace ofH.

Here, M2w([0, T],H)[20] denote the space of H-valued measurable processes which

satisfy

(a) φ(t) is Ft measurable, a.e. in t,

(b) E∫T0 ‖φ(t)‖Hdt <∞.

The main goal of this study is to construct and investigate the difference schemesfor the multipoint nonlocal-boundary value problems (1.1). The outline of the paper is asfollows. In Section 2, the exact single-step difference scheme for the solution of the problem(1.1) is presented. In Section 3, the 1/2-th order of accuracy Rothe difference scheme isconstructed and investigated for the approximate solution of the problem (1.1). The estimateof convergence for the solution of this difference scheme is obtained. In applications, theconvergence estimates for the solution of difference schemes for the numerical solution oftwo multipoint nonlocal-boundary value problems for stochastic parabolic equations areobtained. In Section 4, the numerical application for one-dimensional stochastic parabolicequation is presented.

2. The Exact Single-Step Difference Scheme

Now, let us give some lemmas we need in the sequel. Throughout this paper, let H be aHilbert space, let A be a positive definite self-adjoint operator with A ≥ δI, where δ > 0.

Lemma 2.1. The following estimate holds:

∥∥∥e−tA∥∥∥H→H

≤ e−δt (t ≥ 0),∥∥∥Ae−tA

∥∥∥H→H

≤ 1t

(t > 0). (2.1)

Abstract and Applied Analysis 3

Lemma 2.2. Suppose that assumption

J∑

k=1

|αk| ≤ 1 (2.2)

holds. Then, the operator

I −J∑

k=1

αke−λkA (2.3)

has an inverse

Υ =

(I −

J∑

k=1

αke−λkA

)−1

, (2.4)

and the following estimate is satisfied:

‖Υ‖H→H ≤ 11 − e−λ1δ ≤ C(δ, λ1). (2.5)

Proof. The proof follows from the triangle inequality, assumption (2.2), and estimate

∥∥∥∥∥∥

(I −

J∑

k=1

αke−λkA

)−1∥∥∥∥∥∥H→H

≤ supδ≤μ<∞

1∣∣∣1 −∑Jk=1 αke

−λkμ∣∣∣. (2.6)

Let us now obtain the formula for the mild solution of problem (1.1). It is clear that under theassumptions (i)-(ii) and

E‖v(0)‖2H2<∞, H2 ⊂ H, (2.7)

the Cauchy problem

dv(t) = −Av(t)dt + f(t)dwt, 0 < t < T, v(0) is given (2.8)

and has a unique mild solution, which is represented by the following formula:

v(t) = e−Atv(0) +∫ t

0e−A(t−s)f(s)dws. (2.9)

Then from this formula and the multipoint nonlocal-boundary condition

v(0) =J∑

j=1

αjv(λj)+ ϕ(wλ1 , . . . , wλJ

), (2.10)


we get

v(0) =J∑

j=1

αje−Aλjv(0) +

J∑

j=1

αj

∫λj

0e−A(λj−s)f(s)dws + ϕ

(wλ1 , . . . , wλJ

). (2.11)

By Lemma 2.2 the operator I −∑Jj=1 αje

−Aλj has a bounded inverse Υ = (I −∑Jj=1 αje

−Aλj )−1.

Then

v(0) = Υ

⎧⎨

⎩

J∑

j=1

αj

∫λj


(wλ1 , . . . , wλJ

)⎫⎬

⎭. (2.12)

Therefore, we have formulas (2.9) and (2.12) for the solution of problem (1.1).

Now, we will consider the single-step exact difference scheme. On the segment [0, T]we consider the uniform grid space

[0, T]τ = {tk = kτ, k = 0, 1, . . . ,N, Nτ = T} (2.13)

with step τ > 0. HereN is a fixed positive integer.

Theorem 2.3. Let v(tk) be the solution of (1.1) at the grid points t = tk. Then {v(tk)}N0 is thesolution of the multipoint nonlocal-boundary value problem for the following difference equation (see[16]):

v(tk) − v(tk−1) +(I − e−τA

)v(tk−1) =

∫ tk

tk−1e−(tk−s)Af(s)dws, 1 ≤ k ≤N, (2.14)

v(0) = Υ

⎧⎨

⎩

J∑

j=1

αj

∫λj


(wλ1 , . . . , wλJ

)⎫⎬

⎭. (2.15)

Proof. Putting t = tk and t = tk−1 into the formula (2.9), we can write

v(tk) = e−tkAv(0) +∫ tk

0e−(tk−s)Af(s)dws,

v(tk−1) = e−tk−1Av(0) +∫ tk−1

0e−(tk−1−s)Af(s)dws.

(2.16)

Hence, we obtain the following relation between v(tk) and v(tk−1):

v(tk) = e−τAv(tk−1) +∫ tk

tk−1e−(tk−s)Af(s)dws. (2.17)

Last relation and equality (2.14) are equivalent. Theorem 2.3 is proved.


Note that problem (2.14) is called the single-step exact difference scheme for thesolution of the problem (1.1).

3. Rothe Difference Scheme

In this section, the 1/2-th order of accuracy Rothe difference scheme is constructed andinvestigated for the approximate solution of the problem (1.1). The estimate of convergencefor the solution of this difference scheme is established. In applications, the convergenceestimates for the solution of difference schemes for the numerical solution of two multipointnonlocal-boundary value problems for stochastic parabolic equations are obtained.

3.1. 1/2-th Order-of-Accuracy Rothe Difference Scheme

Let us give some lemmas we need in the sequel.

Lemma 3.1. The following estimates hold:

∥∥∥AαRk∥∥∥H→H

≤ 1(kτ)α

, 1 ≤ k ≤N, 0 ≤ α ≤ 1, (3.1)

∥∥∥A−α(Rk − e−kτA

)∥∥∥H→H

≤ 2τα

k1−α, 1 ≤ k ≤N, 0 ≤ α ≤ 2, (3.2)

where R = (I + τA)−1.

Lemma 3.2. Suppose that assumption (2.2) holds. Then, the operator

I −J∑

j=1

αjR[λj/τ] (3.3)

has a bounded inverse

Υτ =

⎛

⎝I −J∑

j=1

αjR[λj/τ]

⎞

⎠−1

, (3.4)

and the following estimate is satisfied:

‖Υτ‖H→H ≤ C(δ, λ1). (3.5)

Proof. The proof follows from the triangle inequality, assumption (2.2), and estimate

∥∥∥∥∥∥∥

⎛

⎝I −J∑

j=1

αjR[λj/τ]

⎞

⎠−1∥∥∥∥∥∥∥

H→H

≤ supδ≤μ<∞

1∣∣∣1 −∑Jj=1 αk

(1/(1 + μτ

)[λk/τ])∣∣∣. (3.6)


From (2.14) it is clear that for the approximate solution of the multipoint nonlocal-boundaryvalue problem (1.1) it is necessary to approximate the expressions

e−τA,1τ

∫ tk

tk−1e−(tk−s)Af(s)dw, (3.7)

and multipoint nonlocal-boundary condition

v(0) =J∑

j=1

αjv(λj)+ ϕ(wλ1 , . . . , wλJ

). (3.8)

It is possible under stronger assumption than (ii) for f(t). Assume that

maxt∈[0,T]

∥∥∥A−1/2f ′(t)∥∥∥H+ maxt∈[0,T]

∥∥∥A1/2f(t)∥∥∥H

≤ C. (3.9)

Replacing the expressions e−τA, e−(tk−s)A with R = (I + τA)−1, the expression v(λj) withv([λj/τ]τ), and the function f(s) with f(tk−1), we get the implicit Rothe difference scheme:

uk − uk−1 + τAuk = f(tk−1)(wtk −wtk−1), 1 ≤ k ≤N,

u0 =J∑

j=1

αju[λj/τ] + ϕ(wλ1 , . . . , wλJ

).

(3.10)

Let us now obtain the formula for the solution of problem (3.10). It is clear that the Rothedifference scheme

uk − uk−1 + τAuk = f(tk−1)(wtk −wtk−1), 1 ≤ k ≤N, u0 is given, (3.11)

for the solution of the Cauchy problem (2.8) has a unique solution, which is represented bythe following formula:

uk = Rku0 +k∑

s=1

Rk−s+1f(ts−1)(wts −wts−1), 1 ≤ k ≤N. (3.12)

Then from this formula and the multipoint nonlocal-boundary condition

u0 =J∑

j=1

αju[λj/τ] + ϕ(wλ1 , . . . , wλJ

), (3.13)


we get

u0 =J∑

j=1

αjR[λj/τ]u0 +

J∑

j=1

αj

[λj/τ]∑

s=1

R[λj/τ]−s+1f(ts−1)(wts −wts−1) + ϕ(wλ1 , . . . , wλJ

). (3.14)

By Lemma 3.2 the operator I−∑Jj=1 αjR

[λj/τ] has a bounded inverseΥτ = (I −∑Jj=1 αjR

[λj/τ])−1.

Then

u0 = Υτ

⎧⎨

⎩

J∑

j=1

αj

[λj/τ]∑

s=1

R[λj/τ]−s+1f(ts−1)(wts −wts−1) + ϕ(wλ1 , . . . , wλJ

)⎫⎬

⎭. (3.15)

Therefore, we have formulas (3.12) and (3.15) for the solution of problem (3.10). Now, wewill study the convergence of difference scheme (3.10).

Theorem 3.3. Assume that

E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H≤ C. (3.16)

Then the estimate of convergence

max0≤k≤N

(E‖v(tk) − uk‖2H

)1/2 ≤ C1(δ, λ1)τ1/2 (3.17)

holds. Here C and C1(δ, λ1) do not depend on τ .

Proof. Using formulas (2.12) and (3.15), we can write

v(0) − u0 = (Υ − Υτ)ϕ(wλ1 , . . . , wλJ

)+ Υ

J∑

j=1

αj

∫λj

0e−A(λj−s)f(s)dws

− ΥτJ∑

j=1

αj

[λj/τ]∑

s=1

R[λj/τ]−s+1f(ts−1)(wts −wts−1)

=P1,J + P2,J + P3,J + P4,J + P5,J + P6,J + P7,J ,

(3.18)


where

P1,J = (Υ − Υτ)ϕ(wλ1 , . . . , wλJ

), (3.19)

P2,J = (Υ − Υτ)J∑

j=1

αj

∫λj

0e−A(λj−s)f(s)dws, (3.20)

P3,J = ΥτJ∑

j=1

αj

∫λj

[λj/τ]τe−A(λj−s)f(s)dws, (3.21)

P4,J = ΥτJ∑

j=1

αj

[λj/τ]∑

p=1

∫ tp

tp−1

(e−(λj−s)A − e−([λj/τ]τ−s)A

)f(s)dws, (3.22)

P5,J = ΥτJ∑

j=1

αj

[λj/τ]∑

p=1

(e−([λj/τ]τ−pτ)A − R[λj/τ]−p

)∫ tp

tp−1e−(tp−s)Af(s)dws, (3.23)

P6,J = ΥτJ∑

j=1

αj

[λj/τ]∑

p=1

R[λj/τ]−p

⎛

⎝∫ tp

tp−1e−(tp−s)Af(s)dws −

∫ tp

tp−1e−τAf

(tp−1

)dws

⎞

⎠, (3.24)

P7,J = ΥτJ∑

j=1

αj

[λj/τ]∑

p=1

R[λj/τ]−p(e−τA − R

)f(tp−1

)Δwtp . (3.25)

Let us estimate Pk,J for all k = 1, . . . , 7, separately. We start with P1,J . Using formulas (2.4) and(3.4), we obtain

Υ − Υτ = ΥΥτ

⎛

⎝J∑

j=1

αj(e−Aλj − R[λj/τ]

)⎞

⎠, (3.26)

and also the expression in the above sum can be written in the following formula:

e−Aλj − R[λj/τ] = −∫1

0d(R[λj/τ](x)e−(1−x)Aλj

)

= −∫1

0

{[λj

τ

]R[λj/τ]−1(x)

−τA(1 + xτA)2

+AλjR[λj/τ](x)

}e−(1−x)Aλj dx

= −∫1

0R[λj/τ]+1(x)e−(1−x)Aλj

{−[λj

τ

]τA + (1 + xτA)Aλj

}dx

= −∫1


{(λj −

[λj

τ

]τ

)A + xτA2λj

}dx.

(3.27)


Here R(x) = (I + τxA)−1. Using formulas (3.26), (3.27), and (3.19), we can write

P1,J = ΥΥτJ∑

j=1

αj

×(−∫1


{(λj −

[λj

τ

]τ

)A + xτA2λj

}dx

)ϕ(wλ1 , . . . , wλJ

).

(3.28)

Let us estimate expected value of P1,J . Since

(λj −

[λj

τ

]τ

)≤ τ, (3.29)

we have that

(E∥∥P1,J

∥∥2H

)1/2 ≤ ‖Υ‖H→H‖Υτ‖H→H

×

⎛⎜⎝E

∥∥∥∥∥∥

J∑

j=1

αjτ

(∫1

0A1/2R[λj/τ]+1(x)e−(1−x)AλjA1/2ϕ

(wλ1 , . . . , wλJ

)dx

)∥∥∥∥∥∥

2

H

+λjτE

∥∥∥∥∥

∫1

0A3/2R[λj/τ]+1(x)e−(1−x)AλjA1/2ϕ

(wλ1 , . . . , wλJ

)dx

∥∥∥∥∥

2

H

⎞

⎠1/2

.

(3.30)

In the same manner by using the triangle inequality and estimates (3.2) and (3.1), we get

(E∥∥P1,J

∥∥2H

)1/2 ≤ C1(δ, λ1)

⎛⎜⎝

J∑

j=1

∣∣αj∣∣∫1

0τ

dx√[

λj/τ + 1]τx

×(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2

+J∑

j=1

∣∣αj∣∣λj∫1/2

0τ

dx

(1 − x)3/2λ3/2j


×(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2

+J∑

j=1

∣∣αj∣∣λj∫1

1/2τ

dx([λj/τ

]τx)3/2

×(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2⎞

⎠

≤ C1(δ, λ1)(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2

⎛⎜⎝

J∑

j=1

C2τ1√λj

+J∑

j=1

C3τ1√λj

⎞⎟⎠

≤ C4(δ, λ1)J∑

j=1

C4τ1/2 1

λ1/2j

(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2

≤ C5(δ, λ1)τ1/2(E∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

)1/2

.

(3.31)

Now, let us estimate P2,J . Using formula (3.20), the triangle inequality, and estimates (3.5),(3.2), and (3.1), we get

(E∥∥P2,J

∥∥2H

)1/2 ≤ C5(δ, λ1)τ1/2

⎛⎜⎜⎝

J∑

j=1

E

∥∥∥∥∥∥∥

∫λj

0e−A(λj−s)A1/2f(s)dws

∥∥∥∥∥∥∥

2

H

⎞⎟⎟⎠

1/2

≤ C5(δ, λ1)τ1/2

⎛⎜⎝

J∑

j=1

∫λj

0

∥∥∥A1/2f(s)∥∥∥2

Hds

⎞⎟⎠

1/2

≤ C6(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H.

(3.32)

Let us estimate P3,J . Using formula (3.21), the triangle inequality, and estimates (3.5), (3.2),and (3.1), we get

(E∥∥P3,J

∥∥2H

)1/2 ≤ ‖Υτ‖H→H

J∑

j=1

∣∣αj∣∣(∥∥∥e−A(λj−s)

∥∥∥2

H→H

∥∥∥A−1/2∥∥∥2

H→H

×∫λj

[λj/τ]τ

∥∥∥A1/2f(s)∥∥∥2

Hds

)1/2

,


≤ C6(δ, λ1)J∑

j=1

∣∣αj∣∣

⎛⎜⎝∫λj

[λj/τ]τ

∥∥∥A1/2f(s)∥∥∥2

Hds

⎞⎟⎠

1/2

,

≤ C6(δ, λ1)J∑

j=1

∣∣αj∣∣(λj −

[λj

τ

]τ

)1/2

max0≤s≤T

(∥∥∥A1/2f(s)∥∥∥2

H

)1/2

≤ C7(δ, λ1)τ1/2max0≤s≤T

∥∥A1/2f(s)∥∥H.

(3.33)

Next, let us estimate P4,J . Using formula (3.22), the triangle inequality, and estimates (3.5),(3.2), and (3.1), we get

(E∥∥P4,J

∥∥2H

)1/2 ≤ ‖Υτ‖H→H

J∑

j=1

∣∣αj∣∣

×

⎛⎜⎝

[λj/τ]∑

p=1

∫ tp

tp−1

∥∥∥A−1/2(e−(λj−s)A − e−([λj/τ]τ−s)A

)∥∥∥2

H→H

∥∥∥A1/2f(s)∥∥∥2

Hds

⎞⎟⎠

1/2

≤ C1(δ, λ1)

⎛⎜⎝

[λj/τ]∑

p=1

τ

∫ tp

tp−1

∥∥∥A1/2f(s)∥∥∥2

Hds

⎞⎟⎠

1/2

≤C1(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H.

(3.34)


(E∥∥P5,J

∥∥2H

)1/2 ≤ ‖Υτ‖H→H

⎛

⎝J∑

j=1

∣∣αj∣∣[λj/τ]∑

p=1

∥∥∥A−1/2(e−([λj/τ]τ−pτ)A − R[λj/τ]−p

)∥∥∥2

H→H

×E∫ tp

tp−1

∥∥∥e−(tp−s)A∥∥∥2

H→H

∥∥∥A1/2f(s)∥∥∥2

Hds

⎞

⎠1/2

≤ C2(δ, λ1)τ1/2(∫T

0

∥∥∥A1/2f(s)∥∥∥2

Hds

)1/2

≤ C2(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H.

(3.35)



(E∥∥P6,J

∥∥2H

)1/2 ≤ ‖Υτ‖H→H

J∑

j=1

∣∣αj∣∣

×⎛

⎝[λj/τ]∑

p=1

∫ tp

tp−1

∥∥∥R[λj/τ]−p∥∥∥2

H→HE∥∥∥e−(tp−s)Af(s) − e−τAf(tp−1)

∥∥∥2

Hds

⎞

⎠1/2

≤ C1(δ, λ1)J∑

j=1

⎛⎜⎝

[λj/τ]∑

p=1

∫ tp

tp−1E∥∥∥e−(tp−s)Af(s) − e−(tp−tp−1)Af(tp−1)

∥∥∥2

Hds

⎞⎟⎠

1/2

= C1(δ, λ1)J∑

j=1

⎛

⎝[λj/τ]∑

p=1

∫ tp

tp−1E∥∥∥(e−(tp−s)A − e−(tp−tp−1)A

)f(s)

+ e−(tp−tp−1)A(f(s) − f(tp−1

))∥∥∥2

Hds

⎞

⎠1/2

≤ C2(δ, λ1)J∑

j=1

⎛⎜⎝

[λj/τ]∑

p=1

∫ tp

tp−1

∥∥∥A−1/2(e−(tp−s)A − e−(tp−tp−1)A

)A1/2f(s)

∥∥∥2

H

+∥∥∥e−(tp−tp−1)A

(f(s) − f(tp−1)

)∥∥∥2

Hds

⎞

⎠1/2

≤ C3(δ, λ1)J∑

j=1

⎛

⎝[λj/τ]∑

p=1

∫ tp

tp−1

(τ∥∥∥A1/2f(s)

∥∥∥2

H+∥∥f(s) − f(tp−1)

∥∥2H

)ds

⎞

⎠1/2

≤ C4(δ, λ1)J∑

j=1

⎛⎜⎝

[λj/τ]∑

p=1

E

∫ tp

tp−1

(τ∥∥∥A1/2f(s)

∥∥∥2

H+∥∥f ′(s)τ

∥∥2H

)ds

⎞⎟⎠

1/2

≤ C4(δ, λ1)τ1/2

⎛⎜⎝∫T

0

∥∥∥A1/2f(s)∥∥∥2

Hds +

∫T

0

∥∥f ′(s)∥∥2Hds

⎞⎟⎠

1/2

≤ C5(δ, λ1)τ1/2(max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H+ max

0≤s≤T

∥∥f ′(s)∥∥H

).

(3.36)


Finally, let us estimate P7,J . Using formula (3.25), the triangle inequality, and estimates (3.5),(3.2), and (3.1), we get

(E∥∥P5,J

∥∥2H

)1/2 ≤ ‖Υτ‖H→H

J∑

j=1

∣∣αj∣∣

×⎛

⎝[λj/τ]∑

p=1

∥∥∥R[λj/τ]−p∥∥∥2

H→H

∥∥∥A−1/2(e−τA − R

)∥∥∥2

H→H

∥∥∥A1/2f(tp−1

)∥∥∥2

H

× E∥∥∥Δwtp

∥∥∥2

H

⎞

⎠1/2

≤ C(δ, λ1)

⎛

⎝[λj/τ]∑

p=1

τ∥∥∥A1/2f

(tp−1

)∥∥∥2

HE∥∥∥Δwtp

∥∥∥2

H

⎞

⎠1/2

.

(3.37)

Since Δwtp is a Wiener process and

E∥∥∥Δwtp

∥∥∥2

H≤ Δtp = τ, (3.38)

we have that

(E∥∥P5,J

∥∥2H

)1/2 ≤ C(δ, λ1)τ1/2⎛

⎝[λj/τ]∑

p=1

∥∥∥A1/2f(tp−1

)∥∥∥2

Hτ

⎞

⎠1/2

≤ C1(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H.

(3.39)

Applying estimates for Pk,J , k = 1, . . . , 7, we get the estimate:

(E‖v(t0) − u0‖2H

)1/2 ≤ C4(δ, λ1)τ1/2. (3.40)

To prove the Theorem 3.3 it suffices to establish the following estimate:

max1≤k≤N

(E‖v(tk) − uk‖2H

)1/2 ≤ C2(δ, λ1)τ1/2. (3.41)


Using formulas (2.9) and (3.12), we can write

v(tk) − uk = e−kτAv(0) +k∑

s=1

e−(k−s)τA∫ ts

ts−1e−A(ts−p)f

(p)dwp

−Rku0 −k∑

s=1

Rk−s+1f(ts)(wts −wts−1) = P1,k + P2,k + P3,k + P4,k + P5,k,

(3.42)

where

P1,k =(e−kτA − Rk

)Υ

⎧⎪⎨

⎪⎩

J∑

j=1

αj

∫λj


(wλ1 , . . . , wλJ

)⎫⎪⎬

⎪⎭,

P2,k = Rk(v(0) − u0),

P3,k =k−1∑

s=1

[e−(k−s)τA − Rk−s

] ∫ ts


(p)dwp,

P4,k =k∑

s=1

Rk−s∫ ts


(p)dwp − e−τAf(ts−1)(wts −wts−1),

P5,k =k∑

s=1

Rk−s[e−τA − R

]f(ts−1)(wts −wts−1).

(3.43)

Let us estimate Pm,k for all m = 1, . . . , 5, separately. We start with P1,k. Using the triangleinequality and estimates (3.5), (3.2), and (3.1), we get

(E‖P1,k‖2H

)1/2 ≤⎛

⎝∥∥∥(e−kτA − Rk

)A−1/2

∥∥∥2

H→H

× E

∥∥∥∥∥∥A1/2Υ

⎧⎨

⎩

J∑

j=1

αj

∫λj


(wλ1 , . . . , wλJ

)⎫⎬

⎭

∥∥∥∥∥∥

2

H

⎞⎟⎠

1/2

≤ C1(δ, λ1)τ1/2⎛

⎝E

∥∥∥∥∥∥Υ

J∑

j=1

αj

∫λj

0e−A(λj−s)A1/2f(s)dws

+ A1/2ϕ(wλ1 , . . . , wλJ

)∥∥∥∥∥∥

2

H

⎞⎟⎠

1/2


≤ C2(δ, λ1)τ1/2‖Υ‖H→H

⎛⎜⎝

⎛

⎝J∑

j=1

∣∣αj∣∣⎞

⎠2 ∫λj

0

∥∥∥e−A(λj−s)∥∥∥2

H→H

∥∥∥A1/2f(s)∥∥∥2

Hds

+∥∥∥A1/2ϕ

(wλ1 , . . . , wλJ

)∥∥∥2

H

⎞

⎠1/2

≤ C3(δ, λ1)τ1/2(∫T

0

∥∥∥A1/2f(s)∥∥∥2

Hds + E

∥∥∥A1/2ϕ∥∥∥2

H

)1/2

≤ C4(δ, λ1)τ1/2(max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H+(E∥∥∥A1/2ϕ

∥∥∥2

H

)1/2).

(3.44)

Now, we estimate P2,k. Using estimate (3.1), we get

(E‖P2,k‖2H

)1/2 ≤(∥∥∥Rk

∥∥∥2

H→HE‖v(0) − u0‖2H

)1/2

≤(E‖v(0) − u0‖2H

)1/2. (3.45)

Applying the estimate (3.40), we obtain

(E‖P2,k‖2H

)1/2 ≤ C(δ, λ1)τ1/2. (3.46)

Now, we estimate P3,k. Using the triangle inequality and estimates (3.5), (3.2), and (3.1), weget

(E‖P3,k‖2H

)1/2 ≤ C(δ, λ1)

(k−1∑

s=1

∥∥∥A−1/2[e−(k−s)τA − Rk−s

]∥∥∥2

H→H

×∥∥∥e−A(ts−p)

∥∥∥2

H→H

∫ ts

ts−1

∥∥∥A1/2f(p)∥∥∥

2

Hdp

)1/2

≤ C(δ, λ1)

(k−1∑

s=1

τ

∫ ts

ts−1

∥∥∥A1/2f(p)∥∥∥

2

Hdp

)1/2

≤ C(δ, λ1)τ1/2(∫T

0

∥∥∥A1/2f(p)∥∥∥

2

Hdp

)1/2

≤ C(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H.

(3.47)


Now, we estimate P4,k. We denote that

bj =∫ tj

tj−1

∫s

tj−1

(e−τAA−1/2f ′(z) +A1/2e−(tj−z)Af(s)

)dzdws,

b∗j =

{bj , 1 ≤ j ≤ k − 1,0, otherwise.

(3.48)

Then

P4,k =k∑

s=1

Rk−s∫ ts

ts−1

(e−A(ts−p)f

(p) − e−τAf(ts−1)

)dwp

=k∑

s=1

Rk−s∫ ts

ts−1

((e−A(ts−p) − e−τA

)f(p)+ e−τA

(f(p) − f(ts−1)

))dwp

=k∑

s=1

A1/2Rk−s∫ tj

tj−1

∫s

tj−1


)dzdws

=N∑

i=1

RiA1/2b∗k−i,

(E‖P4,k‖2H

)1/2=

⎛

⎝E

∥∥∥∥∥

N∑

i=1

RiA1/2b∗k−i

∥∥∥∥∥

2

H

⎞

⎠1/2

.

(3.49)

Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

(E‖P4,k‖2

)1/2 ≤(

N∑

i=1

E∥∥∥RiA1/2b∗k−i

∥∥∥2

H

)1/2

≤(

N∑i=1

∥∥A1/2Ri∥∥H→HE

∥∥b∗k−i∥∥2H

)1/2

.

(3.50)

Since

(E∥∥∥b∗j∥∥∥2

H

)1/2

=

⎛⎜⎜⎝E

∥∥∥∥∥∥∥

∫ tj

tj−1

∫s

tj−1


)dzdws

∥∥∥∥∥∥∥

2

H

⎞⎟⎟⎠

1/2

≤∫ tj

tj−1

(∫ s

tj−1

∥∥∥(e−τAA−1/2f ′(z) +A1/2e−(tj−zA)f(s)

)∥∥∥2

Hdz

)1/2

ds


≤∫ tj

tj−1

∫ s

tj−1

∥∥∥(e−τAA−1/2f ′(z) +A1/2e−(tj−z)Af(s)

)∥∥∥2

Hdzds

≤ C(δ, λ1)τ3/2(max0≤s≤T

∥∥A1/2f(s)∥∥H + max

0≤s≤T

∥∥A−1/2f ′(s)∥∥H

),

(3.51)

we have that

(E‖P4,k‖2H

)1/2 ≤N∑

i=1

C1√iτ

(E∥∥b∗k−i

∥∥2H

)1/2

≤N∑

i=1

C1√iτC(δ, λ1)τ3/2

(max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H+ max

0≤s≤T

∥∥∥A−1/2f ′(s)∥∥∥H

)

≤ C1(δ, λ1)τ1/2(max0≤s≤T

∥∥A1/2f(s)∥∥H + max

0≤s≤T

∥∥A−1/2f ′(s)∥∥H

).

(3.52)

Finally, we estimate P5,k. We denote that

qj = A1/2f(tj−1)Δwtj ,

a∗j =

{qj , 1 ≤ j ≤ k − 1,0, otherwise.

(3.53)

Therefore,

P5,k =N∑

i=1

A1/2RiA−1(e−τA − R

)q∗k−i. (3.54)

Using the triangle inequality and estimates (3.5), (3.2), and (3.1), we get

(E‖P5,k‖2H

)1/2 ≤N∑

i=1

∥∥∥A1/2Ri∥∥∥H→H

∥∥∥A−1(e−τA − R

)∥∥∥H→H

(E∥∥q∗k−i

∥∥2H

)1/2

≤N∑

i=1

2τ√iτ

(E∥∥q∗k−i

∥∥2H

)1/2 ≤ Cmax1≤j≤N

(E∥∥qj∥∥2H

)1/2.

(3.55)

Since

(E∥∥qj∥∥2H

)1/2 ≤(E∥∥∥A1/2f

(tj−1)Δwtj

∥∥∥2

H

)1/2

≤ C1(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H, (3.56)


we have that

(E‖P5,k‖2H

)1/2 ≤ C2(δ, λ1)τ1/2max0≤s≤T

∥∥∥A1/2f(s)∥∥∥H. (3.57)

Combining estimates P1,k, P2,k, P3,k, P4,k, and P5,k, we obtain (3.41). Theorem 3.3 is proved.

3.2. Applications

Now, we consider applications of Theorem 3.3. First, let us consider the nonlocal-boundaryvalue problem for one-dimensional stochastic parabolic equation:

du(t, x) − (a(x)ux)xdt + δu(t, x)dt = f(t, x)dwt, 0 < t < T, 0 < x < 1,

u(0, x) =J∑

j=1

αju(λj , x

)+ ϕ(wλ1 , . . . , wλJ , x

), 0 ≤ x ≤ 1,

J∑

j=1

∣∣αj∣∣ ≤ 1, 0 < λ1 < · · · < λJ ≤ T, wt =

√tξ, ξ ∈N(0, 1), 0 ≤ t ≤ T,

u(t, 0) = u(t, 1), ux(t, 0) = ux(t, 1), 0 ≤ t ≤ T,

(3.58)

where δ > 0, a(x) ≥ a > 0 (x ∈ (0, 1)), ϕ(wλ1 , . . . , wλJ , x) (x ∈ [0, 1]) and f(t, x) (t, x ∈ [0, 1])are smooth functions with respect to x.

The discretization of problem (3.58) is carried out in two steps. In the first step, wedefine the grid space

[0, 1]h = {x = xn : xn = nh, 0 ≤ n ≤M, Mh = 1}. (3.59)

Let us introduce the Hilbert space L2h = L2([0, 1]h) of the grid functions ϕh(x) ={ϕn}M−1

1 defined on [0, 1]h, equipped with the norm

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈[0,1]h

∣∣ϕ(x)∣∣2h

⎞

⎠1/2

. (3.60)

To the differential operator A generated by problem (3.58), we assign the differenceoperator Ax

hby the formula

Axhϕ

h(x) ={−(a(x)ϕx

)x,n + δϕn

}M−1

1(3.61)


acting in the space of grid functions ϕh(x) = {ϕn}M0 satisfying the conditions ϕ0 = ϕM, ϕ1 −ϕ0 = ϕM − ϕM−1. It is well known that Ax

his a self-adjoint positive definite operator in L2h.

With the help of Axh, we arrive at the nonlocal-boundary value problem:

duh(t, x) +Axhu

h(t, x)dt = fh(t, x)dwt, 0 < t < T, x ∈ [0, 1]h,

uh(0, x) =J∑

j=1

αjuh(λj , x

)+ ϕ(wλ1 , . . . , wλJ , x

), x ∈ [0, 1]h.

(3.62)

In the second step, we replace (3.62)with the difference scheme (3.10):

uhk(x) − uhk−1(x) + τAxhu

hk(x) = f

hk−1(x)(wtk −wtk−1), 1 ≤ k ≤N,

fhk−1(x) = fh(tk−1, x), tk = kτ, 1 ≤ k ≤N, x ∈ [0, 1]h,

uh0(x) =J∑

j=1

αjuh[λj/τ]

(x) + ϕ(wλ1 , . . . , wλJ , x

), x ∈ [0, 1]h.

(3.63)

Theorem 3.4. Let τ and h be sufficiently small positive numbers. Then, the solutions of differencescheme (3.63) satisfy the following convergence estimate:

max0≤k≤N

(E∥∥∥u(tk) − uhk

∥∥∥2

L2h

)1/2

≤ C(δ, λ1)(τ1/2 + h

), (3.64)

where C(δ, λ1) do not depend on τ and h. Here, one puts u(tk) = {u(tk, xn)}M0 as the grid function ofexact solution of problem (3.58) at the grid points t = tk, 0 ≤ k ≤N and x = xn, 0 ≤ n ≤M.

Proof. Let us introduce the Banach space C([0, 1],H) of abstract mesh functions uk = uhk(x)

defined on [0, 1]τ with values in H = L2h. Then, difference scheme (3.63) can be reduced tothe abstract difference scheme:

(uk − uk−1) + τAuk = fk,

fk = f(tk−1), tk = kτ, 1 ≤ k ≤N,

u0 =J∑

j=1

αju[λj/τ] + ϕ(wλ1 , . . . , wλJ

),

(3.65)

in a Hilbert space L2h with the operator A = Axh by formula (3.62). It is clear that A = A∗ and

A ≥ δI in H = L2h. Hence, Axh is a self-adjoint positive definite operator in L2h. Therefore,

Theorem 3.3 applies to this case, and Theorem 3.4 is proved.


Second, let Ω be the unit open cube in the n-dimensional Euclidean space Rn = {x =

(x1, . . . , xn) : 0 < xi < 1, i = 1, . . . , n} with boundary S, Ω = Ω ∪ S. In [0, T] ×Ω, the nonlocalboundary value problem for the multidimensional parabolic equation

du(t, x) −n∑

r=1

(ar(x)uxr )xrdt = f(t, x)dwt, 0 < t < T,

x = (x1, . . . , xn) ∈ Ω,

u(0, x) =J∑

j=1

αju(λj , x

)+ ϕ(wλ1 , . . . , wλJ , x

), x ∈ Ω,

J∑

j=1

∣∣αj∣∣ ≤ 1, 0 < λ1 < · · · < λJ ≤ T, wt =

√tξ, ξ ∈N(0, 1), 0 ≤ t ≤ T,

u(t, x) = 0, x ∈ S, 0 ≤ t ≤ T

(3.66)

with the Dirichlet condition is considered. Here ar(x), (x ∈ Ω), ϕ(x) (x ∈ Ω), and f(t, x) (t ∈(0, 1), x ∈ Ω) are given smooth functions with respect to x and ar(x) ≥ a > 0.

The discretization of problem (3.66) is carried out in two steps. In the first step, definethe grid space Ωh = {x = xm = (h1m1, . . . , hnmn); m = (m1, . . . , mn), 0 ≤ mr ≤ Nr, hrNr =1, r = 1, . . . , n},Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.

Let L2h denote the Hilbert space

L2h = L2

(Ωh

)=

⎧⎪⎨

⎪⎩ϕh(x) :

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

<∞

⎫⎪⎬

⎪⎭. (3.67)

The differential operator A in (3.66) is replaced with

Axhu

h(x) = −n∑

r=1

(ar(x)uhxr

)

xr ,jr, (3.68)

where the difference operator Axhis defined on those grid functions uh(x) = 0, for all x ∈ Sh.

It is well known that Axhis a self-adjoint positive definite operator in L2h.

Using (3.66) and (3.68), we get

duh(t, x) +Axhu

h(t, x)dt = fh(t, x)dwt, 0 < t < T, x ∈ Ωh,

uh(0, x) =J∑

j=1

αjuh(λj , x

)+ ϕ(wλ1 , . . . , wλJ , x

), x ∈ Ωh.

(3.69)


In the second step, we replace (3.69)with the difference scheme (3.10):

uhk(x) − uhk−1(x) + τAxhu

hk(x) = f

hk−1(x)(wtk −wtk−1), 1 ≤ k ≤N,

fhk−1(x) = fh(tk−1, x), tk = kτ, 1 ≤ k ≤N, x ∈ Ωh,

uh0(x) =J∑

j=1

αjuh[λj/τ]

(x) + ϕ(wλ1 , . . . , wλJ , x

), x ∈ Ωh.

(3.70)

Theorem 3.5. Let τ and |h| =√h21 + · · · + h2n be sufficiently small positive numbers. Then, the

solution of difference scheme (3.70) satisfies the following convergence estimate:

max0≤k≤N

(E∥∥∥u(tk) − uhk

∥∥∥2

L2h

)1/2

≤ C(δ, λ1)(τ1/2 + |h|2

), (3.71)

where C(δ, λ1) do not depend on τ and |h|. Here, one puts u(tk) = u(tk, x)|x∈Ωhas the grid function

of exact solution of problem (3.66) at the grid points t = tk, 0 ≤ k ≤N and x ∈ Ωh.

The proof of Theorem 3.5 is based on the abstract Theorem 3.3 and the symmetryproperties of the difference operator Ax

hdefined by formula (3.68).

4. Numerical Application

Now, we consider the numerical application of nonlocal boundary value problem:

dv − vxxdt = e−t sinxdwt, 0 < t < 1, 0 < x < π,

v(0, x, 0) = v(1, x,w1) + sinx − e−1 sinxw1 − e−1 sinx, 0 ≤ x ≤ π,v(t, 0, wt) = v(t, π,wt) = 0, 0 ≤ t ≤ 1,

wt =√tξ, ξ ∈N(0, 1), 0 ≤ t ≤ 1,

(4.1)

for one-dimensional stochastic parabolic equation. For numerical solution of (4.1), weconsider the difference scheme 1/2-th order of accuracy in t and second order of accuracyin x for the approximate solution of the nonlocal boundary value problem (4.1):

ukn − uk−1n − ukn+1 − 2ukn + ukn−1

h2τ = f(tk, xn)τ

(√kτ −√(k − 1)τ

)ξ,

f(tk, xn) = e−tk sinxn, tk = kτ, xn = nh, 1 ≤ k ≤N, 1 ≤ n ≤M − 1,

uk0 = ukM = 0, 0 ≤ k ≤N,

u0n = uNn + sinxn − e−1 sinxnw1 − e−1 sinxn, 0 ≤ n ≤M.

(4.2)


We will write it in the matrix form

Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = 0, UM = 0.(4.3)

Here

ϕn =

⎡⎢⎢⎢⎢⎢⎣

0ϕ1n

·ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

, 0 =

⎡⎢⎢⎢⎢⎢⎣

00·00

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

ϕkn = f(tk, xn)τ(√

kτ −√(k − 1)τ

)ξ, 1 ≤ k ≤N, 1 ≤ n ≤M − 1,

A =

⎡⎢⎢⎢⎢⎢⎣

0 0 · 0 00 a · 0 0· · · · ·0 0 · a 00 0 · 0 a

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, B =

⎡⎢⎢⎢⎢⎢⎣

1 0 · 0 −1b c · 0 0· · · · ·0 0 · c 00 0 · b c

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

a =(− τ

h2

), b = (−1), c =

(1 +

2τh2

), C = A,

D =

⎡⎢⎢⎢⎢⎢⎣

1 0 · 0 00 1 · 0 0· · · · ·0 0 · 1 00 0 · 0 0

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, us =

⎡⎢⎢⎢⎢⎢⎣

u0su1s·

uN−1s

uNs

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(1)

,

s = n − 1, n, n + 1.

(4.4)

For the solution of the last matrix equation, we use the modified Gauss eliminationmethod (see [17]). We seek a solution of the matrix equation by the following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 1, uM = 0, (4.5)

where αj , are (N + 1) × (N + 1) square matrices and βj , are (N + 1) × 1 column matrices and(j = 1, . . . ,M − 1) defined by formulas

αn+1 = −(B + Cαn)−1A,

βn+1 = (B + Cαn)−1(Dϕn − Cβn

), n = 1, . . . ,M − 1.

(4.6)


Table 1: Error analysis.

N/M 10/30 20/60 40/120Difference scheme (4.2) 0.0929 0.0401 0.0187

Here

α1 =

⎡⎢⎢⎢⎢⎢⎣

0 0 . 0 00 0 . 0 0. . . . .0 0 . 0 00 0 . 0 0

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, β1 = 0. (4.7)

The error between the exact solution and the solutions derived by difference schemesis shown in Table 1. To obtain the results we simulated the 1,000 sample paths of Brownianmotion for each level of discretization. The estimate (3.71) in Theorem 3.5 suggests that if wedouble the number of nodes, then the error should be decreased by a factor of 1/

√2. The

theoretical statement for the solution of this difference scheme is supported by the results ofthe numerical experiment. In fact, we doubleN andM; the error is even less than half of theprevious error.

Acknowledgment

The authors wish to thank Professor A. Lukashev (Istanbul, Turkey), for his valuablesuggestions which helped us to improve the present paper.

References

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Research ArticleFDM for Elliptic Equations withBitsadze-Samarskii-Dirichlet Conditions

Allaberen Ashyralyev1, 2 and Fatma Songul Ozesenli Tetikoglu1

1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey2 Department of Mathematics, ITTU, 74400 Ashgabat, Turkmenistan

Correspondence should be addressed to Fatma Songul Ozesenli Tetikoglu, [email protected]

Received 8 April 2012; Accepted 6 May 2012


Copyright q 2012 A. Ashyralyev and F. S. Ozesenli Tetikoglu. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

A numerical method is proposed for solving nonlocal boundary value problem for themultidimensional elliptic partial differential equation with the Bitsadze-Samarskii-Dirichletcondition. The first and second-orders of accuracy stable difference schemes for the approximatesolution of this nonlocal boundary value problem are presented. The stability estimates, coercivity,and almost coercivity inequalities for solution of these schemes are established. The theoreticalstatements for the solutions of these nonlocal elliptic problems are supported by results ofnumerical examples.

1. Introduction

Many problems in fluid mechanics, dynamics, elasticity, and other areas of engineering,physics, and biological systems lead to partial differential equations of elliptic type. The roleplayed by coercive inequalities in the study of local boundary-value problems for elliptic andparabolic differential equations is well known (see, e.g., [1, 2]).

In the present paper, we consider the Bitsadze-Samarskii type nonlocal boundaryvalue problem

−d2u(t)dt2

+Au(t) = f(t), (0 < t < 1),

u′(0) = ϕ,

u′(1) = βu′(λ) + ψ,∣∣β∣∣ ≤ 1, 0 ≤ λ < 1

(1.1)


for elliptic differential equations in a Hilbert space H with self-adjoint positive definiteoperator A. It is known (see, e.g., [3–11]) that various nonlocal boundary value problems forelliptic equations can be reduced to the boundary value problem (1.1). The simply nonlocalboundary value problem was presented and investigated for the first time by Bitsadzeand Samarskii [12]. Further, methods of solutions of Bitsadze-Samarskii nonlocal boundaryvalue problems for elliptic differential equations have been studied extensively by manyresearchers (see [13–21] and the references given therein).

A function u(t) is called a solution of problem (1.1) if the following conditions aresatisfied.

(i) u(t) is twice continuously differentiable on the segment [0, 1]. Derivatives at theendpoints of the segment are understood as the appropriate unilaterial derivatives.

(ii) The element u(t) belongs to D(A) for all t ∈ [0, 1], and the function Au(t) iscontinuous on [0, 1].

(iii) u(t) satisfies the equation and nonlocal boundary condition in (1.1).

Let Ω be the open unit cube in Rn(x = (x1, . . . , xn) : 0 < xk < 1, 1 ≤ k ≤ n) withboundary S, Ω = Ω ∪ S. In present paper, we are interested in studying the stable differenceschemes for the numerical solution of the following nonlocal boundary value problem for themultidimensional elliptic equation

−utt −n∑

r=1

(ar(x)uxr )xr + δu = f(t, x), 0 < t < 1,

x = (x1, . . . , xn) ∈ Ω,

ut(0, x) = ϕ(x),

ut(1, x) = βut(λ, x) + ψ(x), x ∈ Ω,∣∣β∣∣ ≤ 1, 0 ≤ λ < 1,

u(t, x) = 0, 0 ≤ t ≤ 1, x ∈ S, S = ∂Ω.

(1.2)

Here ψ(x), ϕ(x) (x ∈ Ω), and f(t, x) (t ∈ (0, 1), x ∈ Ω) are given smooth functions, δ is alarge positive constant and ar(x) ≥ a > 0.

In the present paper, the first and second-orders of accuracy difference schemes arepresented for the approximate solution of problem (1.2). The stability and coercive stabilityestimates for the solution of these difference schemes are obtained. A numerical method isproposed for solving nonlocal boundary value problem for the multidimensional ellipticpartial differential equation with the Bitsadze-Samarskii-Dirichlet condition. A procedure ofmodified Gauss elimination method is used for solving these difference schemes in the caseof two-dimensional elliptic partial differential equations.


2. Difference Schemes: Well-Posedness

The discretization of problem (1.2) is carried out in two steps. In the first step let us definethe grid sets as follows:

Ωh = {x : x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn),

0 ≤ mr ≤Nr, hrNr = L, r = 1, . . . , n}

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.

(2.1)

We introduce the Hilbert space L2h = L2(Ωh) of the grid functions ϕh(x) = {ϕ(h1m1, . . . ,

hnmn)} defined on Ωh, equipped with the norm

∥∥∥ϕh∥∥∥L2(Ωh)

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

(2.2)

and the Hilbert spaceW22 (Ωh) defined on Ωh, equipped with the norm

∥∥∥ϕh∥∥∥W2

2 (Ωh)=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣ϕhxr ,mr

∣∣∣2h1 · · ·hn

⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣ϕhxrxr ,mr

∣∣∣2h1 · · ·hn

⎞

⎠1/2

.

(2.3)

Finally, we introduce the Banach spaces C([0, 1]τ , L2h) and Cα([0, 1]τ , L2h) of grid abstractfunction {ϕh

k(x)}N−1

1 defined on [0, 1]τ with values L2h, equipped with the following norms:

∥∥∥∥{ϕhk

}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

= max1≤k≤N−1

∥∥∥ϕhk∥∥∥L2h,

∥∥∥∥{ϕhk

}N−1

1

∥∥∥∥Cα([0,1]τ ,L2h)

= max1≤k≤N−1

∥∥∥ϕhk∥∥∥L2h

+ sup1≤k<k+r≤N−1

∥∥ϕk+r − ϕk∥∥L2h

(rτ)α.

(2.4)

To the differential operator A generated by the problem (1.2) we assign the differenceoperator Ax

h by the formula

Axhu

h = −n∑

r=1

(ar(x)uhxr

)

xr ,mr

+ δuhxr (2.5)

acting in the space of grid functions uh(x), satisfying the condition uh(x) = 0 for all x ∈ Sh.It is known that Ax

his a self-adjoint positive definite operator in L2(Ωh). With the help of


Axh, we arrive at the nonlocal boundary value problem for an infinite system of the following

ordinary differential equations:

−d2uh(t, x)dt2

+Axhu

h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,

uht (0, x) = ϕh(x), uht (1, x) = βu

ht (λ, x) + ψ

h(x), x ∈ Ωh.

(2.6)

In the second step, we replaced problem (2.6) by the first-order of accuracy difference schemeas follows:

−uhk+1(x) − 2uhk(x) + u

hk−1(x)

τ2+Ax

huhk(x) = f

hk (x),

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uh1(x) − uh0(x)τ

= ϕh(x), x ∈ Ωh,

uhN(x) − uhN−1(x)τ

= βuh[λ/τ]+1(x) − uh[λ/τ](x)

τ+ ψh(x), x ∈ Ωh

(2.7)

and the second order of accuracy difference scheme as follows:

−uhk+1(x) − 2uh

k(x) + uhk−1(x)

τ2+Ax

huhk(x) = f

hk (x),


−3uh0(x) + 4uh1(x) − uh2(x)2τ

= ϕh(x), x ∈ Ωh,

uhN−2(x) − 4uhN−1(x) + 3uhN(x)2τ

= β

⎡

⎣uh[λ/τ]−1(x) − 4uh[λ/τ](x) + 3uh[λ/τ]+1(x)

2τ

+uh[λ/τ]+1(x) − 2uh[λ/τ](x) + u

h[λ/τ]−1(x)

τ

(λ

τ−[λ

τ

])⎤

⎦

+ ψh(x), x ∈ Ωh.

(2.8)

Now, we will study the well-posedness of (2.7) and (2.8). We have the following theorem onstability of (2.7) and (2.8).


Theorem 2.1. Let τ and |h| be sufficiently small positive numbers. Then the solutions of differenceschemes (2.7) and (2.8) satisfy the following stability estimate:

max1≤k≤N

∥∥∥uhk∥∥∥L2h

≤M1

[∥∥∥ϕh∥∥∥L2h

+∥∥∥ψh∥∥∥L2h

+ max1≤k≤N−1

∥∥∥fhk∥∥∥L2h

], (2.9)

whereM1 does not depend on τ, h, ψh(x), ϕh(x) and fhk(x), 1 ≤ k ≤N − 1.

Proof. The proof of (2.9) is based on the following formula:

uhk(x) =(I − R2N

)−1{(

Rk − R2N−k)uh0(x) +

(RN−k − RN+k

)uhN(x)

−(RN−k − RN+k

)(2I + τBxh

)−1(Bxh)−1N−1∑

i=1

(RN−1−i − RN−1+i

)fhi (x)τ

}

+(2I + τBxh

)−1(Bxh)−1N−1∑

i=1

(R|k−i|−1 − Rk+i−1

)fhi (x)τ for k = 1, . . . ,N − 1,

(2.10)

where

uh0(x) = Pτ(I + τBxh

)(2I + τBxh

)−1(Bxh)−1

×[(I + R)RN−2

N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

+ (I + R)[I + R2N−2 − β

(RN−[λ/τ]−1 + RN+[λ/τ]

)]

×N−1∑

i=1

Ri−1fhi (x)τ − β(RN + RN−1

)[λ/τ]−1∑

i=1

R[λ/τ]−ifhi (x)τ

+ β(RN−2 + RN−1

) N−1∑

i=[λ/τ]+1

Ri−[λ/τ]fhi (x)τ

+β(RN + RN−1

)N−1∑

i=1

R[λ/τ]+ifhi (x)τ − β(RN−1 + RN

)fh[λ/τ](x)τ

]

− Pτ(I − R)−1[I + R2N−1 − β

(RN−[λ/τ]−1 + RN+[λ/τ]

)]ϕh(x)τ

+ Pτ(I − R)−1(RN−1 + RN

)ψh(x)τ,


uhN(x) = Pτ(I + τBxh

)(2I + τBxh

)−1(Bxh)−1

×[[R−1(I + R) − β

(RN−[λ/τ]−1 + RN+[λ/τ]−1

)]N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

− β(I + R2N−1

)[λ/τ]−1∑

i=1

R[λ/τ]−ifhi (x)τ + β(I + R2N−1

)R−1

N−1∑

i=[λ/τ]+1

Ri−[λ/τ]fhi (x)τ

+ β(I + R2N−1

)N−1∑

i=1

R[λ/τ]+ifhi (x)τ − β(I + R2N−1

)fh[λ/τ](x)τ

+(I + R)[RN + RN−1 − β

(R[λ/τ] + R2N−[λ/τ]−1

)]N−1∑

i=1

Ri−1fhi (x)τ

]

− Pτ(I − R)−1[RN + R2N−1 − β

(R[λ/τ] + R2N−[λ/τ]−1

)]ϕh(x)τ

+ Pτ(I − R)−1(I + R2N−1

)ψh(x)τ,

Pτ =[I − R2N−2 − β

(RN−[λ/τ]−1 + RN+[λ/τ]−1

)]−1,

(2.11)

for (2.7), and

uh0(x) = Dτ

(I + τBxh

)(2I + τBxh

)−1(Bxh)−1

×{(I + R)

(4R − I − R2

)(I − 3R)RN−4

(I − R2N

)N−2∑

i=1

(RN−i − RN+i

)fhi (x)τ

− (I + R)(4R − I − R2

)(I − R2N

)

×[3I − R − R2N−2(I − 3R)

− β[RN−[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN+[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]

×N−1∑

i=2

Ri−2fhi (x)τ − β(I + R)RN−2(4R − I − R2

)(I − R2N

)

×[(

I − 3R + 2(λ

τ−[λ

τ

])(I − R)

)[λ/τ]−1∑

i=1

R[λ/τ]−i−1fhi (x)τ

+(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

) N−1∑

i=[λ/τ]+2

Ri−[λ/τ]−1fhi (x)τ

−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)N−1∑

i=1

R[λ/τ]+i−1fhi (x)τ

]


− β(I − R2N

)(I + R)RN−2

(4R − R2 − I

)(4 + 4

(λ

τ−[λ

τ

]))

×(fh[λ/τ]+1(x)τ − fh[λ/τ](x)τ

)− (I + R)(4I − R)

(I − R2N

)

×[3I − R − R2N−2(I − 3R)

− β(RN−[λ/τ]−1

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN+[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

))]

× fh1 (x)τ − (I + R)RN−1(4R − R2 − I

)(I − R2N

)(4I + R2N−3(I − 3R)

)

×fhN−1(x)τ

}

−Dτ

(I − R2N

)(I − R)−1(I + R)RN−2

(4R − I − R2

)2τψh(x)

+Dτ

(I − R2N

)(I − R)−1

×[3I − R − R2N−2(I − 3R)

− β[RN−[λ/τ]−1

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN+[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)]]2τϕh(x),

(2.12)

uhN(x) = Dτ

(I + τBxh

)(2I + τBxh

)−1(Bxh)−1

×{(

I − R2N)[

(I + R)(4R − I − R2

)R−2(R − 3I)

+ β[(R − 3I)

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)RN−[λ/τ]−1

−(3R − I)(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)RN+[λ/τ]−1

]]

×N−2∑

i=1

(RN−i − RN+i

)fhi (x)τ + (I + R)

(I − R2N

)(4R − I − R2

)

×[(I + R)RN−2

(4R − I − R2

)

− β[R[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−R2N−[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]N−1∑

i=2

Ri−2fhi (x)τ


+ β(I − R2N

)(R − 3I + R2N−2(I − 3R)

)

×[(

I − 3R + 2(λ

τ−[λ

τ

])(I − R)

)[λ/τ]−1∑

i=1

R[λ/τ]−i−1fhi (x)τ

+(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

) N−1∑

i=[λ/τ]+2

Ri−[λ/τ]−1fhi (x)τ

−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)N−1∑

i=1

R[λ/τ]+i−1fhi (x)τ

]

+ β(R − 3I + R2N−2(3I − R)

)(I − R2N

)

×(4 + 4

(λ

τ−[λ

τ

])(I − R)

)(fh[λ/τ]+1(x)τ − fh[λ/τ](x)τ

)

+ (I + R)(I − R2N

)

×[(I + R)RN−2

(I − 4R + R2

)

− β[R[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−R2N−[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]fh1 (x)τ

+ R(I − R2N−1

)[R − 4I + R2N−4

(I + R2 − 3R

)(3I − R − R2(4I − R)

)

− β[RN−[λ/τ]−1

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

×(3R − I + R2N(3I − R)

)

− RN+[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

×(3I − R + R2N(I − 3R)

)]]fhN−1(x)τ

}

+Dτ

(I − R2N

)(I − R)−1

(R − 3I + R2N(I − 3R)

)2τϕh(x)

−Dτ

(I − R2N

)(I − R)−1

×[RN−2(I + R)

(R2 − 4R + I

)

− β[R[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−R2N−[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]2τψh(x),


Dτ =(I − R2N

)−1

×{[

−(3I − R)2 + (I − 3R)2]

− β[−RN+[λ/τ]−3(I − 3R)

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN−[λ/τ]−1(3I − R)(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]}−1,

R =(I + τBxh

)−1,

Bxh =12

(τAx

h +√4Ax

h + τ2(Axh

)2)

(2.13)

for (2.8), and the symmetry properties of the difference operator Axh defined by the formula

(2.5).

Difference schemes (2.7) and (2.8) are ill-posed in C([0, 1]τ , L2h). We have thefollowing theorem on almost coercive stability.

Theorem 2.2. Let τ and |h| be sufficiently small positive numbers. Then the solutions of differenceschemes (2.7) and (2.8) satisfy the following almost coercive stability estimate:

max1≤k≤N−1

∥∥∥∥∥uhk+1 − 2uh

k+ uh

k−1τ2

∥∥∥∥∥L2h

+ max1≤k≤N−1

∥∥∥uhk∥∥∥W2

2h

≤M2

[∥∥∥ϕh∥∥∥W2

2h

+∥∥∥ψh∥∥∥W2

2h

+ ln1

τ + |h| max1≤k≤N−1


],

(2.14)

whereM2 does not depend on τ, h, ψh(x), ϕh(x), and fhk (x), 1 ≤ k ≤N − 1.

Proof. The proof of (2.14) is based on the formulas (2.11), (2.12), (2.13), the symmetryproperties of the difference operator Ax

hdefined by the formula (2.5) and on the following

theorem on well-posedness of the elliptic difference problem.

Theorem 2.3. For the solutions of the elliptic difference problem

Axhu

h(x) = wh(x), x ∈ Ωh,

uh(x) = 0, x ∈ Sh(2.15)

the following coercivity inequality holds (see [21]):

n∑

r=1

∥∥∥∥(uh)

xrxr , mr

∥∥∥∥L2h

≤M∥∥∥wh

∥∥∥L2h, (2.16)

where M does not depend on h and wh(x).


Theorem 2.4. Let ϕh(x) = ψh(x) = 0. Then the difference problems (2.7) and (2.8) are well-posed inHolder spaces Cα([0, 1]τ , L2h) and the following coercivity inequality holds:

max1≤k≤N−1

∥∥∥∥∥∥

{uhk+1 − 2uhk + u

hk−1

τ2

}N−1

1

∥∥∥∥∥∥Cα([0,1]τ , L2h)

+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα([0,1]τ , W

22h)

≤ M3

α(1 − α) max1≤k≤N−1

∥∥∥∥{fhk

}N−1

1

∥∥∥∥Cα([0,1]τ ,L2h)

.

(2.17)

HereM3 does not depend on τ, h, and fhk (x), 1 ≤ k ≤N − 1.

Proof. The proof of (2.17) is based on the formula

Axhu

h0(x) − fh1 (x) = Pτ

(I + τBxh

)(2I + τBxh

)−1(I + R)

×[RN−1

N−1∑

i=1

τBxhRN−i(fhi − fhN−1

)

− RN−1N−1∑

i=1

τBxhRN+i(fhi (x) − fh1 (x)

)

−βRN[λ/τ]−1∑i=1

τBxhR[λ/τ]−i

(fhi (x) − fh[λ/τ](x)

)

+[I + R2N−2 − β

(RN−[λ/τ]−1 + RN+[λ/τ]

)]

×N−1∑

i=1

τBxhRi(fhi (x) − fh1 (x)

)

+ βRNN−1∑

i=[λ/τ]+1

τBxhRi−[λ/τ]


)

+ βRNN−1∑

i=1

τBxhR[λ/τ]+i

(fhi (x) − fh1 (x)

)

+ RN−1(I − RN−1

)fhN−1(x)

+ β[RN+[λ/τ]−1 − R2N−[λ/τ]−1 − τBxhRN

]fh[λ/τ](x)

+

[R2N−2 − R2N−1 + R2N−2 − R3N−3 + R3N−2 − RN−1

+ β(R2N−[λ/τ]−2 + R2N+[λ/τ]−1

+R2N+[λ/τ] − RN+[λ/τ]−1)]fh1 (x)

]


− Pτ(I − R)−1τ[I + R2N−1 − β

(RN−[λ/τ]−1 + RN+[λ/τ]

)]Axhϕ

h

+ Pτ(I − R)−1τ(RN−1 + RN

)Axhψ

h,

Ahxu

hN(x) − fhN−1(x) = Pτ

(I + τBxh

)(2I + τBxh

)−1

×{[R(I + R) − β

(RN−[λ/τ]+1 + RN+[λ/τ]+1

)]

×N−1∑

i=1

τBxhRN−i(fhi (x) − fhN−1(x)

)

−[R(I + R) − β

(RN−[λ/τ]+1 + RN+[λ/τ]+1

)]

×N−1∑

i=1

τBxhRN+i(fhi (x) − fh1 (x)

)

− βR2(I + R2N−1

)[λ/τ]−1∑

i=1

τBxhR[λ/τ]−i


)

+ βR(I + R2N−1

) N−1∑

i=[λ/τ]+1

τBxhRi−[λ/τ]


)

+ βR2(I + R2N−1

)N−1∑

i=1

τBxhR[λ/τ]+i

(fhi (x) − fh1 (x)

)

− β(I + R2N−1

)

×[R(I − R[λ/τ]−1

)−(I − RN−[λ/τ]−1

)− τBxhR

]

× fh[λ/τ](x)

+[(I + R)

(R2N−2 − RN − I + R

)

− β(RN−[λ/τ]+1

+ RN+[λ/τ]+1 − R2N−[λ/τ]

− R2N+[λ/τ] − RN−[λ/τ]−1

+RN+[λ/τ]−1 − RN−[λ/τ] + RN+[λ/τ])]

× fhN−1(x) +[(I + R)

(RN − R2N−1

)

+ β(RN+[λ/τ] + RN+[λ/τ]+1 − RN+[λ/τ]+2

+ R2N+[λ/τ]+1 + R2N+[λ/τ]+1 − R3N+[λ/τ]

− R3N+[λ/τ]+1 + R[λ/τ]+1 − R2N−[λ/τ]

+R3N−[λ/τ]−1)]fh1 (x)

}


+ Pτ(I − R)−1Rτ(I + R2N−1

)Ahxψ

− Pτ(I − R)−1Rτ[RN−1 + RN − β

(R[λ/τ] + R2N−[λ/τ]−1

)]Ahxϕ

(2.18)

for (2.7) and

Axhu

h0(x) − fh1 (x) = Dτ

(I + τBxh

)(2I + τBxh

)−1

×{(I + R)

(4R − I − R2

)(I − 3R)RN−3

(I − R2N

)

×[N−2∑

i=1

BxhτRN−i(fhi (x)−fhN−1(x)

)

−N−2∑

i=1

BxhτRN+i(fhi (x)−fh1 (x)

)]

− (I + R)(4R − I − R2

)(I − R2N

)

×[3I − R − R2N−2(I − 3R)

− β[RN−[λ/τ]−1

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN+[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)]]

×N−1∑

i=2

BxhτRi−1(fhi (x) − fh1 (x)

)

− β(I + R)RN−2(4R − I − R2

)(I − R2N

)

×[(

I − 3R + 2(λ

τ−[λ

τ

])(I − R)

)

×[λ/τ]−1∑

i=1

BxhτR[λ/τ]−i


)

+(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

×N−1∑

i=[λ/τ]+2

BxhτRi−[λ/τ]


)


−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

×N−1∑

i=1

BxhτR[λ/τ]+i

(fhi (x) − fh1 (x)

)]

− β(I − R2N

)(I + R)RN−1

(4R − R2 − I

)

×(4 + 4

(λ

τ−[λ

τ

]))τBxhf

h[λ/τ]+1(x)

+ β(I − R2N

)(I + R)RN−2

(4R − R2 − I

)

×[2R(R + 2

(λ

τ−[λ

τ

]))

+ R[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)+ RN−[λ/τ]−2

×(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]fh[λ/τ](x)

×[(I + R)

(I − R2N

)

×[(I − 3R)

(3R − I − R2N−5(I − R)

×(R2 − R2N(I + R)

(R2 − 4R + I

)))

− (3I − R)RN−1(R2 − 4R + I

)− β(4R − I − R2

)

×((

3I − R + 2(λ

τ−[λ

τ

])(I − R)

)RN−[λ/τ]−3

−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

×RN+[λ/τ]−3)]]

fh1 (x)

+ (I + R)(4R − I − R2

)(I − R2N

)RN−2

×(R − 3I − RN−2(I − 3R)

(I + RN(I − R)

))fhN−1(x)

}

−Dτ

(I − R2N

)(I − R)−1(I + R)RN−2

(4R − I − R2

)2τAx

hψh(x)

+Dτ

(I − R2N

)(I − R)−1


×[3I − R − R2N−2(I − 3R)

− β[RN−[λ/τ]

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

−RN+[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)]]2τAx

hϕh(x),

AxhRu

hN(x)− fhN−1(x)

= Dτ

(I + τBxh

)(2I + τBxh

)−1

×{(

I − R2N)

×[(I + R)

(4R − I − R2

)(R − 3I)

+ β[(R − 3I)

(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

× RN−[λ/τ]+1 − (3R − I)

×(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)RN+[λ/τ]+1

]]

×(

N−2∑

i=1

BxhτRN−i(fhi (x) − fhN−1(x)

)−N−2∑

i=1

BxhτRN+i(fhi (x) − fh1 (x)

))

+ (I + R)(I − R2N

)(4R − I − R2

)

×[(I + R)RN−2

(−4R + I + R2

)

− β[R[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−R2N−[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]

×N−1∑

i=2

BxhτRi(fhi (x) − fh1 (x)

)

+ βR(R − 3I + R2N−2(I − 3R)

)(I − R2N

)

×[(

I − 3R + 2(λ

τ−[λ

τ

])(I − R)

)

×[λ/τ]−1∑

i=1

BxhτR[λ/τ]−i


)


+(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

×N−1∑

i=[λ/τ]+2

BxhτR[λ/τ]+i

(fhi (x) − fh1 (x)

)

+(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

×N−1∑

i=1

BxhτR[λ/τ]+i

(fhi (x) − fh1 (x)

)]

+ βR2(R − 3I + R2N−2(I − 3R)

)(I − R2N

)(4 + 4

(λ

τ−[λ

τ

]))

× fh[λ/τ]−1(x) − βR(I − R2N

)

×[(I − R + RN−[λ/τ]

)(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)

+R[λ/τ]−1(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)]

× fh[λ/τ](x) +(I − R2N

)

×[(I + R)

(4R − R2 − I

)RN+1

× (R − 3I)(I − RN−2

)

+ β[(

3I − R + 2(λ

τ−[λ

τ

])(I − R)

)

× R2N−[λ/τ](R2(3I − R) + RN−2

(R3 − 4R2 + I

))

−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)R[λ/τ]

×(R2(3I − R) + (I − 3R)RN−3

(R + RN(I − R)

))]]

× fh1 (x) +(I − R2N

)

×[(I − 3R)

(4R − R2 − I

)(3I − R + R2N−2

)

− β[(

3I − R + 2(λ

τ−[λ

τ

])(I − R)

)RN−[λ/τ]−1(3I − R)

×(R + I + RN

(I − R + RN−2(I − R) + R3N−2(2I − R)

))

−(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)RN+[λ/τ]−1(3R − I)


×(I − R − R2N−3

×(1 + R − RN

(3I − R2

)+ R2N

(R3 − 4R2 + I

)))]]

×fhN−1(x)}

+Dτ

(I − R2N

)(I − R)−1R

(R − 3I + R2N−2(I − 3R)

)2τAx

hϕh(x)

−Dτ

(I − R2N

)(I − R)−1R

×[RN−2(I + R)

(R2 − 4R + I

)

− β[R[λ/τ]−1

(I − 3R + 2

(λ

τ−[λ

τ

])(I − R)

)

−R2N−[λ/τ]−1(3I − R + 2

(λ

τ−[λ

τ

])(I − R)

)]]2τAx

hψh(x)

(2.19)

for (2.8), the symmetry properties of the difference operator Axh defined by the formula (2.5)

and on Theorem 2.3 on well-posedness of the elliptic difference problem.

3. Numerical Analysis

We have not been able to obtain a sharp estimate for the constants figuring in the stabilityinequality. Therefore, we will give the following results of numerical experiments of theBitsadze-Samarskii-Dirichlet problem:

−∂2u(t, x)∂t2

− ∂2u(t, x)∂x2

+ u = f(t, x), 0 < t < 1, 0 < x < 1,

f(t, x) = exp(−πt) sin(πx), 0 < t < 1, 0 < x < 1,

ut(0, x) = −π sin(πx), 0 ≤ x ≤ 1,

ut(1, x) = ut(12, x

)+ π sin(πx)

(exp(−π2

)− exp(−π)

), 0 ≤ x ≤ 1,

u(t, 0) = u(t, 1) = 0, 0 ≤ t ≤ 1

(3.1)

for the two-dimensional elliptic equation.


The exact solution of this problem is

u(t, x) = exp(−πt) sin(πx). (3.2)

For the approximate solution of problem (3.1), we consider the set [0, 1]τ × [0, 1]h of a familyof grid points depending on small parameters τ and h as follows:

[0, 1]τ × [0, 1]h = {(tk, xn) : tk = kτ, 0 ≤ k ≤N, Nτ = 1,

xn = nh, 0 ≤ n ≤M, Mh = 1}.(3.3)

Applying (2.7), we present the following first-order of accuracy difference scheme forthe approximate solution of problem (3.1):

−unk+1 − 2unk + u

nk−1

τ2− un+1

k− 2un

k+ un−1

k

h2+ unk = exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,

1 ≤ n ≤M − 1,

un1 − un0τ

= −π sin(πxn), 0 ≤ n ≤M,

unN − unN−1τ

= βunN/2 − unN/2−1

τ+ π sin(πxn)

×(exp(−π2

)− exp(−π)

), 0 ≤ n ≤M,

u0k = uMk = 0, 0 ≤ k ≤N.

(3.4)

We have (N + 1) × (M + 1) system of linear equations in (3.4) and we will write themin the following matrix form:

Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,

u0 = uM = 0.(3.5)


Here,

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · 0 0 · 0 0 00 a 0 · 0 0 · 0 0 00 0 a · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · a 0 00 0 0 · 0 0 · 0 a 00 0 0 · 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1 1 0 · 0 0 · 0 0 0c b c · 0 0 · 0 0 00 c b · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · b c 00 0 0 · 0 0 · c b c0 0 0 · 1 −1 · 0 −1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

C = A, D = [I](N+1)×(N+1), us =

⎡⎢⎢⎢⎢⎢⎣

u0su1s·

uN−1s

uNs

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

(3.6)

where s = n − 1, n, n + 1 and

ϕn =

⎡⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

.ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

a =1h2, b = − 2

τ2− 2h2

− 1, c =1τ2,

ϕkn =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

−τπ sin(πxn), k = 0,

− exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,

τπ sin(πxn)(exp(−π2

)− exp(−π)

), k =N.

(3.7)

We seek a solution of the matrix equation in following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 1,

uM = 0,(3.8)


where αn(n = 1, . . . ,M) are (N + 1) × (N + 1) square matrices and βn(n = 1, . . . ,M) are(N + 1) × 1 column matrices defined by (see, [17]) as follows:

αn+1 = − (B + Cαn)−1A,


), n = 1, . . . ,M − 1,

(3.9)

where

α1 = [0](N+1)×(N+1), β1 = [0](N+1)×1. (3.10)

Now, applying (2.8) for N even number, we can present the following second-order ofaccuracy difference scheme:

−unk+1 − 2un

k+ un

k−1τ2

− un+1k − 2unk + un−1k

h2+ unk = − exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,

1 ≤ h ≤M − 1,

−3un0 + 4un1 − un22τ

= −π sin(πxn), 0 ≤ n ≤M,

unN−2(x) − 4unN−1(x) + 3unN(x)2τ

=unN/2−2 − 4un

N/2−1 + 3unN/2

2τ

+ π sin(πxn)(exp(−π2

)− exp(−π)

), 0 ≤ n ≤M,

u0k = uMk = 0, 0 ≤ k ≤N(3.11)

for the approximate solution of problem (3.1).So, again we have (N + 1) × (M + 1) system of linear equations in (3.11) and we will

write them in the following matrix form:

Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,

u0 = uM = 0,(3.12)


where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · 0 0 0 · 0 0 00 a 0 · 0 0 0 · 0 0 00 0 a · 0 0 0 · 0 0 0· · · · · · · · · · ·0 0 0 · 0 0 0 · a 0 00 0 0 · 0 0 0 · 0 a 00 0 0 · 0 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−3 4 −1 · 0 0 0 · 0 0 0b c b · 0 0 0 · 0 0 00 b c · 0 0 0 · 0 0 0· · · · · · · · · · ·0 0 0 · 0 0 0 · c b 00 0 0 · 0 0 0 · b c b0 0 0 · −1 4 −3 · 1 −4 3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

C = A, D = [I](N+1)×(N+1),

us =

⎡⎢⎢⎢⎢⎢⎣

u0su1s·

uN−1s

uNs

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

(3.13)

where s = n − 1, n, n + 1 and

ϕkn =

⎡⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

·ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

. (3.14)

Here,

a =1h2, b =

1τ2, c = − 2

h2− 2τ2

− 1, (3.15)

ϕkn =

⎧⎪⎨

⎪⎩

−2τπ sin(πxn), k = 0,− exp(−πtk) sin(πxn), 1 ≤ k ≤N − 1,

2τπ sin(πxn)(exp(−π2

)− exp(−π)

), k =N.

(3.16)

So, we have the second-order difference equation with respect to n with matrix coefficients.To solve this difference equation, we use the same algorithm (3.8) and (3.9).

Now, we will give the results of the numerical experiments.


Table 1: Comparison of the errors of difference schemes.

Difference schemes N =M = 20 N =M = 40 N =M = 60First-order difference scheme (2.7) 0.05384197882300 0.02631633782987 0.01740639813932Second-order difference scheme (2.8) 0.00631619894867 0.00164455475890 7.4144589892985e -004

The errors in numerical solutions are computed by

ENM = max1≤k≤N−1

(M−1∑

n=1

∣∣∣u(tk, xn) − ukn∣∣∣2h

)1/2

(3.17)

for different values of M and N, where u(tk, xn) represents the exact solution and uknrepresents the numerical solution at (tk, xn). The results are shown in Table 1 forN =M = 20,N =M = 40, andN =M = 60.

Thus, second-order of accuracy difference scheme is more accurate compared with thefirst-order of accuracy difference scheme.

4. Conclusion

The first and second-orders of accuracy difference schemes for approximate solutions ofthe Bitsadze-Samarskii-Dirichlet type nonlocal boundary value problem for the multidimen-sional elliptic partial differential equation are presented. Theorems on the stability, almostcoercive stability, and coercive stability estimates for the solution of these difference schemesare established. Numerical experiments are given.

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Research ArticleRegularity for Variational EvolutionIntegrodifferential Inequalities

Yong Han Kang1 and Jin-Mun Jeong2

1 Institute of Liberal Education, Catholic University of Daegu, Daegue 712-702, Republic of Korea2 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Correspondence should be addressed to Jin-Mun Jeong, [email protected]


Academic Editor: Sergey Piskarev

Copyright q 2012 Y. H. Kang and J.-M. Jeong. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We deal with the regularity for solutions of nonlinear functional integrodifferential equations gov-erned by the variational inequality in a Hilbert space. Moreover, by using the simplest definitionof interpolation spaces and the known regularity result, we also prove that the solution mappingfrom the set of initial and forcing data to the state space of solutions is continuous, which veryoften arises in application. Finally, an example is also given to illustrate our main result.

1. Introduction

In this paper, we deal with the regularity for solutions of nonlinear functional integrodiffer-ential equations governed by the variational inequality in a Hilbert spaceH:

(x′(t) +Ax(t), x(t) − z) + φ(x(t)) − φ(z)

≤(∫ t

0k(t − s)g(s, x(s))ds + h(t), x(t) − z

), a.e., 0 < t ≤ T, z ∈ H,

x(0) = x0,

(VIP)

where A is a unbounded linear operator associated with a sesquilinear form satisfyingGarding’s inequality and φ : H → (−∞,+∞] is a lower semicontinuous, proper convexfunction. The nonlinear mapping g is a Lipschitz continuous from R×V intoH in the secondcoordinate, where V is a dense subspace ofH.


The background of these problems has emerged vigorously in such applied fields asautomatic control theory, network theory, and the dynamic systems.

By using the subdifferential operator ∂φ, the control system (VIP) is represented bythe following nonlinear functional differential equation onH:

x′(t) +Ax(t) + ∂φ(x(t)) �∫ t

0k(t − s)g(s, x(s))ds + h(t), 0 < t ≤ T,

x(0) = x0.

(NDE)

In Section 4.3.2 of Barbu [1] (also see Section 4.3.1 in [2]) is widely developed theexistence of solutions for the case g ≡ 0. Recently, the regular problem for solutions of thenonlinear functional differential equations with a nonlinear hemicontinuous and coerciveoperator A was studied in [3]. Some results for solutions of a class of semilinear equationswith the nonlinear terms have been dealt with in [3–7]. As for nontrivial physical examplesfrom the field of visco-elastic materials modeled by integrodifferential equations on Banachspaces, we refer to [8].

In this paper, we will define φε : H → H(ε > 0) such that the function φε is Frechetdifferentiable onH and its Frechet differential ∂φε is a single valued and Lipschitz continuousonH with Lipschitz constant ε−1, where ∂φε = ε−1(I−(I+ε∂φ)−1) as is seen in Corollary 2.2 in[1, Chapter II]. It is also well-known results that limε→ 0φε = φ and limε→ 0∂φε(x) = (∂φ)0(x)for every x ∈ D(∂φ), where (∂φ)0 is the minimal segment of ∂φ. Now, we introduce thesmoothing system corresponding to (NDE) as follows:

x′(t) +Ax(t) + ∂φε(x(t)) =∫ t

0k(t − s)g(s, x(s))ds + h(t), 0 < t ≤ T,

x(0) = x0.

(SDE 1)

First we recall some regularity results and a variation of constant formula for solutionsof the semilinear functional differential equation (in the case g ≡ 0 in (SDE 1)):

x′(t) +Ax(t) + ∂φε(x(t)) = h(t) (1.1)

in a Hilbert spaceH.Next, based on the regularity results for (1.1), we intend to establish the regularity

for solutions of (NDE). Here, our approach is that results of a class of semilinear equationsas (1.1) on L2-regularity remain valid under the above formulation perturbed of nonlinearterms. Here, we note that sine A is not bounded operator H into itself, the Lipschitzcontinuity of nonlinear terms must be defined on some adjusted spaces (see Section 3).Moreover, using the simplest definition of interpolation spaces and known regularity, wehave that the solution mapping from the set of initial and forcing data to the state space ofsolutions is continuous, which very often arises in application. Finally, an example is alsogiven to illustrate our main result.


2. Preliminaries

Let V andH be complex Hilbert spaces forming Gelfand triple V ⊂ H ⊂ V ∗ with pivot spaceH. The norms of V , H and V ∗ are denoted by || · ||, | · |, and || · ||∗, respectively. The innerproduct inH is defined by (·, ·). The embeddings

V ↪→ H ↪→ V ∗ (2.1)

are continuous. Then the following inequality easily follows:

‖u‖∗ ≤ |u| ≤ ‖u‖, ∀u ∈ V. (2.2)

Let a(·, ·) be a bounded sesquilinear form defined in V × V and satisfying Garding’sinequality

Re a(u, u) ≥ ω1‖u‖2 −ω2|u|2, ω1 > 0, ω2 ≥ 0. (2.3)

Let A be the operator associated with the sesquilinear form a(·, ·):

(Au, v) = a(u, v), u, v ∈ V. (2.4)

Then A is a bounded linear operator from V to V ∗ and −A generates an analytic semigroupin both of H and V ∗ as is seen in [9, Theorem 6.1]. The realization for the operator A in Hwhich is the restriction of A to

D(A) = {u ∈ V ; Au ∈ H} (2.5)

is also denoted by A. From the following inequalities:

ω1‖u‖2 ≤ Re a(u, u) +ω2|u|2 ≤ C|Au| |u| +ω2|u|2 ≤ max{C,ω2}‖u‖D(A)|u|, (2.6)

where

‖u‖D(A) =(|Au|2 + |u|2

)1/2(2.7)

is the graph norm of D(A), it follows that there exists a constant C1 > 0 such that

‖u‖ ≤ C1‖u‖1/2D(A)|u|1/2. (2.8)

Thus, we have the following sequence:

D(A) ⊂ V ⊂ H ⊂ V ∗ ⊂ D(A)∗, (2.9)

where each space is dense in the next one and continuous injection.


Lemma 2.1. With the notations (2.8), (2.9), one has

(D(A),H)1/2,2 = V, (2.10)

where (D(A),H)1/2,2 denotes the real interpolation space between D(A) and H (Section 2.4 of [10]or [11]).

The following abstract linear parabolic equation:

x′(t) +Ax(t) = h(t), 0 < t ≤ T,x(0) = x0,

(LE)

has a unique solution x ∈ L2(0, T ;D(A)) ∩ W1,2(0, T,H) for each T > 0 if x0 ∈ V ≡(D(A),H)1/2,2 and h ∈ L2(0, T ;H). Moreover, one has

‖x‖L2(0,T ;D(A))∩W1,2(0,T,H) ≤ C2

(‖x0‖(D(A),H)1/2,2

+ ‖h‖L2(0,T ;H)

), (2.11)

where C2 depends on T andM (see [12, Theorem 2.3], [13]).In order to substituteH for the intermediate space V considering A as an operator in

B(V, V ∗) instead of B(D(A),H) one proves the following result.

Lemma 2.2. Let T > 0. Then

H =

{x ∈ V ∗ :

∫T

0

∥∥∥AetAx∥∥∥2

∗dt <∞

}. (2.12)

Hence, it implies that H = (V, V ∗)1/2,2 in the sense of intermediate spaces generated by an analyticsemigroup.

Proof. Put u(t) = etAx for x ∈ H. From the result of Theorem 2.3 in [12] it follows

u ∈ L2(0, T ;V ) ∩W1,2(0, T ;V ∗), (2.13)

hence

∫T

0


∗dt =∫T

0

∥∥u′(t)∥∥2∗dt <∞. (2.14)

Conversely, suppose that x ∈ V ∗ and∫T0 ||AetAx||2∗dt < ∞. Put u(t) = etAx. Then since

A is an isomorphism from V to V ∗ there exists a constant c > 0 such that

∫T

0‖u(t)‖2dt ≤ c

∫T

0‖Au(t)‖2∗dt = c

∫T

0


∗dt. (2.15)


Thus, we have u ∈ L2(0, T ;V ) ∩W1,2(0, T ;V ∗). By using the definition of real interpolationspaces by trace method, it is known that the embedding L2(0, T ;V ) ∩ W1,2(0, T ;V ∗) ↪→C([0, T];H) is continuous. Hence, it follows x = u(0) ∈ H.

In view of Lemma 2.2 we can apply (2.11) to (LE) in the space V ∗ as follows.

Proposition 2.3. Let x0 ∈ H and h ∈ L2(0, T ;V ∗), T > 0. Then there exists a unique solution x of(LE) belonging to

L2(0, T ;V ) ∩W1,2(0, T ;V ∗) ↪→ C([0, T];H) (2.16)

and satisfying

‖x‖L2(0,T ;V )∩W1,2(0,T ;V ∗) ≤ C2

(|x0| + ‖h‖L2(0,T ;V ∗)

), (2.17)

where C2 is a constant depending on T .

Let φ : V → (−∞,+∞] be a lower semicontinuous, proper convex function. Then thesubdifferential operator ∂φ of φ is defined by

∂φ(x) ={x∗ ∈ V ∗;φ(x) ≤ φ(y) + (x∗, x − y), y ∈ V }. (2.18)

First, let us concern with the following perturbation of subdifferential operator:

x′(t) +Ax(t) + ∂φ(x(t)) � h(t), 0 < t ≤ T,x(0) = x0.

(VE)

Using the regularity for the variational inequality of parabolic type in case where φ :V → (−∞,+∞] is a lower semicontinuous, proper convex function as is seen in [1, Section4.3] one has the following result on (VE).

Proposition 2.4. (1) Let h ∈ L2(0, T ;V ∗) and x0 ∈ V satisfying that φ(x0) < ∞. Then (VE) has aunique solution:

x ∈ L2(0, T ;V ) ∩W1,2(0, T ;V ∗) ↪→ C([0, T];H), (2.19)

which satisfies

x′(t) =(h(t) −Ax(t) − ∂φ(x(t)))0,

‖x‖L2∩W1,2∩C ≤ C3

(1 + ‖x0‖ + ‖h‖L2(0,T ;V ∗)

),

(2.20)

where C3 is a constant and L2 ∩W1,2 ∩ C = L2(0, T ;V ) ∩W1,2(0, T ;V ∗) ∩ C([0, T];H).


(2) Let A be symmetric and let us assume that there exist g ∈ H such that for every ε > 0 andany y ∈ D(φ)

Jε(y + εg

) ∈ D(φ), φ(Jε(y + εg

)) ≤ φ(y). (2.21)

Then for h ∈ L2(0, T ;H) and x0 ∈ D(φ) ∩ V , (VE) has a unique solution:

x ∈ L2(0, T ;D(A)) ∩W1,2(0, T ;H) ↪→ C([0, T];H), (2.22)

which satisfies

‖x‖L2∩W1,2∩C ≤ C3

(1 + ‖x0‖ + ‖h‖L2(0,T ;H)

). (2.23)

Remark 2.5. When the principal operator A is bounded from H to itself, we assume that φ :H → (−∞,+∞] is a lower semicontinuous, proper convex function and g : [0, T] ×H → Hbe a nonlinear mapping satisfying the following:

∣∣g(t, x1) − g(t, x2)∣∣ ≤ L|x1 − x2|, ∀x1, x2 ∈ H. (2.24)

Then it is easily seen that the result of (2) of Proposition 2.4. is immediately obtained.

Remark 2.6. Here, we remark that if V is compactly embedded in H and x ∈ L2(0, T ;V )) (orthe semigroup operator S(t) is compact), the following embedding:

L2(0, T ;V ) ∩W1,2(0, T ;V ∗) ↪→ L2(0, T ;H) (2.25)

is compact in view of Theorem 2 of Aubin [14]. Hence, the mapping (x0, f) �→ x is compactfrom V × L2(0, T ;V ∗) to L2(0, T ;H), which is also applicable to optimal control problem.

3. Regularity for Solutions

We start with the following assumption.

Assumption (F). Let g : [0, T] × V → H be a nonlinear mapping satisfying the following:

∣∣g(t, x) − g(t, y)∣∣ ≤ L∥∥x − y∥∥, g(t, 0) = 0 ∀x, y ∈ V (3.1)

for a positive constant L.For x ∈ L2(0, T ;V ) we set

f(t, x) =∫ t

0k(t − s)g(s, x(s))ds, (3.2)

where k belongs to L2(0, T).


Lemma 3.1. Let x ∈ L2(0, T ;V ), T > 0. Then f(·, x) ∈ L2(0, T ;H). And

∥∥f(·, x)∥∥L2(0,T ;H) ≤ L‖k‖L2

√T‖x‖L2(0,T ;V ). (3.3)

Moreover, if x1, x2 ∈ L2(0, T ;H), then

∥∥f(·, x1) − f(·, x2)∥∥L2(0,T ;H) ≤ L‖k‖

√T‖x1 − x2‖L2(0,T ;V ). (3.4)

The proof is immediately obtained from Assumption (F).For every ε > 0, define

φε(x) = inf

{‖x − Jεx‖2∗

2ε+ φ(Jεx) : x ∈ H

}, (3.5)

where Jε = (I + ε∂φ)−1. Then the function φε is Frechet differentiable on H and its Frechetdifferential ∂φε is Lipschitz continuous onH with Lipschitz constant ε−1 where ∂φε = ε−1(I −(I + ε∂φ)−1) as is seen in Corollary 2.2 in [1, Chapter II]. It is also well-known results thatlimε→ 0 φε = φ and limε→ 0 ∂φε(x) = (∂φ)0(x) for every x ∈ D(∂φ), where (∂φ)0 is the minimalsegment of ∂φ.

Now, one introduces the smoothing system corresponding to (NDE) as follows:

x′(t) +Ax(t) + ∂φε(x(t)) = f(t, x) + h(t), 0 < t ≤ T,x(0) = x0.

(SDE 2)

Since −A generates a semigroup S(t) on H, the mild solution of (SDE 2) can be representedby

xε(t) = S(t)x0 +∫ t

0S(t − s){f(s, xε) + h(s) − ∂φε(xε(s))

}ds. (3.6)

One will use a fixed point theorem and a step and step method to get the globalsolution for (NDE). Then one needs the following hypothesis.

Assumption (A). (∂φ)0 is uniformly bounded, that is,

∣∣∣(∂φ)0x∣∣∣ ≤M1, x ∈ V. (3.7)

Lemma 3.2. For given ε, λ > 0, let xε and xλ be the solutions of (SDE 2) corresponding to ε and λ,respectively. Then there exists a constant C independent of ε and λ such that

‖xε − xλ‖C([0,T];H)∩L2(0,T ;V ) ≤ C(ε + λ), 0 < T. (3.8)


Proof. From (SDE 2)we have

x′ε(t) − x′

λ(t) +A(xε(t) − xλ(t)) + ∂φε(xε(t)) − ∂φλ(xλ(t)) = f(t, xε) − f(t, xλ), (3.9)

and hence, from (2.3) and multiplying by xε(t) − xλ(t), it follows that

12d

dt|xε(t) − xλ(t)|2 +ω1‖xε(t) − xλ(t)‖2 +

(∂φε(xε(t)) − ∂φλ(xλ(t)), xε(t) − xλ(t)

)

≤ (f(t, xε) − f(t, xλ), xε(t) − xλ(t))+ω2|xε(t) − xλ(t)|2.

(3.10)

Here, we note

∣∣f(t, xε) − f(t, xλ)∣∣ ≤ L‖k‖L2‖xε(·) − xλ(·)‖L2(0,t;V )

∫T

0‖xε(·) − xλ(·)‖2L2(0,t;V )dt = T

∫T

0‖xε(t) − xλ(t)‖2dt.

(3.11)

Thus, we have

(f(t, xε) − f(t, xλ), xε(t) − xλ(t)

)

≤ ∣∣f(t, xε) − f(t, xλ)∣∣ · |xε(t) − xλ(t)|

≤ ω1

2T(L‖k‖L2)2∣∣f(t, xε) − f(t, xλ)

∣∣2 +T(L‖k‖L2)2

2ω1|xε(t) − xλ(t)|2

≤ ω1

2T‖xε(·) − xλ(·)‖2L2(0,t;H) +

T(L‖k‖L2)2

2ω1|xε(t) − xλ(t)|2.

(3.12)

Therefore, by using the monotonicity of ∂φ and integrating (3.10) over [0, T] it holds

12|xε(t) − xλ(t)|2 + ω1

2

∫T

0‖xε(t) − xλ(t)‖2dt

≤∫T

0

(∂φε(xε(t)) − ∂φλ(xλ(t)), λ∂φλ(xλ(t)) − ε∂φε(xε(t))

)dt

+

{T(L||k||L2)2

2ω1+ω2

}∫T

0|xε(t) − xλ(t)|2dt.

(3.13)

Here, we used that

∂φε(xε(t)) = ε−1(xε(t) −

(I + ε∂φ

)−1xε(t)). (3.14)


Since |∂φε(x)| ≤ |(∂φ)0x| for every x ∈ D(∂φ) it follows from Assumption (A) and usingGronwall’s inequality that

‖xε − xλ‖C([0,T];H)∩L2(0,T ;V ) ≤ C(ε + λ), 0 < T. (3.15)

Let x ∈ L1(0, T ;V ). Then it is well known that

limh→ 0

h−1∫h

0‖x(t + s) − x(t)‖ds = 0 (3.16)

for almost all point of t ∈ (0, T).

Definition 3.3. The point t which permits (3.16) to hold is called the Lebesgue point of x.

We establish the following results on the solvability of (NDE).

Theorem 3.4. Let Assumptions (F) and (A) be satisfied. Then for every (x0, h) ∈ V × L2(0, T ;V ∗),(NDE) has a unique solution:

x ∈ L2(0, T ;V ) ∩W1,2(0, T ;V ∗) ∩ C([0, T];H), (3.17)

and there exists a constant C4 depending on T such that

‖x‖L2∩W1,2∩C ≤ C4

(1 + ‖x0‖ + ‖h‖L2(0,T ;V ∗)

). (3.18)

Proof. Let us fix T0 > 0 such that

C1C2

(ε−1 +√T0 L‖k‖L2

)( T0√2

)1/2

< 1. (3.19)

Let y ∈ L2(0, T0;V ). Then f(·, y(·)) ∈ L2(0, T0;H) from Assumption (F). Set

(Fx)(t) = f(t, x(t)) − ∂φε(x(t)), 0 ≤ t ≤ T0. (3.20)

Then from Lemma 3.1 it follows that

|(Fx1)(t) − (Fx2)(t)| ≤(ε−1 +√T0L‖k‖L2

)‖x1(t) − x2(t)‖. (3.21)

For i = 1, 2, we consider the following equation:

x′i(t) +Axi(t) =

(Fyi)(t) + h(t), 0 < t ≤ T0,

xi(0) = x0.(3.22)


Then

d

dt(x1(t) − x2(t)) +A(x1(t) − x2(t)) =

(Fy1)(t) − (Fy2

)(t), t > 0,

x1(0) − x2(0) = 0.(3.23)

From (2.11) it follows that

‖x1 − x2‖L2(0,T0;D(A0))∩W1,2(0,T0;H) ≤ C2∥∥Fy1 − Fy2

∥∥L2(0,T0;H). (3.24)

Using the Holder inequality we also obtain that

‖x1 − x2‖L2(0,T0;H) =

{∫T0

0|x1(t) − x2(t)|2dt

}1/2

=

⎧⎨

⎩

∫T0

0

∣∣∣∣∣

∫ t

0(x1(τ) − x2(τ))dτ

∣∣∣∣∣

2

dt

⎫⎬

⎭

1/2

≤{∫T0

0t

∫ t

0|x1(τ) − x2(τ)|2dτdt

}1/2

≤√T0

2‖x1 − x2‖W1,2(0,T0;H).

(3.25)

Therefore, in terms of (2.8) and (3.25)we have

‖x1 − x2‖L2(0,T0;V ) ≤ C1‖x1 − x2‖1/2L2(0,T0;D(A0))‖x1 − x2‖1/2L2(0,T0;H)

≤ C1‖x1 − x2‖1/2L2(0,T0;D(A0))

(T0√2

)1/2

‖x1 − x2‖1/2W1,2(0,T0;H)

≤ C1

(T0√2

)1/2

‖x1 − x2‖L2(0,T0;D(A0))∩W1,2(0,T0;H)

≤ C1C2

(T0√2

)1/2∥∥Fy1 − Fy2∥∥L2(0,T0:H)

≤ C1C2

(ε−1 +√T0L‖k‖L2

)( T0√2

)1/2∥∥y1 − y2∥∥L2(0,T0;V ).

(3.26)

So by virtue of the condition (3.19) the contraction principle gives that (SDE 2) has a uniquesolution in [0, T0]. Thus, letting λ → 0 in Lemma 3.1 we can see that there exists a constant Cindependent of ε such that

‖xε − x‖C([0,T0];H)∩L2(0,T0;V ) ≤ Cε, 0 < T0, (3.27)


and hence, limε→ 0xε(t) = x(t) exists inH. From Assumption (F) and (3.27) it follows that

f(·, xε) −→ f(·, x), strongly in L2(0, T0;H),

Axn −→ Ax, strongly in L2(0, T0;V ∗).(3.28)

Since ∂φε(xε) is uniformly bounded by Assumption (A), from (3.27), (3.28) we have that

d

dtxε −→ d

dtx, weakly in L2(0, T0;V ∗), (3.29)

therefore

∂φε(xε) −→ f(·, x) + h − x′ −Ax, weakly in L2(0, T0;V ∗). (3.30)

Since (I + ε∂φ)−1xε → x strongly and ∂φ is demiclosed, we have that

f(·, x) + h − x′ −Ax ∈ ∂φ(x) in L2(0, T0;V ∗). (3.31)

Thus we have proved that x(t) satisfies a.e. on (0, T0) the equation (NDE).Let y be the solution of

y′(t) +Ay(t) + ∂φ(y(t)) � 0, 0 < t ≤ T0,

y(0) = x0,(3.32)

then, it implies

d

dt

(x(t) − y(t)) +A(x(t) − y(t)) + ∂φ(x(t)) − ∂φ(y(t)) � f(t, x) + h(t). (3.33)

Noting that || · || ≤ | · | ≤ || · ||, by multiplying by x(t) − y(t) and using the monotonicity of ∂φand (2.3), we obtain

12d

dt

∣∣x(t) − y(t)∣∣2 +ω1∥∥x(t) − y(t)∥∥2

≤ ω2∣∣x(t) − y(t)∣∣2 + ∣∣f(t, x) + h(t)∣∣ · ∥∥x(t) − y(t)∥∥.

(3.34)

Since

∣∣f(t, x) + h(t)∣∣ · ∥∥x(t) − y(t)∥∥ ≤ 1

2ω1

∣∣f(t, x) + h(t))∣∣2 +

ω1

2∥∥x(t) − y(t)∥∥2 (3.35)


for every c > 0 and by integrating on (3.34) over (0, t) we have

∣∣x(t) − y(t)∣∣2 +ω1

∫ t

0

∥∥x(s) − y(s)∥∥2ds

≤ 1ω1

∥∥f(·, x) + h∥∥L2(0,T0;V ∗) + 2ω2

∫ t

0

∣∣x(s) − y(s)∣∣2ds(3.36)

and by Gronwall’s inequality:

∣∣x(t) − y(t)∣∣2 +ω1

∫ t

0

∥∥x(s) − y(s)∥∥2ds ≤ ω−11 e

2ω2T0∥∥f(·, x) + h∥∥2L2(0,T0;V ∗). (3.37)

Let us fix T0 > T1 > 0 so that T1 is a Lebesgue point of x, φ(x(T1)) <∞, and

ω−11 e

2ω2T1√T1L‖k‖L2 < ω1. (3.38)

Put

N =√ω−2

1 eω2T1 , (3.39)

then from Assumption (F) it follows

∥∥x − y∥∥L2(0,T1;V ) ≤N∥∥f(·, x) + h∥∥L2(0,T1;V ∗)

≤N√T1 L‖k‖L2‖x‖L2(0,T1;V ) +N‖h‖L2(0,T1:V ∗)

(3.40)

and hence, from (2.17) in Proposition 2.3, we have that

‖x‖L2(0,T1;V )

≤ 1

1 −N√T1L‖k‖L2

(∥∥y∥∥L2(0,T1;V ) +N‖h‖L2(0,T1:V ∗)

)

≤ 1

1 −N√T1L‖k‖L2

{C2(1 + ‖x0‖) +N‖h‖L2(0,T1:V ∗)

}

≤ C4

(1 + ‖x0‖ + ‖h‖L2(0,T1:V ∗)

)

(3.41)

for some positive constant C4. Since the condition (3.38) is independent of initial values,noting the Assumption (A), the solution of (NDE) can be extended to the internal [0, nT1] fornatural number n, that is, for the initial x(nT1) in the interval [nT1, (n + 1)T1], as analogous


estimate (3.41) holds for the solution in [0, (n+ 1)T1]. The norm estimate of x inW1,2(0, T ;H)can be obtained by acting on both side of (NDE) by x′(t) and by using

d

dtφ(x(t)) =

(g(t),

d

dtx(t)), a.e., 0 < t, (3.42)

for all g(t) ∈ ∂φ(x(t)). Furthermore, the estimate (3.18) is immediately obtained from (3.41).

Theorem 3.5. Let Assumptions (F) and (A) be satisfied and (x0, h) ∈ V × L2(0, T ;V ∗), then thesolution x of (NDE) belongs to x ∈ L2(0, T ;V ) ∩W1,2(0, T ;V ∗) and the mapping:

V × L2(0, T ;V ∗) � (x0, h) �−→ x ∈ L2(0, T ;V ) ∩ C([0, T];H) (3.43)

is continuous.

Proof. If (x0, h) ∈ V × L2(0, T ;V ∗) then x belongs to L2(0, T ;V ) ∩ W1,2(0, T ;V ∗) formTheorem 3.4. Let (x0i, hi) ∈ V × L2(0, T ;V ∗) and xi be the solution of (NDE) with (x0i, hi)in place of (x0, h) for i = 1, 2. Multiplying on (NDE) by x1(t) − x2(t), we have

12d

dt|x1(t) − x2(t)|2 +ω1‖x1(t) − x2(t)‖2

≤ ω2|x1(t) − x2(t)|2 +∣∣f(t, x1) − f(t, x2)

∣∣‖x1(t) − x2(t)‖+ ‖h1(t) − h2(t)‖∗‖x1(t) − x2(t)‖.

(3.44)

Let us fix T1 > T2 > 0 so that T2 is a Lebesgue point of x, φ(x(T2) <∞, and

ω1 −ω−11 e

2ω2T2√T2 L‖K‖L2 > 0. (3.45)

Since

‖h1(t) − h2(t))‖∗‖x1(t) − x2(t)‖ ≤ 1ω1

‖h1(t) − h2(t)‖2∗ +ω1

4‖x1(t) − x2(t)‖2, (3.46)

by integrating on (3.44) over [0, T2]where T2 < T and as is seen in (3.37), it follows

‖x1 − x2‖2C([0,T2];H) +ω1

2‖x1 − x2‖2L2(0,T2;V )

≤ ‖x01 − x02‖2 + 1ω1

∥∥f(t, x1) − f(t, x2)∥∥2L2(0,T2;H) +

2ω1

‖h1 − h2‖L2(0,T2;V ∗)

≤ ‖x01 − x02‖2 +ω−11

√T2 L‖K‖L2‖x1 − x2‖2L2(0,T2;V ) +

2ω1

‖h1 − h2‖L2(0,T2;V ∗).

(3.47)


Putting that

N1 ≡ min[1,{ω1

2−ω−1

1

√T2L‖K‖L2

}]1/2, N2 ≡ max

{1,

2ω1

}, (3.48)

we have

‖x1 − x2‖L2∩C ≤ 2N2√1 −N1

(‖x01 − x02‖ + ‖h1 − h2‖). (3.49)

Suppose (x0n, hn) → (x0, h) in V × L2(0, T ;V ∗), and let xn and x be the solutions (SDE 2)with (x0n, hn) and (x0, h), respectively. Then, by virtue of (3.44) and (3.49), we see that xn →x in L2(0, T2, V ) ∩ W1,2(0, T2, V ∗) ↪→ C([0, T2];H). This implies that xn(T2) → x(T2) in H.Therefore the same argument shows that xn → x in

L2(T2,min{2T2, T};V ) ∩ C([T2,min{2T2, T}];H). (3.50)

Repeating this process, we conclude that xn → x in L2(0, T ;V ) ∩ W1,2(0, T2, V ∗) ↪→C([0, T2];H).

4. Example

Let Ω be bounded domain in Rn with smooth boundary ∂Ω. We define the following spaces:

H1(Ω) ={u : u,

∂u

∂xi∈ L2(Ω), i = 1, 2, . . . , n

},

H2(Ω) =

{u : u,

∂x

∂xi,∂2u

∂xi∂xj∈ L2(Ω), i, j = 1, 2, . . . , n

},

H10(Ω) =

{u : u ∈ H1(Ω), u|∂Ω = 0

}= the closure of C∞

0 (Ω) in H1(Ω),

(4.1)

where ∂/∂xiu and ∂2/∂xi∂xju are the derivative of u in the distribution sense. The norm ofH1

0(Ω) is defined by

‖u‖ =

{∫

Ω

n∑

i=1

(∂u(x)∂xi

)2

dx

}1/2

. (4.2)

Hence H10(Ω) is a Hilbert space. Let H−1(Ω) = H1

0(Ω)∗ be a dual space of H10(Ω). For any

l ∈ H−1(Ω) and v ∈ H10(Ω), the notation (l, v) denotes the value l at v. In what follows, we

consider the regularity for given equations in the spaces:

V = H10(Ω) =

{u ∈ H1(Ω); u = 0 on ∂Ω

}, H = L2(Ω), V ∗ = H−1(Ω) (4.3)

as introduced in Section 2. We deal with the Dirichlet condition’s case as follows.


Assume that aij = aji are continuous and bounded on Ω and {aij(x)} is positivedefinite uniformly in Ω, that is, there exists a positive number δ such that

n∑

i,j=1

aij(x)ξiξj ≥ δ|ξ|2, ∀ξ ∈ Ω. (4.4)

Let

bi ∈ L∞(Ω), c ∈ L∞(Ω), βi =n∑

j=1

∂aij

∂xj+ bi. (4.5)

For each u, v ∈ H10(Ω), let us consider the following sesquilinear form:

a(u, v) =∫

Ω

⎧⎨

⎩

n∑

i,j=1

aij∂u

∂xi

∂v

∂xj+

n∑

j=1

βi∂u

∂xiv + cuv

⎫⎬

⎭dx. (4.6)

Since {aij} is real symmetric, by (4.4) the inequality:

n∑

i,j=1

aij(x)ξiξj ≥ δ|ξ|2 (4.7)

holds for all complex vectors ξ = (ξ1, . . . , ξn). By hypothesis, there exists a constant K suchthat |βi(x)| ≤ K and c(x) ≤ K hold a.e., hence

Re a(u, u) ≥∫

Ωδ

n∑

i=1

∣∣∣∣∂u

∂xi

∣∣∣∣2

dx −K∫

Ω

n∑

i=1

∣∣∣∣∂u

∂xi

∣∣∣∣|u|dx −K∫

Ω|u|2dx

≥ δ∫

Ω

n∑

i=1

∣∣∣∣∂u

∂xi

∣∣∣∣2

dx −K∫

Ω

n∑

i=1

(ε

2

∣∣∣∣∂u

∂xi

∣∣∣∣2

+12ε

|u|2)dx −K

∫

Ω|u|2dx

=(δ − ε

2K) n∑

i=1

∫

Ω

∣∣∣∣∂u

∂xi

∣∣∣∣2

dx −(nK

2ε+K)∫

Ω|u|2dx.

(4.8)

By choosing ε = δK−1, we have

Re a(u, u) ≥ δ

2

n∑

i=1

∫

Ω

∣∣∣∣∂u

∂xi

∣∣∣∣2

dx −(nK2

2δ+K

)∫

Ω|u|2dx

=δ

2‖u‖21 −

(nK2

2δ+K +

δ

2

)‖u‖2.

(4.9)


By virtue of Lax-Milgram theorem, we know that for any v ∈ V there exists f ∈ V ∗ such that

a(u, v) =(f, v). (4.10)

Therefore, we know that the associated operator A : V → V ∗ defined by

(Au, v) = −a(u, v), u, v ∈ V (4.11)

is bounded and satisfies conditions (2.3) in Section 2.Let g : [0, T] × V → H be a nonlinear mapping defined by

g(t, u(t, x)) =∫ t

0

n∑

i=1

∂

∂xiσi(s,∇u(s, x))ds. (4.12)

We assume the following.

Assumption (F1). The partial derivatives σi(s, ξ), ∂/∂t σi(s, ξ) and ∂/∂ξjσi(s, ξ), exist andcontinuous for i = 1, 2, j = 1, 2, . . . , n, and σi(s, ξ) satisfies an uniform Lipschitz conditionwith respect to ξ, that is, there exists a constant L > 0 such that

∣∣∣σi(s, ξ) − σi(s, ξ)∣∣∣ ≤ L

∣∣∣ξ − ξ∣∣∣, (4.13)

where | · | denotes the norm of L2(Ω).

Lemma 4.1. If Assumption (F1) is satisfied, then the mapping t �→ g(t, ·) is continuously differen-tiable on [0, T] and u �→ g(·, u) is Lipschitz continuous on V .

Proof. Put

g1(s, u) =n∑

i=1

∂

∂xiσi(s,∇u), (4.14)

then we have g1(s, u) ∈ H−1(Ω). For each w ∈ H10(Ω), we satisfy the following that

(g1(s, u), w

)= −

n∑

i=1

(σi(s,∇u), ∂

∂xiw

). (4.15)

The nonlinear term is given by

g(t, u) =∫ t

0g1(s, u)ds. (4.16)


For any w ∈ H10(Ω), if u and u belong toH1

0(Ω), by Assumption (F1) we obtain

∣∣(g(t, u) − g(t, u)), w∣∣ ≤ LT‖u − u‖‖w‖. (4.17)

We set

f(t, u) =∫ t

0k(t − s)

∫ s

0

n∑

i=1

∂

∂xiσi(τ,∇u(τ, x))dτds, (4.18)

where k belongs to L2(0, T). Let φ : H10(Ω) → (−∞,+∞] be a lower semicontinuous, proper

convex function. Now in virtue of Lemma 4.1, we can apply the results of Theorem 3.4 asfollows.

Theorem 4.2. Let Assumption (F1) be satisfied. Then for any u0 ∈ H10(Ω) and h ∈ L2(0, T ;

H−1(Ω)), the following nonlinear problem:

(u′(t) +Au(t), u(t) − z) + φ(u(t)) − φ(z)

≤ (f(t, u) + h(t), u(t) − z), a.e., 0 < t ≤ T, z ∈ L2(Ω),

u(0) = u0

(4.19)

has a unique solution:

u ∈ L2(0, T ;H1

0(Ω))∩W1,2

(0, T ;H−1(Ω)

)↪→ C([0, T];L2(Ω)

). (4.20)

Furthermore, the following energy inequality holds: there exists a constant CT depending on Tsuch that

‖u‖L2∩W1,2 ≤ CT

(1 + ‖u0‖ + ‖h‖L2(0,T ;H−1(Ω))

). (4.21)

Acknowledgment

This research was supported by Basic Science Research Program through the NationalResearch Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (2011-0026609).

References

[1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhoff Leiden, TheNetherlands, 1976.

[2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, vol. 190 of Mathematics inScience and Engineering, Academic Press, Boston, Mass, USA, 1993.

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[4] N. U. Ahmed and X. Xiang, “Existence of solutions for a class of nonlinear evolution equations withnonmonotone perturbations,” Nonlinear Analysis. Series A, vol. 22, no. 1, pp. 81–89, 1994.

[5] J.-M. Jeong and J.-Y. Park, “Nonlinear variational inequalities of semilinear parabolic type,” Journal ofInequalities and Applications, vol. 6, no. 2, Article ID 896837, pp. 227–245, 2001.

[6] J. M. Jeong, Y. C. Kwun, and J. Y. Park, “Approximate controllability for semilinear retardedfunctional-differential equations,” Journal of Dynamical and Control Systems, vol. 5, no. 3, pp. 329–346,1999.

[7] Y. Kobayashi, T. Matsumoto, and N. Tanaka, “Semigroups of locally Lipschitz operators associatedwith semilinear evolution equations,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2,pp. 1042–1067, 2007.

[8] N. U. Ahmed, “Optimal control of infinite-dimensional systems governed by integrodifferentialequations,” in Differential Equations, Dynamical Systems, and Control Science, vol. 152 of Lecture Notesin Pure and Applied Mathematics, pp. 383–402, Dekker, New York, NY, USA, 1994.

[9] H. Tanabe, Equations of Evolution, vol. 6 of Monographs and Studies in Mathematics, Pitman, London,UK, 1979.

[10] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problmes and Applications, Springer,Berlin, Germany, 1972.

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[13] J. L. Lions and E. Magenes, Problemes Aux Limites Non Homogenes Et Applications, vol. 3, Dunod, Paris,France, 1968.

[14] J. P. Aubin, “Un Theoreme de Compacite,” Comptes Rendus de l’Academie des Sciences, vol. 256, pp.5042–5044, 1963.


Research ArticleWell-Posedness of the First Order ofAccuracy Difference Scheme for Elliptic-ParabolicEquations in Holder Spaces

Okan Gercek

Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

Correspondence should be addressed to Okan Gercek, [email protected]

Received 30 March 2012; Accepted 17 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright q 2012 Okan Gercek. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A first order of accuracy difference scheme for the approximate solution of abstract nonlocalboundary value problem −d2u(t)/dt2 + sign(t)Au(t) = g(t), (0 ≤ t ≤ 1), du(t)/dt + sign(t)Au(t) =f(t), (−1 ≤ t ≤ 0), u(0+) = u(0−), u′(0+) = u′(0−), andu(1) = u(−1) + μ for differential equationsin a Hilbert space H with a self-adjoint positive definite operator A is considered. The well-posedness of this difference scheme in Holder spaces without a weight is established. Moreover,as applications, coercivity estimates in Holder norms for the solutions of nonlocal boundary valueproblems for elliptic-parabolic equations are obtained.

1. Introduction

Nonlocal boundary value problems for partial differential equations have been applied byvarious researchers in order to model numerous processes in different fields of appliedsciences when they are unable to determine the boundary values of the unknown function(see, e.g., [1–15] and the references therein).

Well-posedness of difference schemes of elliptic-parabolic equations with nonlocalboundary conditions in Holder spaces with a weight was studied in [16–19].

In paper [20], the well-posedness of abstract nonlocal boundary value problem

−d2u(t)dt2

+ sign(t)Au(t) = g(t), (0 ≤ t ≤ 1),

du(t)dt

+ sign(t)Au(t) = f(t), (−1 ≤ t ≤ 0),


u(0+) = u(0−), u′(0+) = u′(0−),u(1) = u(−1) + μ

(1.1)

in Holder spaces without a weight was established. The coercivity inequalities for solutionsof the boundary value problem for elliptic-parabolic equations were obtained.

In the present paper, the first order of accuracy difference scheme

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk,

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1,

τ−1(uk − uk−1) −Auk−1 = fk, fk = f(tk−1),

tk−1 = (k − 1)τ, −N + 1 ≤ k ≤ −1,

uN = u−N + μ, u1 − u0 = u0 − u−1

(1.2)

for the approximate solution of problem (1.1) is considered. The well-posedness of differencescheme (1.2) in Holder spaces without a weight is established. As an application, coercivityinequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.

Throughout the paper, H denotes a Hilbert space and A is a self-adjoint positivedefinite operator with A ≥ δI for some δ > δ0 > 0. Then, it is wellknown that B = (1/2)(τA +√A(4 + τ2A)) is a self-adjoint positive definite operator and B ≥ δ1/2I. Furthermore, R =

(I + τB)−1 and P = P(τA) = (I + τA)−1 which are defined on the whole spaceH, are boundedoperators, where I is the identity operator.

2. Well-Posedness of (1.2)

First of all, let us start with some auxiliary lemmas that are used throughout the paper.

Lemma 2.1. The following estimates are satisfied [19, 21, 22]:

∥∥∥Pk∥∥∥H→H

≤M(δ)(1 + δτ)−k, kτ∥∥∥APk

∥∥∥H→H

≤M(δ),

∥∥∥Rk∥∥∥H→H

≤M(δ)(1 + δτ)−k, kτ∥∥∥BRk

∥∥∥H→H

≤M(δ),

∥∥∥Pk − e−kτA∥∥∥H→H

≤ M(δ)k

,∥∥∥Rk − e−kτA1/2

∥∥∥H→H

≤ M(δ)k

,

∥∥∥∥(I − R2N

)−1∥∥∥∥H→H

≤M(δ), k ≥ 1, δ > 0,

(2.1)

for someM(δ) > 0, which is independent of τ is a positive small number.


Let Fτ(H) = F([a, b]τ ,H) be the linear space of mesh functions ϕτ = {ϕk}Nb

Nadefined

on [a, b]τ = {tk = kh,Na ≤ k ≤ Nb,Naτ = a,Nbτ = b} with values in the Hilbert space H.Next, C([a, b]τ ,H), Cα([−1, 1]τ ,H), Cα/2([−1, 0]τ ,H), and Cα([0, 1]τ ,H) (0 < α < 1) denoteBanach spaces on Fτ(H) with norms:

∥∥ϕτ∥∥C([a,b]τ ,H) = max

Na≤k≤Nb

∥∥ϕk∥∥H,

∥∥ϕτ∥∥Cα([−1,1]τ ,H) =

∥∥ϕτ∥∥C([−1,1]τ ,H) + sup

−N≤k<k+r≤0

∥∥ϕk+r − ϕk∥∥Hr

−α/2



−α,

∥∥ϕτ∥∥Cα/2([−1,0]τ ,H) =

∥∥ϕτ∥∥C([−1,0]τ ,H) + sup

−N≤k<k+r≤0


−α/2,

∥∥ϕτ∥∥Cα([0,1]τ ,H) =

∥∥ϕτ∥∥C([0,1]τ ,H) + sup

1≤k<k+r≤N−1


−α.

(2.2)

With the help of the self-adjoint positive definite operator B in a Hilbert space H, theBanach space Eα = Eα(B,H) (0 < α < 1) consists of those v ∈ H for which the norm (see[22, 23]):

‖v‖Eα = supz>0

zα∥∥∥B(z + B)−1v

∥∥∥H+ ‖v‖H, (2.3)

is finite. By the definition of Eα(B,H),

D(B) ⊂ Eα(B,H) ⊂ Eβ(B,H) ⊂ H, (2.4)

for all β < α.

Lemma 2.2. For 0 < α < 1, the norms of the spaces Eα(B,H) and Eα/2(A,H) are equivalent (see[24]).

Theorem 2.3. Suppose μ ∈ D(A), Aμ ∈ Eα(B,H), f0 +g0 ∈ Eα/2(A,H), f−N +gN ∈ Eα(B,H),g(t) ∈ Cα([0, 1]τ ,H), and f(t) ∈ Cα/2([−1, 0]τ ,H), 0 < α < 1. Boundary value problem (1.2) iswellposed in Holder space Cα([−1, 1]τ ,H) and the following coercivity inequality holds:

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα([0,1]τ ,H)

+∥∥∥{Auk}N−1

−N∥∥∥Cα([−1,1]τ ,H)

+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα/2([−1,0]τ ,H)

≤M[∥∥Aμ

∥∥Eα(B,H) +

1α(1 − α)

[∥∥fτ∥∥Cα/2([−1,0]τ ,H) +

∥∥gτ∥∥Cα([0,1]τ ,H)

]

+∥∥(I + τB)

(f0 + g0

)∥∥Eα/2(A,H) +

∥∥(I + τB)(f−N + gN

)∥∥Eα(B,H)

],

(2.5)

whereM is independent of not only fτ , gτ , and μ but also of τ and α.


Proof. First of all, let us get the formulae for solution of problem (1.2). By [21, 25],

uk =(I − R2N

)−1{[Rk − R2N−k

]ξ +[RN−k − RN+k

]ψ

−[RN−k − RN+k

](I + τB)(2I + τB)−1B−1

N−1∑

s=1

[RN−s − RN+s

]gsτ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

s=1

[R|k−s| − Rk+s

]gsτ, 1 ≤ k ≤N

(2.6)

is the solution of boundary value difference problem:

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1, u0 = ξ, uN = ψ,(2.7)

uk = P−kξ − τ0∑

s=k+1

Ps−kfs, −N ≤ k ≤ −1 (2.8)

is the solution of inverse Cauchy problem:

τ−1(uk − uk−1) −Auk−1 = fk, fk = f(tk−1),

tk−1 = (k − 1)τ, −(N − 1) ≤ k ≤ 0, u0 = ξ.(2.9)

Combining the conditions ψ = u−N +μ, ξ = u0 and formulas (2.6), (2.8), we get formu-las

uk =(I − R2N

)−1{[Rk − R2N−k

]u0 +

[RN−k − RN+k

][PNu0 − τ

0∑

s=−N+1

Ps+Nfs + μ

]

−[RN−k − RN+k

](I + τB)(2I + τB)−1B−1

N−1∑

s=1

[RN−s − RN+s

]gsτ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

s=1

[R|k−s| − Rk+s

]gsτ, 1 ≤ k ≤N,

(2.10)

uk = P−ku0 − τ0∑

s=k+1

Ps−kfs, −N ≤ k ≤ −1. (2.11)


Operator equation

2u0 − Pu0 + τPf0 =(I − R2N

)−1{[R − R2N−1

]u0 +

[RN−1 − RN+1

]

×[PNu0 − τ

0∑

s=−N+1

Ps+Nfs + μ

]

−[RN−1−RN+1

](I+τB)(2I+τB)−1B−1

N−1∑

s=1

[RN−s−RN+s

]gsτ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

s=1

[Rs−1 − R1+s

]gsτ

(2.12)

follows from formulas (2.10), (2.11), and the condition u1 − u0 = u0 − u−1. As the operator

I + (I + τA)(I + 2τA)−1R2N−1 + B−1A(I + 2τA)−1(I − R2N−1

)

− (2I + τB)(I + 2τA)−1RNPN−1(2.13)

has an inverse

Tτ =(I + (I + τA)(I + 2τA)−1R2N−1 + B−1A(I + 2τA)−1

(I − R2N−1

)

−(2I + τB)(I + 2τA)−1RNPN−1)−1

,

(2.14)

it follows that

u0 = Tτ(I + 2τA)−1(I + τA)

{{(2 + τB)RN

[−τ

0∑

s=−N+1

Ps+Nfs + μ

]

−RN−1B−1N−1∑

s=1

[RN−s − RN+s

]gsτ

}

+(I − R2N

)B−1

N−1∑

s=1

Rs−1gsτ −(I − R2N

)(I + τB)B−1Pf0

}

(2.15)

for the solution of operator equation (2.12). Hence, we have formulas (2.10), (2.11), and (2.15)for the solution of difference problem (1.2).


Using formulae (2.10) and (2.15), we can get

Au0 = Tτ(I + 2τA)−1(I + τA)

×{{

(2 + τB)RN

[−τ

0∑

s=−N+1

APs+N(fs − f−N+1

)+Aμ

]

−RN−1AB−2{N−1∑

s=1

BRN−s(gs − gN−1)τ +

N−1∑

s=1

BRN+s(g1 − gs)τ

}}

+(I − R2N

)AB−2

N−1∑

s=1

BRs−1(gs − g1)τ

}

+ Tτ(I + 2τA)−1(I + τA)

×{{

(2 + τB)RN(PN − I

)f−N+1

−RN−1AB−2{(I − RN−1

)gN−1 −

(RN−2 − R2N−1

)g1}}

+(I − R2N

)AB−2

(I − RN−1

)g1 −

(I − R2N

)(I + τB)B−1APf0

},

(2.16)

AuN = PN{Tτ(I + 2τA)−1(I + τA)

×{{

(2 + τB)RN

[−τ

0∑

s=−N+1


)+Aμ

]

−RN−1AB−2{N−1∑

s=1

BRN−s(gs − gN−1)τ +

N−1∑

s=1

BRN+s(g1 − gs)τ

}}

+(I − R2N

)AB−2

N−1∑

s=1

BRs−1(gs − g1)τ

}}

− τ0∑

s=−N+1


)+Aμ +

(PN − I

)f−N+1

+ PN{Tτ(I + 2τA)−1(I + τA)

×{{

(2 + τB)RN(PN − I

)f−N+1

−RN−1AB−2{(I − RN−1

)gN−1 −

(RN−2 − R2N−1

)g1}}

+(I − R2N

)AB−2

(I − RN−1

)g1 −

(I − R2N

)(I + τB)B−1APf0

}}.

(2.17)


Finally, we will get coercivity estimate (2.5). It is based on estimates

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα([0,1]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥Cα([0,1]τ ,H)

≤M[

1α(1 − α)

∥∥gτ∥∥Cα([0,1]τ ,H) +

∥∥Au0 − g0∥∥Eα(B,H) +

∥∥AuN − gN∥∥Eα(B,H)

] (2.18)

for the solution of boundary value difference problem (2.7),

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα/2([−1,0]τ ,H)

+∥∥∥{Auk}0−N

∥∥∥Cα/2([−1,0]τ ,H)

≤M[

1(α/2)(1 − α/2)

∥∥fτ∥∥Cα/2([−1,0]τ ,H) +

∥∥Au0 + f0∥∥Eα(A,H)

],

(2.19)

for the solution of inverse Cauchy difference problem (2.9), and

∥∥Au0 + f0∥∥Eα/2(A,H) ≤

M

α(1 − α)[∥∥g∥∥Cα([0,1],H) +

∥∥f∥∥Cα/2([−1,0],H)

]

+M[∥∥Aμ

∥∥Eα(B,H) +

∥∥f0 + g0∥∥Eα/2(A,H)

],

∥∥Au0 − g0∥∥Eα(B,H) ≤

M

α(1 − α)[∥∥f∥∥Cα/2([−1,0],H) +

∥∥g∥∥Cα([0,1],H)

]

+M[∥∥Aμ

∥∥Eα(B,H) +

∥∥f0 + g0∥∥Eα/2(A,H)

],

∥∥AuN − gN∥∥Eα(B,H) ≤

M

α(1 − α)[∥∥f∥∥Cα/2([−1,0],H) +

∥∥g∥∥Cα([0,1],H)

]

+M[∥∥Aμ

∥∥Eα(B,H) +

∥∥f0 + g0∥∥Eα/2(A,H) +

∥∥f−N + gN∥∥Eα(B,H)

]

(2.20)

for the solution of problem (1.2). Estimates (2.18) and (2.19) were established in [21, 25],respectively.


Estimates (2.20) are derived from the formulas (2.16) and (2.17) for the solution ofproblem (1.2), estimates (2.1) and following estimates

∥∥∥Rk(τB)∥∥∥H→H

≤ M, 1 ≤ k ≤N,

∥∥∥∥(I − R2N(τB)

)−1∥∥∥∥H→H

≤ M, k ≥ 1,

∥∥∥Rk+r(τB) − Rk(τB)∥∥∥H→H

≤ M(r)α

(k + r)α, 1 ≤ k < k + r ≤N, 0 ≤ α ≤ 1,

∥∥∥(I − R(τB))2(τB)−2∥∥∥H→H

≤ M,

∥∥∥(I + R(τB))−1∥∥∥H→H

≤ M,

‖Tτ‖H→H ≤M, ‖BRPTτ‖H→H ≤M,

(2.21)

which were established in [26]. This finalizes the proof of Theorem 2.3.

3. An Application

In this section, an application of the abstract Theorem 2.3 is considered. First, letΩ be the unitopen cube in the n-dimensional Euclidean space R

n (0 < xk < 1, 1 ≤ k ≤ n) with boundaryS, Ω = Ω ∪ S. In [−1, 1] ×Ω, the mixed boundary value problem for multidimensional mixedequation:

−utt −n∑

r=1

(ar(x)uxr )xr = g(t, x), 0 < t < 1, x ∈ Ω,

ut +n∑

r=1

(ar(x)uxr )xr = f(t, x), −1 < t < 0, x ∈ Ω,

f(0, x) + g(0, x) = 0, f(−1, x) + g(1, x) = 0, x ∈ Ω,

u(t, x) = 0, x ∈ S, −1 ≤ t ≤ 1; u(1, x) = u(−1, x) + μ(x), x ∈ Ω,

u(0+, x) = u(0−, x), ut(0+, x) = ut(0−, x), x ∈ Ω

(3.1)

is considered. Here, ar(x) (x ∈ Ω), μ(x) (μ(x) = 0, x ∈ S), g(t, x) (t ∈ (0, 1), x ∈ Ω),and f(t, x) (t ∈ (−1, 0), x ∈ Ω) are given smooth functions and ar(x) ≥ a > 0.


The discretization of problem (3.1) is carried out in two steps. In the first step, the gridsets

Ωh = {x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn) ,

0 ≤ mr ≤Nr, hrNr = 1, r = 1, . . . , n},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S

(3.2)

are defined. To the differential operatorA generated by problem (3.1), the difference operatorAxh is assigned by formula:

Axhu

h = −n∑

r=1

(ar(x)uhxr

)

xr ,mr

(3.3)

acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ Sh.With the help of Ax

h, we arrive at the nonlocal boundary-value problem

−d2uh(t, x)dt2

+Axhu

h(t, x) = gh(t, x), 0 < t < 1, x ∈ Ωh,

duh(t, x)dt

−Axhu

h(t, x) = fh(t, x), −1 < t < 0, x ∈ Ωh,

uh(−1, x) = uh(1, x) + μh(x), x ∈ Ωh,

uh(0+, x) = uh(0−, x), duh(0+, x)dt

=duh(0−, x)

dt, x ∈ Ωh,

(3.4)

for an infinite system of ordinary differential equations.In the second step, problem (3.4) is replaced by difference scheme (1.2) (see [21]):

−uhk+1(x) − 2uh

k(x) + uhk−1(x)

τ2+Ax

huhk(x) = g

hk (x),

ghk (x) = gh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uhk(x) − uhk−1(x)

τ−Ax

huhk−1(x) = f

hk (x),

fhk (x) = fh(tk, x), tk−1 = (k − 1)τ, −N + 1 ≤ k ≤ −1, x ∈ Ωh,

uh−N(x) = uhN(x) + μh(x), x ∈ Ωh,

uh1(x) − uh0(x) = uh0(x) − uh−1(x), x ∈ Ωh.

(3.5)


To formulate the result, we introduce the Hilbert spaces L2h = L2(Ωh), W12h = W1

2 (Ωh), andW2

2h = W22 (Ωh) of the grid functions ϕh(x) = {ϕ(h1m1, . . . , hnmn)} defined on Ωh, equipped

with the norms:

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W1

2h

=∥∥∥ϕh∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣∣(ϕh)

xr

∣∣∣∣2

h1 · · ·hn⎞

⎠

12,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣∣(ϕh)

xr

∣∣∣∣2

h1 · · ·hn⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣∣(ϕh)

xrxr ,mr

∣∣∣∣2

h1 · · ·hn⎞

⎠1/2

.

(3.6)

Theorem 3.1. Let τ and |h| =√h21 + · · · + h2n be sufficiently small numbers. Then, the solutions of

difference scheme (3.5) satisfy the following coercivity stability estimate:

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥Cα([0,1]τ ,L2h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥Cα/2([−1,0]τ ,L2h)

+∥∥∥∥{uhk

}N−1

−N

∥∥∥∥Cα([−1,1]τ ,W2

2h)

≤M[∥∥∥μh

∥∥∥W2

2h

+1

α(1 − α)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥Cα/2([−1,0]τ ,L2h)

+∥∥∥∥{ghk

}N−1

1

∥∥∥∥Cα([0,1]τ ,L2h)

]],

(3.7)

whereM is not dependent on τ, h, μh(x), ghk (x), 1 ≤ k ≤N − 1, and fhk , −N + 1 ≤ k ≤ 0.

The proof of Theorem 3.1 is based on Theorem 2.3, the symmetry properties of thedifference operator Ax

hdefined by formula (3.3), and along with the following theorem on

the coercivity inequality for the solution of elliptic difference equation in L2h.

Theorem 3.2. For the solution of elliptic difference problem:

Axhu

h(x) = ωh(x), x ∈ Ωh,

uh(x) = 0, x ∈ Sh,(3.8)


the following coercivity inequality holds [27]:

n∑

r=1

∥∥∥∥(uh)

xrxr ,mr

∥∥∥∥L2h

≤M∥∥∥ωh

∥∥∥L2h. (3.9)

Here,M depends neither on h nor wh(x).

Acknowledgments

The author would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey)for his inspirational contributions and to the anonymous referees whose careful reading ofthe paper and valuable comments helped to improve it.

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University Press, Tbilisi, Georgia, 1981.[4] V. N. Vragov, Boundary Value Problems for Nonclassical Equations of Mathematical Physics, Textbook for

Universities, NGU, Novosibirsk, Russia, 1983.[5] D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed

and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.[6] A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, Moskow, Russia, 1995.[7] S. N. Glazatov, “Nonlocal boundary value problems for linear and nonlinear equations of variable

type,” Sobolev Institute of Mathematics SB RAS, no. 46, p. 26, 1998.[8] A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semilinear

hyperbolic-parabolic equations,” Nonlinear Analysis-Theory, Methods and Applications, vol. 47, no. 5,pp. 3585–3592, 2001.

[9] D. Guidetti, B. Karasozen, and S. Piskarev, “Approximation of abstract differential equations,” Journalof Mathematical Sciences, vol. 122, no. 2, pp. 3013–3054, 2004.

[10] J. I. Dıaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, “An elliptic-parabolic equation with anonlocal term for the transient regime of a plasma in a stellarator,” Journal of Differential Equations,vol. 198, no. 2, pp. 321–355, 2004.

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Research ArticleBasis Properties of Eigenfunctions of Second-OrderDifferential Operators with Involution

Asylzat Kopzhassarova and Abdizhakhan Sarsenbi

Department of Mathematics, M. Auezov South Kazakhstan State University, 5 Tauke Han Avenue,160012 Shymkent, Kazakhstan

Correspondence should be addressed to Asylzat Kopzhassarova, asyl [email protected]

Received 8 May 2012; Revised 13 July 2012; Accepted 6 August 2012

Academic Editor: Valery Covachev

Copyright q 2012 A. Kopzhassarova and A. Sarsenbi. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We study the basis properties of systems of eigenfunctions and associated functions for one kindof generalized spectral problems for a second-order ordinary differential operator.

1. Introduction

Let us consider the partial differential equation with involution

wt(t, x) = αwxx(t, x) +wxx(t,−x), −1 < x < 1, t > 0. (1.1)

If the initial conditions

w(0, x) = f(x) (1.2)

and the boundary conditions

αjwx(t,−1) + βjwx(t, 1) + αj1w(t,−1) + βj1w(t, 1) = 0, j = 1, 2 (1.3)

are given, then the solving of this equation by Fourier’s method leads to the problem ofexpansion of function f(x) into series of eigenfunctions of spectral problem

−u′′(−x) + αu′′(x) = λu(x),αju

′(−1) + βju′(1) + αj1u(−1) + βju(1) = 0, j = 1, 2.(1.4)


If the function f(x) ∈ L2(−1, 1), then the question about basis property of eigenfunctionsof spectral problem for second-order ordinary differential operator with involutionraises.

Work of many researchers is devoted to the study of differential equations [1–5].Various aspects of functionally differential equations with involution are studied in [6, 7]. Thespectral problems for the double differentiation operator with involution are studied in [8–11] and the issues Riesz basis property of eigenfunctions in terms of coefficients of boundaryconditions were considered.

This kind of spectral problems arises in the theory of solvability of differentialequations in partial derivatives with an involution [7, page 265].

Results presented below are a continuation of studies of one of the authors in [9–11].

2. General Boundary Value Problem

In this paper, we study the spectral problem of the form

Lu ≡ −u′′(−x) + αu′′(x) + βu′(x) + γu′(−x) + ηu(−x) = λu(x), (2.1)

α1u′(−1) + β1u′(1) + α11u(−1) + β11u(1) = 0,

α2u′(−1) + β2u′(1) + α21u(−1) + β21u(1) = 0,

(2.2)

where α, β, γ, η, αi, βi, αij , βij are some complex numbers.By direct calculation, one can verify that the square of the operator is in the form

L2u =(1 + α2

)uIV (x) − 2αuIV (−x) + 2αγu′′′(−x) + 2αβu′′′(x) + 2αηu′′(−x)

+(−2η + β2 − γ2

)u′′(x) + η2u(x).

(2.3)

Since it is assumed that Lu belongs to domain of operator L also, then function Lusatisfies boundary-value conditions (2.2)

α1(Lu)′(−1) + β1(Lu)′(1) + α11(Lu)(−1) + β11(Lu)(1) = 0,

α2(Lu)′(−1) + β2(Lu)′(1) + α21(Lu)(−1) + β21(Lu)(1) = 0.(2.4)

That is, the operator L2 is generated by previous differential expression and boundary-valueconditions (2.2) and (2.4).

The expression L2u is an ordinary differential expression for α = 0.Therefore, applying the method in [8–10] we can obtain the following statement (the

result).


Theorem 2.1. If α = 0, then the eigenfunctions of the generalized spectral problem (2.1) and (2.2)form a Riesz basis of the space L2(−1, 1) in the following cases:

(1) α1β2 − α2β1 /= 0;

(2) α1β2 − α2β1 = 0, |α1| + |β1| > 0, α21 /= β22, α

221 /= β

221,

(3) α1 = β1 = α2 = β2 = 0; α11β21 − α21β11 /= 0.

The root vectors of operators A and A2 coincide under some conditions (see, forinstance, [10]). Therefore, we can consider the square of the operator L which is an ordinarydifferential operator. It is well known [12–14] that eigenfunctions of ordinary differentialoperator of even order with strongly regular boundary value conditions form a Riesz basis.As in [10], from here it is possible to deduce correctness of Theorem 2.1.

This technique is not applicable for a = 0 since L2u is not an ordinary differentialoperator. Therefore, we consider this case separately.

3. General Solution of Special Type Equation

Let the operator L be given by the differential expression with an involution

Lu = −u′′(−x) + αu′′(x), (3.1)

and boundary conditions (2.2).We consider the spectral problem Lu = λu(x) with periodic, antiperiodic boundary

conditions, with the boundary conditions of Dirichlet and Sturm type. In these cases, itis possible to compute all the eigenvalues and eigenfunctions explicitly. The basis of ourstatements is the following.

Theorem 3.1. If a2 /= 1, then the general solution of equation

−u′′(−x) + αu′′(x) = λu(x), (3.2)

where λ is the spectral parameter, has the form

u(x) = A cos

√λ

1 − αx + B sin

√λ

−1 − αx,(3.3)

where A and B are arbitrary complex numbers.

If α2 = 1 and λ/= 0, then (3.2) has only the trivial solution.

Proof. It is easy to see that functions (3.3) are solutions of (3.2). Let us prove the absence ofother solutions.


Any function u(x) can be represented as a sum of even and odd functions. Substitutingthis representation into (3.2) and into −u′′(x) + αu′′(−x) = λu(−x), we conclude that thefunctions u1(x) and

−(1 − α)u′′1(x) = λu1(x),−(−1 − α)u′′2(x) = λu2(x).

(3.4)

4. The Dirichlet Problem

Consider the spectral problem (3.2) a2 /= 1 with boundary conditions

u(−1) = 0, u(1) = 0. (4.1)

Note that the spectral problem (3.2) and (4.1) is self-adjoint for real α. We calculate theeigenvalues and eigenfunctions of the Dirichlet problem (3.2) and (4.1). Using Theorem 3.1,it is easy to see that the spectral problem (3.2) and (4.1) has two sequences of simpleeigenvalues.

If α /∈ {(8k2 + 4k + 1)/(4k + 1) : k ∈ Z}, then corresponding eigenfunctions are givenby the formulas

uk1(x) = cos(π2+ kπ

)x, k = 0, 1, 2, . . . , uk2(x) = sin kπx, k = 1, 2, . . . . (4.2)

If α /∈ (8k2 + 4k + 1)/(4k + 1) for some k0 ∈ Z, then the eigenfunctions of the spectral problem(3.2) and (4.1) are given by

uk1(x) = cos(π2+ kπ

)x, k = 0, 1, 2, . . . , uk2(x) = sin kπx, k = 1, 2, . . . , k /= k0,

uk01(x) = cos(π2+ k0π

)x + sin

√1 − α−1 − α

(π2+ k0π

)x,

uk02(x) = sin k0πx + cos

√−1 − α1 − α k0πx.

(4.3)

Theorem 4.1. If a2 /= 1, then the system of eigenfunctions of the spectral problem (3.2) and (4.1),which is given above, forms an orthonormal basis of the space L2(−1, 1).

Proof. For real values of α, the spectral problem (3.2) and (4.1) is self-adjoint. Therefore,the system (4.1), as a system of eigenfunctions self-adjoint operator, is an orthonormal.Analogously, the case α = (8k20 + 4k0 + 1)/(4k0 + 1), k0 ∈ Z, is considered. Also note thatevery orthonormal basis is automatically a Riesz basis.

The system (4.2) does not depend on α, hence Theorem 4.1 is proved.


5. Periodic and Antiperiodic Problem

Now consider the spectral problem (3.2) with the periodic boundary conditions

u(−1) = u(1), u′(−1) = u′(1). (5.1)

It follows immediately from Theorem 3.1 that the eigenfunctions of the spectral problem (3.2)and (5.1) are given by

(λk1)2 = −(1 + α)k2π2, (λk2)

2 = (1 − α)k2π2. (5.2)

They are simple and correspond to the eigenfunctions

uk1(x) = sin kπx, k = 0, 1, 2, . . . , uk2(x) = cos kπx, k = 0, 1, 2, . . . . (5.3)

Similarly, the eigenvalues and eigenfunctions of the spectral problem with antiperiodicboundary conditions

u(−1) = −u(1), u′(−1) = −u′(1) (5.4)

are calculated.In this case, there are two series of eigenvalues also

(λk1)2 = (1 − α)

(π2+ kπ

), k = 0, 1, 2, . . . ,

(λk2)2 = (−1 − α)

(π2+ kπ

), k = 0, 1, 2, . . . .

(5.5)

They correspond to the eigenfunctions

uk1 = cos(π2+ kπ

)x, k = 1, 2, . . . , uk2 = sin

(π2+ kπ

)x, k = 0, 1, 2, . . . . (5.6)

Theorem 5.1. If α2 /= 1, then the systems of eigenfunctions of the spectral problem (3.2) with periodicor antiperiodic boundary conditions form orthonormal bases of the space L2(−1, 1).

The proof is analogous to the proof of Theorem 4.1. Also note that for periodicconditions the eigenfunctions form the classical orthonormal basis of L2(−1, 1).

Analogously, it is possible to check that the eigenfunctions of spectral problems (3.2),α2 /= 1, with boundary conditions of Sturm type

u′(−1) = 0, u′(1) = 0 (5.7)


and with nonself-adjoint boundary conditions

u(−1) = 0, u′(−1) = u′(1) (5.8)

form orthonormal bases of L2(−1, 1).

Acknowledgment

The work is carried out under the auspices of Ministry of Education and Science of theRepublic of Kazakhstan (0264/SF, 0753/SF).

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Research ArticleA Note on the Inverse Problem for a FractionalParabolic Equation

Abdullah Said Erdogan and Hulya Uygun


Correspondence should be addressed to Abdullah Said Erdogan, [email protected]

Received 15 May 2012; Accepted 8 July 2012


Copyright q 2012 A. S. Erdogan and H. Uygun. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

For a fractional inverse problemwith an unknown time-dependent source term, stability estimatesare obtained by using operator theory approach. For the approximate solutions of the problem,the stable difference schemes which have first and second orders of accuracy are presented. Thealgorithm is tested in a one-dimensional fractional inverse problem.

1. Introduction

Inverse problems arise in many fields of science and engineering such as ion transportproblems, chromatography, and heat determination problems with an unknown internalenergy source. Different typed of inverse problems have been investigated, and the mainresults obtained in this field of research were given by many researchers (see [1–10]).More than three centuries the theory of fractional derivatives developed mainly as a puretheoretical field of mathematics. Fractional integrals and derivatives appear in the theoryof control of dynamical systems, when the controlled system or/and the controller isdescribed by a fractional differential equation (see [11]). Recently, many application areassuch as bioengineering applications, image and signal processing are also related to fractionalcalculus. Methods of solutions of problems and theory of fractional calculus have beenstudied by many researchers [11–28]. Among them finite difference method is used forsolving several fractional differential equations (see [20, 22, 23, 27] and the referencestherein).

1.1. Statement of the Problem

Many scientists and researchers are trying to enhance mathematical models of real-life casesfor investigating and understanding the behavior of them. Therefore, some phenomena have


been modeled and investigated as fractional inverse problems (see [29–33] and the referencestherein). In this paper, we consider the fractional parabolic inverse problemwith the Dirichletcondition

∂u(t, x)∂t

− a∂2u(t, x)∂x2

−D1/2t u(t, x) + σu(t, x) = p(t)q(x) + f(t, x), 0 < x < π, 0 < t ≤ T,

u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ T,u(0, x) = ϕ(x), 0 ≤ x ≤ π,u(t, x∗) = ρ(t), 0 < x∗ < π,

(1.1)

where u(t, x) and p(t) are unknown functions, a(x) ≥ a > 0, and σ > 0 is a sufficiently largenumber. Here, D1/2

t = D1/20+ is the standard Riemann-Liouville’s derivative of order 1/2.

Theorems on the stability of problem (1.1) are analyzed by assuming that q(x) is asufficiently smooth function, q(0) = q(π) = 0 and q(x∗)/= 0.

2. Main Results

In this section, stability estimates for the solution of (1.1) are investigated. For the mathe-

matical substantiation, we introduce the Banach space◦Cα

[0, π], α ∈ (0, 1), of all continuousfunctions φ(x) defined on [0, π]with φ(0) = φ(π) = 0 satisfying a Holder condition for whichthe following norm is finite

∥∥φ∥∥ ◦Cα

[0,π]=∥∥φ∥∥C[0,π] + sup

0<x<x+h<π

∣∣φ(x + h) − φ(x)∣∣hα

, (2.1)

where C[0, π] is the space of all continuous function φ(x) defined on [0, π] with the norm

∥∥φ∥∥C[0,π] = max

0≤x≤π

∣∣φ(x)∣∣. (2.2)

With the help of a positive operator A, we introduce the fractional spaces Eα, 0 < α < 1,consisting of all v in a Banach space E for which the following norm is finite:

‖v‖Eα = ‖v‖E + supλ>0

λ1−α∥∥A exp{−λA}v∥∥E. (2.3)

Throughout the paper, positive constants will be indicated by Mi (α, β, . . .). Here variablesare used to focus on the fact that the constant depends only on α, β, . . . and the subindex i isused to indicate a different constant.


Theorem 2.1. Let ϕ ∈◦C

2α+2[0, π], F ∈ C([0, T],

◦C

2α[0, π]), and ρ′ ∈ C[0, T]. Then for the

solution of problem (1.1), the following coercive stability estimates

‖ut‖C([0,T],

◦C

2α[0,π])

+ ‖u‖C([0,T],

◦C

2α+2[0,π])

≤ M(x∗, q)∥∥ρ′∥∥C[0,T] +M

(a, δ, σ, α, x∗, q, T

)

×(∥∥ϕ

∥∥ ◦C

2α+2[0,π]

+ ‖F‖C([0,T],

◦C

2α[0,π])

+∥∥ρ∥∥C[0,T]

),

∥∥p∥∥C[0,T] ≤ M

(x∗, q)∥∥ρ′∥∥C[0,T] +M

(a, δ, σ, α, x∗, q, T

)

×[∥∥ϕ∥∥ ◦C

2α+2[0,π]

+ ‖F‖C([0,T],

◦C

2α[0,π])

+∥∥ρ∥∥C[0,T]

]

(2.4)

hold.

Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [8]):

u(t, x) = η(t)q(x) +w(t, x), (2.5)

where

η(t) =∫ t

0p(s)ds. (2.6)

Using the overdetermined condition, we get

η(t) =ρ(t) −w(t, x∗)

q(x∗), (2.7)

p(t) =ρ′(t) −wt(t, x∗)

q(x∗). (2.8)

Using identity (2.8) and the triangle inequality, it follows that

∣∣p(t)∣∣ =∣∣∣∣ρ′(t) −wt(t, x∗)

q(x∗)

∣∣∣∣ ≤M(x∗, q)(∣∣ρ′(t)

∣∣ + |wt(t, x∗)|)

≤ M(x∗, q)(

max0≤t≤T

∣∣ρ′(t)∣∣ +max

0≤t≤Tmax0≤x≤π

|wt(t, x)|)

≤ M(x∗, q)(

max0≤t≤T

∣∣ρ′(t)∣∣ +max

0≤t≤T‖wt(t, ·)‖ ◦

C2α[0,π]

)

(2.9)


for any t, t ∈ [0, T]. Here, w(t, x) is the solution of the following problem:

∂w(t, x)∂t

− a∂2w(t, x)∂x2

− aρ(t) −w(t, x∗)q(x∗)

d2q(x)dx2

−D1/2t w(t, x)

− D1/2t ρ(t) −D1/2

t w(t, x∗)q(x∗)

q(x) + σρ(t) −w(t, x∗)

q(x∗)q(x)

+ σw(t, x) = f(t, x), 0 < x < π, 0 < t ≤ T,

w(t, 0) = w(t, π) = 0, 0 ≤ t ≤ T,w(0, x) = ϕ(x), 0 ≤ x ≤ π.

(2.10)

For simplicity, we assign

F(t, x) =aρ(t)q(x∗)

d2q(x)dx2

− σρ(t)q(x∗)

q(x) +D1/2t ρ(t)q(x∗)

q(x) + f(t, x),

G(t, x) = Q1(q, ρ, x, x∗, t

)w(t, x∗) +Q2

(q, x, x∗)D1/2

t w(t, x∗) +D1/2t w(t, x),

(2.11)

where

Q1(q, ρ, x, x∗, t

)=

1q(x∗)

(−ad

2q(x)dx2

+ σρ(t)

),

Q2(q, x, x∗) = − q(x)

q(x∗).

(2.12)

Note that functions F(t, x), Q1(q, ρ, x, x∗, t) and Q2(q, x, x∗) only contain given functions.Then, we can rewrite problem (2.10) as

∂w(t, x)∂t

− a∂2w(t, x)∂x2

+ σw(t, x) = F(t, x) +G(t, x), 0 < x < π, 0 < t ≤ T,

w(t, 0) = w(t, π) = 0, 0 ≤ t ≤ T,w(0, x) = ϕ(x), 0 ≤ x ≤ π.

(2.13)

So, the end of proof of Theorem 2.1 is based on estimate (2.9) and the following theorem.

Theorem 2.2. For the solution of problem (2.10), the following coercive stability estimate

‖wt‖ ◦C

2α[0,π]

≤ M(a, δ, σ, α, x∗, q, T

)

×(∥∥ϕ

∥∥ ◦C

2α+2[0,π]

+ ‖F‖C([0,T],

◦C

2α[0,π])

+∥∥ρ∥∥C[0,T]

) (2.14)

holds.


Proof. In a Banach space E =◦C[0, π], with the help of the positive operator A defined by

Au = −a(x)∂2u(t, x)∂x2

+ σu, (2.15)

with

D(A) ={u(x) : u, u′, u′′ ∈ C[0, π], u(0) = u(π) = 0

}, (2.16)

where σ is a positive constant, problem (2.10) can be written in the abstract form as an initial-value problem

wt +Aw = F(t) +G(t), 0 < t ≤ T,w(0) = ϕ.

(2.17)

By the Cauchy formula, the solution can be written as

w(t) = e−tAϕ −∫ t

0e−(t−s)A(F(s) +G(s))ds. (2.18)

Applying the formula

D1/2t u(t) =

∫ t

0

u′(ζ)dξ√π(t − ξ)1/2

, (2.19)

we get the following presentation of the solution of abstract problem (2.17):

D1/2w(t) = −∫ t

0

Ae−ξAϕ√π(t − ξ)1/2

dξ −∫ t

0

F(ξ)√π(t − ξ)1/2

dξ

−∫ t

0

G(ξ)√π(t − ξ)1/2

dξ +∫ t

0

∫ ξ

0

Ae−(ξ−s)AF(s)√π(t − ξ)1/2

dsdξ

+∫ t

0

∫ ξ

0

Ae−(ξ−s)AG(s)√π(t − ξ)1/2

dsdξ.

(2.20)


Changing the order of integration, we obtain that

D1/2w(t) = −∫ t

0

Ae−ξAϕ√π(t − ξ)1/2

dξ −∫ t

0

F(ξ)√π(t − ξ)1/2

dξ

+∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξF(s)ds −∫ t

0

G(ξ)√π(t − ξ)1/2

dξ

+∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξG(s)ds =5∑

k=1

Jk,

(2.21)

where

J1(t) = −∫ t

0

Ae−ξAϕ√π(t − p)1/2

dξ,

J2(t) = −∫ t

0

F(ξ)√π(t − ξ)1/2

dξ,

J3(t) =∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξF(s)ds,

J4(t) = −∫ t

0

G(ξ)√π(t − ξ)1/2

dξ,

J5(t) =∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξG(s)ds.

(2.22)

Now, we estimate Jk(t), k = 1, 2, 3, 4, 5 separately. It is known that [13]

∥∥∥Aαe−tA∥∥∥E→E

≤M, 0 ≤ α ≤ 1. (2.23)

Since operators A and exp(−tA) commute,

∥∥∥Ae−tAϕ∥∥∥Eα

≤∥∥∥e−tA

∥∥∥Eα →Eα

∥∥Aϕ∥∥Eα

≤∥∥∥e−tA

∥∥∥E→E

∥∥Aϕ∥∥Eα. (2.24)

Applying the definition of norm of the spaces Eα and (2.23) and (2.24), we get

‖J1(t)‖Eα =∥∥∥∥∥

∫ t

0

Ae−ξAϕ√π(t − p)1/2

dξ

∥∥∥∥∥Eα

≤M1∥∥Aϕ

∥∥Eα (2.25)


for any t, t ∈ [0, T]. Estimation of J2(t) is as follows:

‖J2(t)‖Eα =∥∥∥∥∥

∫ t

0

F(ξ)√π(t − ξ)1/2

dξ

∥∥∥∥∥Eα

≤ ‖F(t)‖C(Eα)∫ t

0

1√π(t − ξ)1/2

dξ ≤M2‖F‖C(Eα).(2.26)

Let us estimate J3(t):

‖J3(t)‖Eα =∥∥∥∥∥

∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξF(s)ds

∥∥∥∥∥Eα

≤∥∥∥∥∥

∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξds

∥∥∥∥∥Eα →Eα

‖F‖C(Eα).

(2.27)

It is proven that (see [28])

∥∥∥∥∥

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξ

∥∥∥∥∥E→E

≤ M√t − s . (2.28)

Using the definition of norm of the spaces Eα, we can obtain that

∥∥∥∥∥

∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξds

∥∥∥∥∥Eα →Eα

=∫ t

0

∥∥∥∥∥

∫ t

s

Ae−(t−s)A√π(t − ξ)1/2

dξ

∥∥∥∥∥E→E

ds

+ supλ>0

∫ t

0

∥∥∥∥∥

∫ t

s

λ1−αAe−λAAe−(t−s)A

√π(t − ξ)1/2

dξ

∥∥∥∥∥E→E

ds.

(2.29)

Using estimates (2.23) and (2.28), we get

‖J3(t)‖Eα ≤∥∥∥∥∥

∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξds

∥∥∥∥∥Eα →Eα

‖F‖C(Eα)

≤ M3‖F‖C(Eα).(2.30)

Expanding G(s), estimation of J4(t) is as follows:

‖J4(t)‖Eα ≤∫ t

0

∥∥∥∥∥Q1(q, ρ, x, x∗, t

)w(ξ, x∗)

√π(t − ξ)1/2

∥∥∥∥∥Eα

dξ

+∫ t

0

∥∥∥∥∥Q2(q, x, x∗)D1/2

t w(ξ, x∗)√π(t − ξ)1/2

∥∥∥∥∥Eα

dξ +∫ t

0

∥∥∥∥∥D1/2t w(ξ, x)

√π(t − ξ)1/2

∥∥∥∥∥Eα

dξ.

(2.31)


It is known that (see [34])

‖w‖Eα ≤M∥∥∥D1/2

t w∥∥∥Eα. (2.32)

Since Q1(q, ρ, x, x∗, t) and Q2(q, x, x∗) are known functions, it is easy to see that

‖J4(t)‖Eα ≤M4(q, ρ, x, x∗, T

) ∫ t

0

1√π(t − ξ)1/2

∥∥∥D1/2t w(ξ)

∥∥∥Eαdξ. (2.33)

Estimation of J5(t) can be given similar to the estimation of J4(t). By (2.23) and (2.32),

‖J5(t)‖Eα ≤∫ t

0

∫ t

s

Ae−(ξ−s)A√π(t − ξ)1/2

dξG(s)ds

≤ M5(q, ρ, x, x∗, T

) ∫ t

0

∥∥∥D1/2t w(s)

∥∥∥Eαds.

(2.34)

Finally combining estimates (2.25), (2.26), (2.30), (2.33), and (2.34), we get

∥∥∥D1/2t w

∥∥∥Eα

≤ M1∥∥Aϕ

∥∥Eα

+ (M2 +M3)‖F‖C(Eα)

+∫ t

0

(M4√

π(t − ξ)1/2+M5

)∥∥∥D1/2t w(s)

∥∥∥Eαds.

(2.35)

Using the Gronwall’s inequality, we can write

∥∥∥D1/2t w

∥∥∥Eα

≤ eM6(M1∥∥Aϕ

∥∥Eα

+M7‖F‖C(Eα)). (2.36)

From the last estimate, we can obtain the estimate forwt(t) by using problem (2.17) and well-posedness of the Cauchy problem in C(Eα) (see [35]). So the following theorem finishes theproof of Theorem 2.2.

Theorem 2.3 (see, [36]). For 0 < α < 1/2, the norms of the spaces Eα(C[0, π], A) and C2α[0, π]are equivalent.


3. Numerical Results

We have not been able to obtain a sharp estimate for the constants figuring in the stabilityinequalities. So we will provide the following results of numerical experiments of thefollowing problem:

∂u(t, x)∂t

=∂2u(t, x)∂x2

− u(t, x) +D1/2t u(t, x) + p(t) sinx + f(t, x),

f(t, x) =(−3t − 1√

πt−1/2 +

2√πt1/2)sinx, x ∈ (0, π), t ∈ (0, 1],

u(0, x) = sinx, x ∈ [0, π],

u(t, 0) = u(t, π) = 0, t ∈ [0, 1],

u(t,π

2

)= 1 − t.

(3.1)

The exact solution of the given problem is u(t, x) = (1 − t) sinx and for the control parameterp(t) is 1 + t.

3.1. The First Order of Accuracy Difference Scheme

For the approximate solution of the problem (3.1), the Rothe difference scheme

ukn − uk−1n

τ=ukn+1 − 2ukn + u

kn−1

h2− ukn +D1/2

τ ukn + pkqn + f(tk, xn),

f(tk, xn) =(−3tk − 1√

πt−1/2k

+2√πt1/2k

)sinxn,

pk = p(tk), qn = sinxn, xn = nh, tk = kτ,

1 ≤ k ≤N, 1 ≤ n ≤M − 1, Mh = π, Nτ = 1,

u0n = sinxn, 0 ≤ n ≤M,

uk0 = ukM = 0, 0 ≤ k ≤N,

uks = ρ(tk), ρ(tk) = 1 − tk, 0 ≤ k ≤N, s =⌊π

2h

⌋,

(3.2)


where �x� denotes greatest integer less than x is constructed. Throughout the paper, let usdenote

ρ(tk) = 1 − tk, qn = sinxn,

tk = {tk = kτ, 0 ≤ k ≤N, Nτ = 1},xn = {xn = nh, 0 ≤ n ≤M − 1, Mh = π},

f(tk, xn) =(−3tk − 1√

πt−1/2k

+2√πt1/2k

)sinxn,

F(tk, xn) =ρ(tk)sin(xs)

(sin(xn+1) − 2 sin(xn) + sin(xn−1)

h2− sin(xn)

)

− 1√π

k∑

m=1

Γ(k −m + (1/2))(k −m)!

τ1/2

sin(xs)sin(xn) + f(tk, xn).

(3.3)

We search the solution of (3.2) in the following form:

ukn = ηkqn +wkn, (3.4)

where

ηk =k∑

i=1

piτ, 1 ≤ k ≤N, η0 = 0. (3.5)

Moreover for the interior grid point uks , we have that

uks = ηkqs +wks = ρ(tk). (3.6)

From (3.4), (3.5), and the condition uks = ρ(tk), it follows that

ηk =ρ(tk) −wk

s

qs, (3.7)

pk =ηk − ηk−1

τ, 1 ≤ k ≤N, (3.8)

ukn =ρ(tk) −wk

s

qsqn +wk

n, 0 ≤ k ≤N, 0 ≤ n ≤M, (3.9)


where wkn, 0 ≤ k ≤N, 0 ≤ n ≤M is the solution of the difference scheme

wkn −wk−1

n

τ=wkn+1 − 2wk

n +wkn−1

h2−wk

n

− wks

sin(xs)


h2− sin(xn)

)

− 1√π

k∑

m=1

Γ(k −m + (1/2))(k −m)!

(wms −wm−1

s

sin(xs)τ1/2sin(xn) −

wmn −wm−1

n

τ1/2

)

+ F(tk, xn), 1 ≤ k ≤N, 1 ≤ n ≤M − 1,

wk0 = wk

M = 0, 0 ≤ k ≤N,

w0n = sin(xn), 0 ≤ n ≤M.

(3.10)

First, applying the first order of accuracy difference scheme (3.10), we obtain (M+1)×(M+1)system of linear equations and we write them in the matrix form

Awk +k−1∑

j=0

Bjwj = Dϕk, 1 ≤ k ≤N, w0 = {sin(xn)}Mn=0, (3.11)

where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 . 0 . 0 0 0x y x 0 . z1 . 0 0 00 x y x . z2 . 0 0 0. . . . . . . . . .0 0 0 0 . zM−1 . x y x0 0 0 0 . 0 . 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

B0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 . 0 . 0 00 a 0 . f1 . 0 00 0 a . f2 . 0 0. . . . . . . .0 0 0 . fs + a . 0 0. . . . . . . .0 0 0 . fM−1 . 0 a0 0 0 . 0 . 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,


Bj =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 . 0 . 0 00 c 0 . d1 . 0 00 0 c . d2 . 0 0. . . . . . . .0 0 0 . ds + c . 0 0. . . . . . . .0 0 0 . dM−1 . 0 c0 0 0 . 0 . 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

(3.12)

for any j = 1, 2, . . . , k − 2, and

Bk−1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 . 0 . 0 00 v 0 . c1 . 0 00 0 v . c2 . 0 0. . . . . . . .0 0 0 . cs + v . 0 0. . . . . . . .0 0 0 . cM−1 . 0 v0 0 0 . 0 . 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

. (3.13)

Here, for any n = 1, 2, . . . ,M − 1,

x = − 1h2, y =

1τ+

2h2

+ 1 +1√τ,

zn =sin(xn+1) − 2 sin(xn) + sin(xn−1)

sin(xs)h2− sin(xn)sin(xs)

− sin(xn)sin(xs)

√τ, in (s + 1)th column,

a =Γ(k − 1/2)√τπ(k − 1)!

, fn = − sin(xn)Γ(k − 1/2)√τπ sin(xs)(k − 1)!

,

c =1√τπ

(Γ(k − j − 1/2

)(k − j − 1

)!

− Γ(k − j + 1/2

)(k − j)!

),

dn =sin(xn)√τπ sin(xs)

(Γ(k − j − 1/2

)(k − j − 1

)!

− Γ(k − j + 1/2

)(k − j)!

),

v = − 1√τ− 1τ, cn =

sin(xn)√τ sin(xs)

,


wr =

⎡⎢⎣

wr0...

wrM

⎤⎥⎦

(M+1)×1

for any r = 0, 1, . . . , k,

ϕk =

⎡⎢⎢⎢⎢⎢⎢⎣

0φk1...

φkM−10

⎤⎥⎥⎥⎥⎥⎥⎦

(M+1)×1

,

φkn =(

ρ(tk)sin(xs)

sin(xn+1) − 2 sin(xn) + sin(xn−1)h2

− ρ(tk)sin(xs)

sin(xn))

− 1√π

k∑

m=1

Γ(k −m + 1/2)(k −m)!

τ1/2

sin(xs)sin(xn) + f(tk, xn),

(3.14)

and D is (M + 1) × (M + 1) identity matrix. Using (3.11), we can obtain that

wk = A−1

⎛

⎝Dϕk −k−1∑

j=0

Bjwj

⎞

⎠, k = 1, 2, . . . ,N, w0 = {sinxn}Mn=0. (3.15)

To solve the resulting difference equations, we apply the method given in (3.15) step by stepfor k = 1, 2, . . . ,N. For the evaluation of wr, r = 2, 3, . . . ,N, wr−1 is needed. It is obtained inthe previous step. Then, the solution pairs (u, p) are obtained by using the last formulas (3.9)and (3.8).

3.2. The Second Order of Accuracy Difference Scheme

For the approximate solution of the problem (3.1), the Crank-Nicholson difference scheme

ukn − uk−1n

τ=ukn+1 − 2ukn + u

kn−1

2h2+uk−1n+1 − 2uk−1n + uk−1n−1

2h2

− ukn + uk−1n

2+pk + pk−1

2qn +D1/2

τ u(tk − τ

2, xn)+ f(tk − τ

2, xn),

pk = p(tk), 1 ≤ k ≤N, 1 ≤ n ≤M − 1

u0n = sin(xn), 0 ≤ n ≤M,

uk0 = ukM = 0, 0 ≤ k ≤N,

uks +uks+1 − uks

h(x∗ − sh) = ρ(tk), 0 ≤ k ≤N, s =

⌊π

2h

⌋

(3.16)

is constructed.


Here,

Γ(k − r + 1

2

)=∫∞

0tk−r+1/2e−t dt. (3.17)

Moreover, applying the second order of approximation formula for

D1/2t u(tk − τ

2

)=

1Γ(1/2)

∫ tk−τ/2

0

(tk − τ

2− s)−1/2

u′(s)ds, (3.18)

it is obtained (see [27])

D1/2τ u =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−d√23

u0 +d√2

3u1 +

dτ

3√2sin(xn), k = 1,

−2d√6

5u0 +

d√6

5u1 +

d√6

5u2 − dτ

√6

10sin(xn), k = 2,

dk−1∑m=2

{[(k −m)b1 + b2]um−2 + [(2m − 2k − 1)b1 − 2b2]um−1

+[(k −m + 1)b1 + b2]um} + d

6√2[−uk−2 − 4uk−1 + 5uk], 3 ≤ k ≤N.

(3.19)

Here and throughout the paper,

D1/2t u = D1/2

τ u(tk − τ

2, xn),

d =2√πτ

, b1 =

√

k −m +12−√

k −m − 12,

b2 = −13

((k −m +

12

)3/2

−(k −m − 1

2

)3/2).

(3.20)

We search the solution of (3.16) in the following form:


where

ηk =k∑

i=1

pi + pi−1

2τ, 1 ≤ k ≤N, η0 = 0. (3.22)


We have that

uks +uks+1 − uks

h(x∗ − sh) = ηk

((1 − x∗ − sh

h

)qs +

x∗ − shh

qs+1

)

+(1 − x∗ − sh

h

)wks +

x∗ − shh

wks+1 = ρ(tk).

(3.23)

Let us denote

y =x∗ − sh

h=x∗

h−⌊x∗

h

⌋h, (3.24)

where 0 ≤ y < 1. Then, one can write

ηk =ρ(tk) −

(1 − y)wk

s − ywks+1(

1 − y)qs + yqs+1. (3.25)

So the values of (p(tk) + p(tk−1))/2, 1 ≤ k ≤N can be obtained by the following formula:

pk + pk−1

2=

(ρ(tk) − ρ(tk−1)

)/τ − (1 − y)((wk

s −wk−1s

)/τ) − y

((wks+1 −wk−1

s+1

)/τ)

(1 − y)qs + yqs+1

.(3.26)

Let wr denote

wr =

⎡⎢⎣

wr0...

wrM

⎤⎥⎦

(M+1)×1

for r = 0, 1, . . . ,N. (3.27)

For k = 1, one can show that w1 is the solution of the difference scheme

w1n −w0

n

τ=w1n+1 − 2w1

n +w1n−1

2h2+w0n+1 − 2w0

n +w0n−1

2h2

− w1n +w

0n

2+(qn+1 − 2qn + qn−1

2h2− qn

2

)


×(ρ(t1) −

(1 − y)w1

s − yw1s+1(

1 − y)qs + yqs+1+ρ(t0) −

(1 − y)w0

s − yw0s+1(

1 − y)qs + yqs+1

)

+d√2

3

(ρ(t1) −

(1 − y)w1

s − yw1s+1(

1 − y)qs + yqs+1q(n) +w1

n

)

− d√2

3

(ρ(t0) −

(1 − y)w0

s − yw0s+1(

1 − y)qs + yqs+1q(n) +w0

n

)

+dτ

3√2q(n) + f

(t1 − τ

2, xn), 1 ≤ n ≤M − 1,

w10 = w1

M = 0,

w0n = sin(xn), 0 ≤ n ≤M.

(3.28)

We have the system of linear equations and we write them in the matrix form

A1w1 + B1w

0 = Dϕ1, (3.29)

where

A1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · 0 · · 0 0 0a y1 a 0 · l1 c1 · 0 0 00 a y1 a · l2 c2 · 0 0 0· · · · · · · · · · ·0 0 0 a · ls + a cs · 0 0 00 0 0 0 · ls+1 + y1 cs+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · lM−1 cM−1 · a y1 a0 0 0 0 · 0 0 · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

B1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 0a v1 a 0 · d1 e1 · 0 0 00 a v1 a · d2 e2 · 0 0 0· · · · · · · · · · ·0 0 0 a · ds + a es · 0 0 00 0 0 0 · ds+1 + v1 es+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · dM−1 eM−1 · a v1 a0 0 0 0 · 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

.

(3.30)


Here, for any n = 1, 2, . . . ,M − 1,

a =(− 12h2

), y1 =

(1τ+

1h2

+12− d

√23

),

v1 =

(− 1τ+

1h2

+12+ d

√23

),

ln =

(qn+1 − 2qn + qn−1

)(1 − y)

2h2((1 − y)qs + yqs+1

) +qn(1 − y)

2((1 − y)qs + yqs+1

) + d√23qn,

cn =

(qn+1 − 2qn + qn−1

)y

2h2((1 − y)qs + yqs+1

) +qny

2((1 − y)qs + yqs+1

) + d√23qn,

dn =

(qn+1 − 2qn + qn−1

)(1 − y)

2h2((1 − y)qs + yqs+1

) +qn(1 − y)

2((1 − y)qs + yqs+1

) − d√23qn,

en =

(qn+1 − 2qn + qn−1

)y

2h2((1 − y)qs + yqs+1

) +qny

2((1 − y)qs + yqs+1

) − d√23qn,

ϕ1 =

⎡⎢⎢⎢⎢⎢⎢⎣

0φ11...

φ1M−10

⎤⎥⎥⎥⎥⎥⎥⎦

(M+1)×1

,

φ1n =(sin(xn+1) − 2 sin(xn) + sin(xn−1)

2h2− sin(xn)

2

)ρ(t1) + ρ(t0)(

1 − y)qs + yqs+1

+d√2qn3

ρ(t1) − ρ(t0)(1 − y)qs + yqs+1

+dτ

3√2qn + f

(t1 − τ

2, xn)

(3.31)

and D is (M + 1) × (M + 1) identity matrix. Using (3.29), we can obtain that

w1 = A−11

(Dϕ1 − B1w

0), w0 = {sinxn}Mn=0. (3.32)

For k = 2, w2 is the solution of the difference scheme

w2n −w1

n

τ=w2n+1 − 2w2

n +w2n−1

2h2− w2

n +w1n

2+w1n+1 − 2w1

n +w1n−1

2h2

+(qn+1 − 2qn + qn−1

2h2− qn

2

) (ρ(t2) −

(1 − y)w2

s − yw2s+1(

1 − y)qs + yqs+1

+ρ(t1) −

(1 − y)w1

s − yw1s+1(

1 − y)qs + yqs+1

)


+d√6

5

(ρ(t2) −

(1 − y)w2

s − yw2s+1(

1 − y)qs + yqs+1q(n) +w2

n

)

+d√6

5

(ρ(t1) −

(1 − y)w1

s − yw1s+1(

1 − y)qs + yqs+1q(n) +w1

n

)

− 2d√6

5

(ρ(t0) −

(1 − y)w0

s − yw0s+1(

1 − y)qs + yqs+1q(n) +w0

n

)

− dτ√6

10q(n) + f

(t2 − τ

2, xn), 1 ≤ n ≤M − 1,

w20 = w2

M = 0,

w0n = sin(xn), 0 ≤ n ≤M.

(3.33)

The system of linear equations given above can be written in the matrix form

A2w2 + B2w

1 + C2w0 = Dϕ2, (3.34)

where

A2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · 0 · · 0 0 0a y2 a 0 · g1 h1 · 0 0 00 a y2 a · g2 h2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · gs + a hs · 0 0 00 0 0 0 · gs+1 + y2 hs+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · gM−1 hM−1 · a y2 a0 0 0 0 · 0 0 · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

B2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 0a v2 a 0 · g1 h1 · 0 0 00 a v2 a · g2 h2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · gs + a hs · 0 0 00 0 0 0 · gs+1 + v2 hs+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · gM−1 hM−1 · a v2 a0 0 0 0 · 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,


C2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 00 z 0 0 · i1 j1 · 0 0 00 0 z 0 · i2 j2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · is + z js · 0 0 00 0 0 0 · is+1 js+1 + z · 0 0 0· · · · · · · · · · ·0 0 0 0 · iM−1 jM−1 · 0 z 00 0 0 0 · 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

.

(3.35)

Here, for any n = 1, 2, . . . ,M − 1,

a = − 12h2

, y2 =1τ+

1h2

+12− d

√6

5,

v2 = − 1τ+

1h2

+12− d

√6

5,

gn =

(qn+1 − 2qn + qn−1

)(1 − y)

2h2((1 − y)qs + yqs+1

) +qn(1 − y)

2((1 − y)qs + yqs+1

)

+d√6qn(1 − y)

5((1 − y)qs + yqs+1

) , in (s + 1)th column,

hn =

(qn+1 − 2qn + qn−1

)y

2h2((1 − y)qs + yqs+1

) +qny

2((1 − y)qs + yqs+1

)

+d√6qny

5((1 − y)qs + yqs+1

) , in (s + 2)th column,

in = − 2d√6qn(1 − y)

5((1 − y)qs + yqs+1

) , jn = − 2√6qny

5((1 − y)qs + yqs+1

) , z =2d

√6

5,

ϕ2 =

⎡⎢⎢⎢⎢⎢⎢⎣

0φ21...

φ2M−10

⎤⎥⎥⎥⎥⎥⎥⎦

(M+1)×1

,

φ2n =


2h2− sin(xn)

2+d√6qn5

)

× ρ(t2) + ρ(t1)(1 − y)qs + yqs+1

− 2d√6qn5

ρ(t0)(1 − y)qs + yqs+1

− dτ√6

10qn + f

(t2 − τ

2, xn).

(3.36)


Using (3.34), we can obtain that

w2 = A−12

(Dϕ2 − B2w

1 − C2w0), w0 = {sinxn}Mn=0. (3.37)

For 3 ≤ k ≤N, we can obtain the following difference scheme:

wkn −wk−1

n

τ=wkn+1 − 2wk

n +wkn−1

2h2+wk−1n+1 − 2wk−1

n +wk−1n−1

2h2

− wkn +w

k−1n

2+(qn+1 − 2qn + qn−1

2h2− qn

2

)

×(ρ(tk) −

(1 − y)wk

s − ywks+1(

1 − y)qs + yqs+1

)+(qn+1 − 2qn + qn−1

2h2− qn

2

)

×(ρ(tk−1) −

(1 − y)wk−1

s − ywk−1s+1(

1 − y)qs + yqs+1

)

+ dk−1∑

m=2

{((k −m)b1 + b2)

×(ρ(tm−2) −

(1 − y)wm−2

s − ywm−2s+1(

1 − y)qs + yqs+1q(n)

)+ ((2m − 2k − 1)b1 − 2b2)

×(ρ(tm−1) −

(1 − y)wm−1

s − ywm−1s+1(


)

+ ((2m − 2k − 1)b1 − 2b2)wm−1n

×(ρ(tm) −

(1 − y)wm

s − ywms+1(


)+ ((k −m)b1 + b2)wm−2

n

+((k −m − 1)b1 + b2)wmn + ((2m − 2k − 1)b1 − 2b2)wm−1

n

}

− d

6√2

(ρ(tk−2) −

(1 − y)wk−2

s − ywk−2s+1(

1 − y)qs + yqs+1q(n) +wk−2

n

)

− 4d

6√2

(ρ(tk−1) −

(1 − y)wk−1

s − ywk−1s+1(

1 − y)qs + yqs+1q(n) +wk−1

n

)

+5d

6√2

(ρ(tk) −

(1 − y)wk

s − ywks+1(

1 − y)qs + yqs+1q(n) +wk

n

)+ f(tk − τ

2, xn),

1 ≤ n ≤M − 1,

wk0 = wk

M = 0, 3 ≤ k ≤N,

w0n = sin(xn), 0 ≤ n ≤M.

(3.38)


This system can be written in matrix form as

A3wk + B3w

k−1 + C3wk−2 +

k−3∑

j=0

Ejwj = Dϕk, (3.39)

where

A3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · 0 · · 0 0 0a y3 a 0 · r1 m1 · 0 0 00 a y3 a · r2 m2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · rs + a ms · 0 0 00 0 0 0 · rs+1 + y3 ms+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · rM−1 sM−1 · a y3 a0 0 0 0 · 0 0 · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

B3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 0a X a 0 · Y1 Z1 · 0 0 00 a X a · Y2 Z2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · Ys + a Zs · 0 0 00 0 0 0 · Ys+1 +X Zs+1 + a · 0 0 0· · · · · · · · · · ·0 0 0 0 · YM−1 ZM−1 · a X a0 0 0 0 · 0 · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

C3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 00 S 0 0 · R1 T1 · 0 0 00 0 S 0 · R2 T2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · Rs + S Ts · 0 0 00 0 0 0 · Rs+1 Ts+1 + S · 0 0 0· · · · · · · · · · ·0 0 0 0 · RM−1 TM−1 · 0 S 00 0 0 0 · 0 · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

E0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 00 G 0 0 · H1 I1 · 0 0 00 0 G 0 · H2 I2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · Hs +G Is · 0 0 00 0 0 0 · Hs+1 Is+1 +G · 0 0 0· · · · · · · · · · ·0 0 0 0 · HM−1 IM−1 · 0 G 00 0 0 0 · 0 · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,


E1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 00 G1 0 0 · J1 K1 · 0 0 00 0 G1 0 · J2 K2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · Js +G1 Ks · 0 0 00 0 0 0 · Js+1 Ks+1 +G1 · 0 0 0· · · · · · · · · · ·0 0 0 0 · JM−1 KM−1 · 0 G1 00 0 0 0 · 0 · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

Ej =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 · · 0 0 00 G2 0 0 · P1 L1 · 0 0 00 0 G2 0 · P2 L2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · Ps +G2 Ls · 0 0 00 0 0 0 · Ps+1 Ls+1 +G2 · 0 0 0· · · · · · · · · · ·0 0 0 0 · PM−1 LM−1 · 0 G2 00 0 0 0 · 0 · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(M+1)×(M+1)

,

(3.40)

for j = 2, 3, . . . k − 3.Here, for any n = 1, 2, . . . ,M − 1,

a = − 12h2

, d =2√πτ

, y3 =1τ+

1h2

+12− 5d

6√2,

v3 = − 1τ+

1h2

+12+

4d

6√2,

αn =dq(n)

(1 − y)

6√2((1 − y)qs + yqs+1

) , βn =dq(n)y

6√2((1 − y)qs + yqs+1

) ,

rn =

(qn+1 − 2qn + qn−1

)(1 − y)

2h2((1 − y)qs + yqs+1

) +qn(1 − y)

2((1 − y)qs + yqs+1

) + 5dq(n)

(1 − y)

6√2((1 − y)qs + yqs+1

) ,

sn =

(qn+1 − 2qn + qn−1

)y

2h2((1 − y)qs + yqs+1

) +qny

2((1 − y)qs + yqs+1

) + 5dq(n)y

6√2((1 − y)qs + yqs+1

) ,

X = v3 − pp × d, Yn−1 = γn−1 + 6√2pp × αn−1,

Zn−1 = δn−1 + 6√2pp × βn−1, S = d

(1

6√2− pp − nn

),

Rn−1 = −αn−1 + 6√2(nn + pp

)αn−1,

Tn−1 = −βn−1 + 6√2(nn + pp

)βn−1,


γn =

(qn+1 − 2qn + qn−1

)(1 − y)

2h2((1 − y)qs + yqs+1

) +qn(1 − y)

2((1 − y)qs + yqs+1

) − 4dq(n)

(1 − y)

6√2((1 − y)qs + yqs+1

) ,

δn =

(qn+1 − 2qn + qn−1

)y

2h2((1 − y)qs + yqs+1

) +qny

2((1 − y)qs + yqs+1

) − 4dq(n)y

6√2((1 − y)qs + yqs+1

) ,

mm = (k −m)b1 + b2, nn = (2m − 2k − 1)b1 − 2b2,

pp = (k −m + 1)b1 − 2b2,

nn1 = (2(m + 1) − 2k − 1)b1 − 2b2,

pp1 = (k − (m + 2) + 1)b1 + b2,

G = −d ×mm, G1 = −d × (mm + nn1),

G2 = −d × (mm + nn1 + pp1),

Hn−1 = 6√2 × α1 ×mm, In−1 = 6

√2 × β1 ×mm,

Jn−1 = 6√2 × αn−1 × (mm + nn1),

Kn−1 = 6√2 × βn−1 × (mm + nn1),

Pn−1 = 6√2 × αn−1 ×

(mm + nn1 + pp1

),

Ln−1 = 6√2 × βn−1 ×

(mm + nn1 + pp1

),

ϕk =

⎡⎢⎢⎢⎢⎢⎢⎣

0φk1...

φkM−10

⎤⎥⎥⎥⎥⎥⎥⎦

(M+1)×1

,

φkn =(sin(xn+1) − 2 sin(xn) + sin(xn−1)

2h2− sin(xn)

2

)ρ(tk) + ρ(tk−1)(1 − y)qs + yqs+1

+5dqnρ(tk)

6√2((1 − y)qs + yqs+1

) − 4dqnρ(tk−1)

6√2((1 − y)qs + yqs+1

)

− dqnρ(tk−2)

6√2((1 − y)qs + yqs+1

) + f(tk − τ

2, xn)+

dqn(1 − y)qs + yqs+1

×k−1∑

m=2

{((k −m)b1 + b2)ρ(tm−2)

+((2m − 2k − 1)b1 − 2b2)ρ(tm−1) + ((k −m + 1)b1 + b2)ρ(tm)}.

(3.41)


Table 1: Comparison of exact solution and approximate solutions.

Method N =M= 15 N =M= 45 N =M= 751st order of accuracy 0.1190 0.0126 0.00452nd order of accuracy 0.0055 6.0917× 10−4 2.1932× 10−4

Finally, from (3.39), it follows that

wk = (A3)−1⎛

⎝Dϕk − B3wk−1 − C3w

k−2 −k−3∑

j=0

Ejwj

⎞

⎠, 3 ≤ k ≤N. (3.42)

Applying the last formula step by step, we can reachwk. Then, using (3.21), (3.25), and (3.26),we reach the approximate solutions of u(t, x) and (p(tk) + p(tk−1))/2.

3.3. Error Analysis

In this part, the results of the numerical analysis is given. The numerical solutions arerecorded for different values of N and M and ukn represents the approximate solution ofu(t, x) at grid points (tk, xn). Table 1 gives the error analysis between the exact solution andthe solutions derived by difference schemes. Table 1 is constructed forN = M = 15, 45, and75, respectively. For their comparison, the errors are computed by

E = max1≤k≤N1≤n≤M

∣∣∣u(tk, xn) − ukn∣∣∣. (3.43)

Thus, the second order of accuracy difference scheme is more accurate comparing withthe first order of accuracy difference scheme.

Acknowledgment

The authors are grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for hiscomments and suggestions to improve the quality of the paper.

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Research ArticleExact Asymptotic Expansion of Singular Solutionsfor the (2 + 1)-D Protter Problem

Lubomir Dechevski,1 Nedyu Popivanov,2 and Todor Popov2

1 Faculty of Technology, Narvik University College, Lodve Langes Gate 2, 8505 Narvik, Norway2 Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria

Correspondence should be addressed to Nedyu Popivanov, [email protected]

Received 29 March 2012; Accepted 24 June 2012


Copyright q 2012 Lubomir Dechevski et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study three-dimensional boundary value problems for the nonhomogeneous wave equation,which are analogues of the Darboux problems in R

2. In contrast to the planar Darboux problem thethree-dimensional version is not well posed, since its homogeneous adjoint problem has an infinitenumber of classical solutions. On the other hand, it is known that for smooth right-hand sidefunctions there is a uniquely determined generalized solution that may have a strong power-typesingularity at one boundary point. This singularity is isolated at the vertex of the characteristic lightcone and does not propagate along the cone. The present paper describes asymptotic expansionof the generalized solutions in negative powers of the distance to this singular point. We derivenecessary and sufficient conditions for existence of solutions with a fixed order of singularity andgive a priori estimates for the singular solutions.

1. Introduction

In the present paper some boundary value problems (BVPs) formulated by M. H. Protter forthe wave equation with two space and one time variables are studied as a multidimensionalanalogue of the classical Darboux problem in the plane. While the Darboux BVP in R

2 iswell posed the Protter problem is not and its cokernel is infinite dimensional. Thereforethe problem is not Fredholm and the orthogonality of the right-hand side function f to thecokernel is one necessary condition for existence of classical solution. Alternatively, to avoidinfinite number of conditions the notion of generalized solution is introduced that allowsthe solution to have singularity on a characteristic part of the boundary. It is known thatfor smooth right-hand side functions there is unique generalized solution and it may havea strong power-type singularity that is isolated at one boundary point. In the present paperwe prove asymptotic expansion formula for the generalized solutions in negative powers ofthe distance to the singular point in the case when f is trigonometric polynomial. We leave


for the next section the precise formulation of the paper’s main results and the comparisonswith recent publications concerning Protter problems, including a semi-Fredholm solvabilityresult in the general case of smooth f but for somewhat easier (3 + 1)-D wave equationproblem. First we give here a short historical survey.

Protter arrived at the multidimensional problems for hyperbolic equations whileexamining BVPs for mixed type equations, starting with planar problems with strongconnection to transonic flow phenomena. In the plane, the problems of Tricomi, Frankl, andGuderley-Morawetz are the classical boundary-value problems that appear in hodographplane for 2D transonic potential flows (see, e.g., the survey of Morawetz [1]). The first two ofthese problems are relevant to flows in nozzles and jets, and the third problem occurs as anapproximation to a respective “exact” boundary-value problem in the study of flows aroundairfoils. For the Gellerstedt equation of mixed type, Protter [2] proposes a 3D analogue to thetwo-dimensional Guderley-Morawetz problem. At the same time, he formulates boundaryvalue problems in the hyperbolic part of the domain, which is bounded by two characteristicsand one noncharacteristic surfaces of the equation. The planar Guderley-Morawetz mixed-type problem is well studied. Existence of weak solutions and uniqueness of strong solutionsin weighted Sobolev spaces were first established by Morawetz by reducing the problem toa first order system which then gives rise to solutions to the scalar equation in the presenceof sufficient regularity. The availability of such sufficient regularity follows from the work ofLax and Phillips [3] who also established that the weak solutions of Morawetz are strong.On the other hand, for the 3D Protter mixed-type problems a general understanding of thesituation is not at hand—even the question of well posedness is surprisingly subtle andnot completely resolved. One has uniqueness results for quasiregular solutions, a class ofsolutions introduced by Protter, but there are real obstructions to existence in this class. Toinvestigate the situation, we study a simpler problem—the Protter problems in the hyperbolicpart Ω of the domain for the mixed-type problem. For the wave equation

�u ≡ ux1x1 + ux2x2 − utt = f(x, t), (1.1)

this is the set

Ω :={(x1, x2, t) : 0 < t <

12, t <

√x21 + x

22 < 1 − t

}. (1.2)

It is bounded, see Figure 1, by two characteristic cones of (1.1)

S1 ={(x1, x2, t) : 0 < t <

12,√x21 + x

22 = 1 − t

},

S2 ={(x1, x2, t) : 0 < t <

12,√x21 + x

22 = t

},

(1.3)

and the disk S0 = {(x1, x2, t) : t = 0, x21 + x

22 < 1}, centered at the origin O(0, 0, 0).

One could think of the Protter problems in Ω as three-dimensional variant of theplanar Darboux problem. The classic Darboux problem involves a hyperbolic equation ina characteristic triangle bounded by two characteristic and one noncharacteristic segments.The data are prescribed on the noncharacteristic part of the boundary and one of


S2

t

S1

x2

x1O

S0

Figure 1: The domain Ω.

the characteristics. Actually, the set Ω could be produced via rotation around the t-axis inR

3 of the flat triangle Ω2 := {(x1, t) : 0 < t < 1/2; t < x1 < 1 − t} ⊂ R2—a characteristic

triangle for the corresponding string equation

ux1x1 − utt = g(x1, t). (1.4)

As mentioned before, the classical Darboux problem for (1.4) is to find solution in Ω2 withdata prescribed on {t = 0} and {t = 1 − x1}, for example. In conformity with this planar BVP,Protter [2, 4] formulated and studied the following problems.

Problems (P1) and (P2)

Find a solution of the wave equation (1.1) in whichΩ satisfies one of the following boundaryconditions:

u|S0 = 0, u|S1 = 0, (P1)

or

ut|S0 = 0, u|S1 = 0. (P2)

Nowadays, it is known that the Protter Problems (P1) and (P2) are not well posed,in contrast to the planar Darboux problem. In fact, in 1957 Tong [5] proved the existenceof infinite number nontrivial classical solutions to the corresponding homogeneous adjointproblem (P1∗). The adjoint BVPs to Problems (P1) and (P2) were also introduced by Protter.

Problems (P1∗) and (P2∗)

Find a solution of the wave equation (1.1) in Ω which satisfies the boundary conditions:

u|S0 = 0, u|S2 = 0 (adjoint to Problem (P1)), (P1∗)


or

ut|S0 = 0, u|S2 = 0 (adjoint to Problem (P2)). (P2∗)

Since [5], for each of the homogeneous Problems (P1∗) and (P2∗) (i.e., f ≡ 0 in (1.1)),an infinite number of classical solutions has been found (see Popivanov, Schneider [6], Khe[7]). According to this fact, a necessary condition for classical solvability of Problem (P1) or(P2) is the orthogonality in L2(Ω) of the right-hand side function f(x, t) to all the solutions ofthe corresponding homogenous adjoint problem (P1∗) or (P2∗). Although Garabedian proved[8] the uniqueness of a classical solution of Problem (P1) (for its analogue in R

4), generally,Problems (P1) and (P2) are not classically solvable. Instead, Popivanov and Schneider [6]introduced the notion of generalized solution. It allows the solution to have singularity onthe inner cone S2 and by this the authors avoid the infinite number of necessary conditionsin the frame of the classical solvability. In [6] some existence and uniqueness results for thegeneralized solutions are proved and some singular solutions of Protter Problems (P1) and(P2) are constructed.

In the present paper we study the properties of the generalized solution for ProtterProblem (P2) in R

3. From the results in [6] it follows that for n ∈ N there exists a smoothright-hand side function f ∈ Cn(Ω), such that the corresponding unique generalized solutionof Problem (P2) has a strong power-type singularity at the originO and behaves like r−n(P,O)there. This feature deviates from the conventional belief that such BVPs are classicallysolvable for very smooth right-hand side functions f . Another interesting aspect is that thesingularity is isolated only at a single point the vertex O of the characteristic light cone,and does not propagate along the bicharacteristics which makes this case different from thetraditional case of propagation of singularity (see, e.g., Hormander [9], Chapter 24.5).

The Protter problems have been studied by different authors using various types oftechniques likeWiener-Hopf method, special Legendre functions, a priori estimates, nonlocalregularization, and others. For recent known results concerning Protter’s problems see thepaper [6] and references therein. For further publications in this area see [7, 10–16]. Onthe other hand, Bazarbekov gives in Ω another analogue of the classical Darboux problem(see [17]) and analogously in R

4 (see [18]) in the corresponding four-dimensional domainΩ. Some different statements of Darboux type problems in R

3 or connected with themProtter problems for mixed type equations (also studied in [2]) can be found in [19–25].Some results concerning the nonexistence principle for nontrivial solution of semilinearmixed type equations in multidimensional case, can be found in [26]. For recent existenceresults concerning closed boundary-value problems formixed type equations see for example[27], and also [28] that studies an elliptic-hyperbolic equation which arises in modelsof electromagnetic wave propagation through zero-temperature plasma. The existence ofbounded or unbounded solutions for the wave equation in R

3 and R4, as well as for the

Euler-Poisson-Darboux equation has been studied in [7, 13–16, 29].Further, we aim to find some exact a priori estimates for the singular solutions of

Problem (P2) and to outline the exact structure and order of singularity. For some otherProtter problems necessary and sufficient conditions for existence of solutions with fixedorder of singularity were found (see [15] in R

3 and [16] in R4) and an asymptotic formula for

the solution of Problem (P1) in R4 was obtained in [30].

Considering Protter Problems, Popivanov and Schneider [6] proved the existence ofsingular solutions for both wave and degenerate hyperbolic equation. First a priori estimatesfor singular solutions of Protter Problems, involving the wave equation in R

3, were obtained


in [6]. In [10] Aldashev mentioned the results of [6] and, for the case of the wave equation inRm+1, he notes the existence of solutions in the domain Ωε (Ωε → Ω and S2,ε approximates

S2 if ε → 0), which blows up on the cone S2,ε like ε−(n+m−2), when ε → 0. It is obvious that form = 2 this results can be compared to the estimates in Corollary 2.4 here. Finally, we point outthat in the case of an equation, which involves the wave operator and nonzero lower terms,Karatoprakliev [24] obtained a priori estimates, but only for the sufficiently smooth solutionsof Protter Problem.

Regarding the ill-posedness of the Protter Problems, there have appeared somepossible regularization methods in the case of the wave equation, involving either lowerorder terms ([11, 31]), or some other type perturbations, like integrodifferential term, ornonlocal one ([12]).

In Section 2 the result of the existence of infinite number of classical solutions tothe homogeneous Problem (P2∗) (Lemma 2.1) and the definition of generalized solution ofProblem (P2) are given. The main results of the paper, concerning the asymptotic expansionof the unique generalized solution u(x, t) of Problem (P2) (Theorem 2.3) are formulated anddiscussed. The expansion of u(P) is given in negative powers of the distance r(P,O) tothe point O of singularity. An estimate for the remainder term and the exact behavior ofthe singularity under the orthogonality conditions imposed on the right-hand side functionof the wave equation is found. Necessary and sufficient conditions for the existence ofonly bounded solutions are given in Corollary 2.4. In Section 3, the auxiliary 2D boundaryvalue Problems (P2.1) and (P2.2), which correspond to the (2 + 1)-D Problem (P2), areconsidered. Actually, these 2D problems are transferred to an integral Volterra equation,which is invertible. Using the special Legendre functions Pν, some exact formulas for thesolution of the Problem (P2.2) are derived in Lemma 3.4. Some figures showing the effectsappearing near the singularity point are also presented. Section 4 contains the most technicalpart of the paper. In this section the results concerning the asymptotic expansions of thegeneralized solution of the 2D Problem (P2.1) are proved and the proof of the main Theorem2.3 is given.

2. Main Results on (2 + 1)-D Protter’s Problem (P2)

Define the functions

Enk(x, t) =k∑

i=0

Bki

(x21 + x

22 − t2

)n−1/2−k−i(x21 + x

22

)n−i , n, k ∈ N ∪ {0}, (2.1)

where the coefficients are

Bki := (−1)i (k − i + 1)i(n + 1/2 − k − i)ii!(n − i)i

, Bk0 = 1, (2.2)

with (a)i := a(a + 1) · · · (a + i − 1), (a)0 := 1. Then for the functions

Wnk,1(x, t) := Enk(x, t)Re

{(x1 + ix2)n

},

Wnk,2(x, t) := E

nk(x, t) Im

{(x1 + ix2)n

},

(2.3)

we have the following lemma.


Lemma 2.1 (see [29]). Let n ∈ N, n ≥ 4. For k = 0, . . .,[(n − 3)/2] and i = 1, 2 the functionsWn

k,i(x, t) are classical C2(Ω) ∩ C∞(Ω) solutions to the homogeneous Problem (P2∗).

A necessary condition for the existence of classical solution for Problem (P2) is theorthogonality of the right-hand side function f to all functionsWn

k,i(x, t), which are solutions

of the homogeneous adjoint Problem (P2∗). To avoid these infinite number necessaryconditions in the framework of classical solvability, one needs to introduce some generalizedsolutions of Problems (P2)with possible singularities on the characteristic cone S2, or only atits vertex O. Popivanov and Schneider in [6] give the following definition.

Definition 2.2. A function u = u(x1, x2, t) is called a generalized solution of the Problem (P2)in Ω if:

(1) u ∈ C1(Ω \O), ut|S0\O = 0, u|S1 = 0,

(2) the identity

∫

Ω

(utwt − ux1wx1 − ux2wx2 − fw

)dx1dx2dt = 0 (2.4)

holds for all w ∈ C1(Ω), wt = 0 on S0, and w = 0 in a neighborhood of S2.

The uniqueness of the generalized solution of Problem (P2) and existence results forf ∈ C1(Ω) can be found in [6].

Further, we fix the right-hand side function f as a trigonometric polynomial of order lwith respect to the polar angle:

f(x1, x2, t) = Re

{l∑

n=2

fn(|x|, t)(x1 + ix2)n}, (2.5)

with some complex-valued function-coefficients fn(|x|, t). For n = 0, . . . , l; k = 0, . . . , [n/2]and i = 1, 2, denote by βn

k,ithe constants

βnk,i :=∫

ΩWn

k,i(x, t)f(x, t)dxdt. (2.6)

Note that actually β0k,i = 0 and β1k,i = 0 in cases of n = 0 and n = 1, due to the specialform of the functions Wn

k,i and the fact that in the representation (2.5) of the function f thesum starts from n = 2.


The main result is as follows.

Theorem 2.3. Suppose that the function f(x, t) ∈ C1(Ω) is a trigonometric polynomial (2.5). Thenthere exist functions Fn(x, t), Fn

k,i(x, t), F(x, t) ∈ C2(Ω \O) with the following properties:

(i) the unique generalized solution u(x, t) of Problem (P2) exists, belongs to C2(Ω \ O) andhas the asymptotic expansion at the origin O:

u(x, t) =l∑

m=0

(|x|2 + t2

)−m/2Fm(x, t) +

(|x|2 + t2

)1/4F(x, t) ln

(|x|2 + t2

), (2.7)

(ii) for the coefficient functions Fm(x, t) the representation

Fm(x, t) =[(l−m)/2]∑

k=0

2∑

i=1

βm+2kk,i Fm+2k

k,i (x, t), m = 0, . . . , l, (2.8)

holds, where the functions Fnk,i(x, t) are bounded and independent of f ,

(iii) if in the expression (2.8) for Fm(x, t) at least one of the constants βm+2kk,i is different from zero

(i.e., the corresponding orthogonality condition is not fulfilled), then there exists a direction(α1, α2, 1) with (α1, α2, 1)t ∈ S2 for 0 < t < 1/2, such that limt→+0F

m(α1t, α2t, t) = cm =const /= 0,

(iv) if in the expression (2.8) for F0(x, t) at least one of the constants β2kk,i is different fromzero (i.e., the corresponding orthogonality condition is not fulfilled), then the generalizedsolution is not continuous at O,

(v) for the function F(x, t) the estimate

|F(x, t)| ≤ C{maxΩ

∣∣f(x, t)∣∣ +max

Ω

∣∣ft(x, t)∣∣}, (x, t) ∈ Ω, (2.9)

holds with a constant C independent of f .

As a consequence of Theorem 2.3 one gets the following results that highlight the twoextreme cases of the assertion. The first part gives rough estimate of the expansion (2.7) anddescribes the “worst” possible singularity. The second part shows that one could control thesolution by making some of the defined by (2.6) constants βnk,i in (2.8) to be zero, that is, bytaking f to be orthogonal in L2(Ω) to the corresponding functionsWn

k,i defined in (2.3).

Corollary 2.4. Suppose that f ∈ C1(Ω) has the form (2.5).

(i) Without any orthogonality conditions imposed, the unique generalized solution u ofProblem (P2) satisfies the a priori estimate

|u(x, t)| ≤ C(|x|2 + t2

)−l/2∥∥f∥∥C1(Ω), (x, t) ∈ Ω. (2.10)


(ii) Let the orthogonality conditions,

βnk,i ≡∫

ΩWn

k,i(x, t)f(x, t)dx dt = 0, (2.11)

be fulfilled for all n = 2, . . . , l; k = 0, . . . , [(n − 1)/2] and i = 1, 2. Then the generalizedsolution u(x, t) belongs to C2(Ω \O), is bounded and the a priori estimate

supΩ

|u| ≤ C{∥∥f

∥∥C(Ω) +

∥∥ft∥∥C(Ω)

}(2.12)

holds.

(iii) In addition to (ii), if the conditions (2.11) are fulfilled for k = [n/2] also, then u ∈ C(Ω) isa classical solution and u(O) = 0.

Let us point out that in the case (ii), the generalized solution u is bounded if andonly if the conditions (2.11) are fulfilled for k ≤ [(n − 1)/2] due to Theorem 2.3(iii). Inaddition, if all the conditions (2.11) are fulfilled for k ≤ [(n − 1)/2], but for some k = [n/2]the corresponding orthogonality condition is not satisfied, then u is not continuous at O,according to Theorem 2.3 (iv). Such a solution is illustrated in Figure 4.

Notice that some of the functionsWnk,i(x, t) involved in the orthogonality conditions in

Corollary 2.4(ii) and (iii) are not classical solutions of the homogenous adjoint Problem (P2∗)in view of Lemma 2.1, although they satisfy the homogenous wave equation in Ω. In fact,for some k, Wn

k,i or their derivatives may be discontinuous at S2. For example when n is anodd number and k = (n − 1)/2, the functionsWn

k,iare not continuous at the origin O. On the

other hand, when n is even and k = n/2,Wnn/2,i are singular on the cone S2 and do not satisfy

the homogeneous adjoint boundary condition there. However, this singularity is integrablein the domain Ω.

To explain the results in Theorem 2.3 and Corollary 2.4 we construct Table 1. Itillustrates the connection between the singularity of the generalized solution and thefunctionsWn

k,i.

Both functionsWnk,i, i = 1, 2 are located in column number n and row number (n − 2k)

in Table 1. Thus, Wn0,i form the rightmost diagonal, the next one is empty—we put in these

cells “diamonds” �,Wn1,i constitute the third one, and so on. The row number designates the

order of singularity of the generalized solution.Corollary 2.4 shows that the generalized solution u(x, t) is bounded, when the right-

hand side function f is orthogonal to the functions in Table 1, except the ones in row number0. If f is orthogonal to all the functions in the Table 1 (including the row 0), then u iscontinuous in Ω. When the right-hand side f satisfies orthogonal conditions (2.11) for allthe functions from the rows in Table 1 with row-number larger then m, 0 < m < l, but thereis a function Wp

q,i with p − 2q = m from m th row which is not orthogonal to f (i.e., βpq,i /= 0),then the solution behaves like r−m at the origin, according to the expansion (2.7). If there areno orthogonality conditions, then the worst case with singularity r−l appears.

Figures 2–5 are created using MATLAB and represent some numerical computationsfor singular solutions of Problem (P2) (actually the behaviour in (r, t)-domain D1, notincluding the terms sinnϕ and cosnϕ). They illustrate different cases according to themain results for the existence of a singularity at the origin O depending on orthogonality


Table 1: The order of singularity of the solution and the functionsWnk,i.

l l − 1 l − 2 l − 3 · · · p · · · 4 3 20 · · · · · · · · · · · · · · · · · · · · · W4

2,i � W21,i

1 · · · · · · · · · · · · · · · · · · · · · � W31,i �

2 · · · · · · · · · · · · · · · · · · · · · W41,i � W2

0,i

3 · · · · · · · · · · · · · · · · · · · · · � W30,i

4 · · · · · · · · · · · · · · · · · · · · · W40,i

... · · · · · · · · · · · · · · · · · · · · ·p − 2q · · · · · · · · · · · · · · · W

p

q,i · · ·· · ·

... · · · · · · · · · · · · · · ·l − 3 � Wl−1

1,i � Wl−30,i

l − 2 Wl1,i � Wl−2

0,i

l − 1 � Wl−10,i

l Wl0,i

−20

−15

−10

−5

0

5

00.1

0.20.3

0.40.5

0 0.2 0.4 0.6 0.8 1 −10−8−6−4−20

2

4

6

8

10

Figure 2: No orthogonality conditions.

conditions. Figure 2 is related to Corollary 2.4(i)—it gives the graph of the solution forthe worst case without any orthogonality conditions fulfilled and the solution is going to−∞ at the singular point O. In Figure 3, only one of orthogonality conditions (2.11) fork ≤ [(n − 1)/2] is not fulfilled and the solution tends to ±∞. Figures 4 and 5 are connectedto Corollary 2.4(ii) and (iii): Figure 4 presents the case when all the orthogonality conditions(2.11) for k ≤ [(n − 1)/2] are satisfied and the solution is bounded but not continuous at(0, 0), while Figure 5 concerns the last part (iii) from Corollary 2.4, when conditions (2.11)are additionally fulfilled for k = [n/2] and the solution is continuous.

Remark 2.5. Wemention some differences between the results given here for the Problem (P2)and some other results in R

3, but for the Problem (P1), like that from [15].

(i) In [15], assuming the right-hand side function f is smooth enough (i.e., f ∈ Cl)only the behavior of the singularities was studied using some weighted norms


−10

−5

0

5

−8−6−4−20246810

00.10.20.30.40.5

0 0.2 0.4 0.6 0.8 1

Figure 3: One orthogonality condition is not fulfilled.

−10

−5

0

5

10

15

0

1−4−2024681012141618

00.10.20.30.40.5 0.5

Figure 4: Orthogonality conditions fulfilled for k ≤ [(n − 1)/2].

0

1

2

3

4

5

6

7

012345678

00.1

0.20.3

0.40.5

00.2

0.40.6

0.81

−1

Figure 5: All orthogonality conditions fulfilled for k ≤ [n/2].


(analogous to the weighted Sobolev norms in corner domains). In the present paperwe need only f ∈ C1 and find in addition the explicit asymptotic expansion of thegeneralized solution. The bounded but not continuous at the origin solutions arealso studied here.

(ii) Comparing the power of singularity of the generalized solution for Problem (P2)here and for Problem (P1) in [15] for the worst case without any orthogonalityconditions one can see that the power in the estimate (2.10) from Corollary 2.4(i)is (|x|2 + t2)−l/2, while in the analogous estimate in Conclusion 1 [15] it is (|x|2 +t2)−(l−1)/2.

(iii) It is interesting to compare the results [14, 15], published in 2002. Going in adifferent way in both cases the authors asked for singular solutions of Problem(P1) in R

3. However, in [14] there are absent any analogues to the orthogonalityconditions presented in [15], and in contrast to [15] in [14] the dependence of theexact order of singularity on the data is not clarified.

Remark 2.6. Let us also compare the present expansion and the results in [30], where anasymptotic expansion of Problem (P1) is found for somewhat easier four-dimensional case.

(i) Both for Problem (P2) in R3 here and Problem (P1) in R

4 as in [30], the studyis based on the properties of the special Legendre functions. Instead of Legendrefunctions Pν with non-integer indices ν = n − 1/2 here, in the four-dimensionalcase one has to deal with integer indices ν = n, that is, simply with the Legendrepolynomials Pn. One can easily modify both these techniques to obtain similarresults for the (m + 1)-dimensional problems: for evenm (analogous to the presentcase R

3) or for odd m (similarly to R4 case). Some different kind of results for the

Protter problems in Rm+1 are presented in [10, 11].

(ii) For the four-dimensional Problem (P1) in [30], the Corollary 3.3 gives only that thesolution is bounded, it could be discontinuous at the origin. On the other hand,here Theorem 2.3 gives us also the control over the bounded but not continuousparts of the generalized solution (through the coefficient F0(x, t) for m = 0 inthe expansion formula (2.7)). As a sequence, Corollary 2.4(iii) guarantees that thesolution is continuous.

(iii) Based on the formulae and the computations from [30], the general case in R4 is also

treated, when the right-hand side f is smooth enough, but not a finite harmonicpolynomial analogous to (2.5). The results are announced and published in [32,33]. For right-hand side functions f ∈ C10(Ω) in [33] the necessary and sufficientconditions for the existence of bounded solution are found. They involve infinitenumber of orthogonality conditions for f that comes from the fact that this is nota Fredholm problem. On the other hand, the results from [33] show that the linearoperator mapping the generalized solution u into f is a semi-Fredholm operatorin C10(Ω). Let us recall that a semi-Fredholm operator is a bounded operator thathas a finite dimensional kernel or cokernel and closed range. Additionally, in [32]a right-hand side function is constructed such that the unique generalized solutionof Protter Problem (P1) in R

4 has exponential type singularity. One expects thatsimilar results could also be obtained for the Problem (P2) in R

3 studied here. Thesequestions correspond to the Open Problem (1) below.


Remark 2.7. Let us mention one obvious consequence of Theorem 2.3 and all the argumentsabove, concerning construction of functions orthogonal to the solutions Wn

k,iof the

homogeneous adjoint Problem (P2∗). Take an arbitrary C2(Ω) function U(x, t) satisfyingthe boundary conditions (P2). Then the function F := �U with the wave operator �, isorthogonal to all the functionsWn

k,i, n = 1, 2, . . ..

Finally, we formulate some still open questions, that naturally arise from the previousworks on the Protter problem and the discussions above.

Open Problems

(1) To study the more general case when the right-hand side function f ∈ Ck(Ω), for anappropriate k. The smooth function f could be represented as a Fourier series rather than,the finite trigonometric polynomial (2.5) in the discussions here.

(i) Find some appropriate conditions for the function f under which there exists ageneralized solution of the Protter problem (P2).

(ii) What kind of singularity can the generalized solution have? The a priori estimates,obtained in [6, 31], which indicate that the generalized solutions of Problem (P2)(including the singular ones), can have at most an exponential growth as ρ → 0.The natural question is as follows: is there a singular solution of these problemswith exponential growth as ρ → 0 or do all such solutions have only polynomialgrowth?

(iii) Is it possible to prove some a priori estimates for generalized solutions of the Problem(P2) with smooth function f which is not a harmonic polynomial?

(iv) Find some appropriate conditions for the function f under which the Problem (P2)has only regular, bounded solutions, or even classical solutions.

(2) To study the Protter problems for degenerate hyperbolic equations. Up to now it isonly known that some singular solutions exist.

(i) We do not know what is the exact behavior of the singular solution even when theright-hand side function f is a finite sum like (2.5). Can we prove some a prioriestimates for generalized solutions?

(ii) Is it possible to find some orthogonality conditions for the function f , as here, underwhich only bounded solutions exist?

(3) Why does there appear a singularity for such smooth right-hand side even for thewave equation? Can we numerically model this phenomenon?

(4) What happens with the ill-posedness of the Protter problems in a more generaldomain (as in [2, 4])when the maximal symmetry is lost if the cone S2 is replaced by anotherlight characteristic one with the vertex away from the origin.


3. Preliminaries

We have a relation between the functionsWnk,i

and the Legendre functions Pν. For ν > −1/2,the functions Pν could be defined by the equality (Section 3.7, formula (6), from Erdelyi et al.[34]),

Pν(z) =1π

∫π

0

(z +

√z2 − 1 cos t

)νdt, z ≥ 0, (3.1)

where for z < 1 in this formula√z2 − 1 := i

√1 − z2.

Let (ρ, ϕ, t) be the cylindrical coordinates in R3, that is, x1 = ρ cosϕ, x2 = ρ sinϕ. For

simplicity, define the function Enk(ρ, t) := En

k(x, t)|x|n. The following result is in connection

with Lemma 5.1 from [15]. Actually, to prove this result one could formally follows thearguments of Lemma 2.3 from [16], where the four-dimensional analogue of Problem (P2)is treated.

Lemma 3.1. For n ∈ N and ν = n − 1/2 define the functions

hνk(ξ, η):=∫ ξ

η

skPν

(ξη + s2

s(ξ + η

))ds, (3.2)

for 0 < η < ξ. Then in {ρ > t} the equality

ρ−1/2∂

∂thνν−2k

(ρ + t2

,ρ − t2

)= ankE

nk

(ρ, t)

(3.3)

holds for k = 0, 1, . . . , [n/2] with some constants ank /= 0.

Proof. Lemma 5.1 from [15] for k ≥ 0 gives

ρ−1/2hνν−2k−2

(ρ + t2

,ρ − t2

)= Cn

kHnk

(ρ, t), (3.4)

where Cnk = const/= 0 and according to Lemma 2.2 from [29]

∂

∂tHn

k

(ρ, t)= 2(n − k − 1)Enk+1

(ρ, t). (3.5)

Therefore the equality (3.3) holds for k ≥ 1. We have to prove it for k = 0. In the proof ofLemma 5.1 from [15] the integrals hνk were calculated using the Mellin transform. In order tocompute hν

k(ξ, η) in the same way let us first introduce the variables x and z:

ξ =ρ + t2

; η =ρ − t2

; x =ρ2

ρ2 − t2 ; z =

(ρ2 − t2)1/2

2. (3.6)


As a consequence after some calculations, formulas (2.2.(4)), (1.10), and (1.4) from Samko etal. [35], show that

z−ν−1hνν(ξ, η)= Cνx

(ν+1)/2I10+

(x−ν−3/2(x − 1)1/2+

)(x), (3.7)

where Iα0+(u)(s) is the Riemann-Liouville fractional integral (for its properties see e.g., [34,35]); in our case we have I10(u)(s) =

∫s0 u(τ)dτ . As usual, we denote also λ+(s) := λ(s) for

s > 0, λ+(s) := 0 for s ≤ 0. The substitution of (3.6) in (3.7) shows that

∂

∂t

{zν+1x(ν+1)/2I10+

(x−ν−3/2(x − 1)−1/2+

)}=∂

∂t

{zν+1x(ν+1)/2

∫x

0τ−ν−3/2(τ − 1)−1/2+ dτ

}

=∂

∂t

{ρν+12−ν−1

∫x

1τ−ν−3/2(τ − 1)−1/2dτ

}= 2−ν−1ρν+1x−ν−3/2(x − 1)−1/2

∂x

∂t

= 2−ν−1ρν+1(ρ2 − t2)ν+3/2

ρ2ν+3

(ρ2 − t2)1/2

t

2tρ2(ρ2 − t2)2

= 2−ν(ρ2 − t2)νρν

(3.8)

and thus

ρ−1/2∂

∂thνν

(ρ + t2

,ρ − t2

)= an0E

n0

(ρ, t). (3.9)

The next result is crucial for construction of solutions of Problem (P2) in thediscussions later.

Lemma 3.2. Let ν ∈ R, ν > 1/2 and the functions F(ξ) ∈ C1(0, 1/2] satisfy F(1/2) = 0. Then allsolutions λ ∈ C1(0, 1/2] of the Volterra integral equation of first kind

∫ ξ

1/2λ′(ξ1)Pν

(ξ

ξ1

)dξ1 = F(ξ) (3.10)

are

λ(ξ) = λ(12

)+ F(ξ) +

∫1/2

ξ

P ′ν

(ξ1ξ

)F(ξ1)ξ1

dξ1. (3.11)


Proof. Formulas (35.17) and (35.28) from Samko et al. [35] state that the solution of theintegral equation (3.10) is given by

λ′(ξ) = −ξ d2

dξ2

(ξ

∫1/2

ξ

Pν

(ξ1ξ

)F(ξ1)ξ21

dξ1

)

= − d

dξ

(ξ2d

dξ

∫1/2

ξ

Pν

(ξ1ξ

)F(ξ1)ξ21

dξ1

).

(3.12)

Then, using that F(1/2) = 0, an integration gives (3.11).

One could use the Mellin transform to calculate the following integral.

Lemma 3.3 (see [16]). Let ν ∈ R, ν > −1/2, then∫1/2

ξ

Pν

(ξ1ξ

)Pν(2ξ1)ξ21

dξ1 =1 − 2ξξ

. (3.13)

According to the existence and uniqueness results in [6], it is sufficient to studyProblem (P2) when the right-hand side f of the wave equation is simply

f(ρ, t, ϕ

)= f1

n

(ρ, t)cosnϕ + f2

n

(ρ, t)sinnϕ, n ∈ N ∪ {0}. (3.14)

Then we seek solutions for the wave equation of the same form:

u(ρ, t, ϕ

)= u1n

(ρ, t)cosnϕ + u2n

(ρ, t)sinnϕ. (3.15)

Thus Problem (P2) reduces to the following one.

Problem (P2.1)

Solve the equation

(un)ρρ +1ρ(un)ρ − (un)tt −

n2

ρ2un = fn

(ρ, t)

(3.16)

in D1 = {0 < t < 1/2; t < ρ < 1 − t} ⊂ R2 with the boundary conditions

(un)t(ρ, 0

)= 0 for 0 < ρ ≤ 1, un

(ρ, 1 − ρ) = 0 for

12≤ ρ ≤ 1. (P2.1)

Let us now introduce new coordinates

ξ =ρ + t2

; η =ρ − t2

, (3.17)


and set

v(ξ, η)= ρ1/2un

(ρ, t); g

(ξ, η)= ρ1/2fn

(ρ, t). (3.18)

Denoting ν = n − 1/2, one transforms Problem (P2.1) into the following.

Problem (P2.2)

Find a solution v(ξ, η) of the equation

vξη − ν(ν + 1)(ξ + η

)2 v = g(ξ, η)

(3.19)

in the domain D = {0 < ξ < 1/2; 0 < η < ξ}with the following boundary conditions:

(vξ − vη

)(η, η

)= 0, v

(12, η

)= 0 for η ∈

(0,

12

). (P2.2)

Problems (P2.1) and (P2.2)were introduced in [6], although the change of coordinatesξ = 1 − ρ − t and η = 1 − ρ + twas used there instead of (3.17). Of course, because the solutionof Problem (P2) may be singular, the same is true for the solutions of (P2.1) and (P2.2). Forthat reason, Popivanov and Schneider [6] defined and proved the existence and uniquenessof generalized solutions of Problems (P2.1) and (P2.2), which correspond to the generalizedsolution of Problem (P2). Further, by “solution” of Problem (P2.1) or (P2.2)we mean exactlythis unique generalized solution.

Lemma 3.4. Let ν ∈ R, ν > 1/2 and g ∈ C1(D). Then the solution v(ξ, η) of Problem (P2.2) isgiven by the following formula:

v(ξ, η)= τ(ξ) +

∫1/2

ξ

τ(ξ1)∂

∂ξ1Pν

((ξ − η)ξ1 + 2ξη

ξ1(ξ + η

))dξ1

−∫1/2

ξ

(∫η

0Pν

((ξ − η)(ξ1 − η1

)+ 2ξ1η1 + 2ξη

(ξ1 + η1

)(ξ + η

))g(ξ1, η1

)dη1

)dξ1,

(3.20)

where

τ(ξ) =∫ ξ

1/2Pν

(ξ1ξ

)G(ξ1)dξ1, (3.21)

G(ξ) =∫1/2

ξ

∫ ξ

0Pν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))(

∂

∂ξ1− ∂

∂η1

)g(ξ1, η1

)dη1dξ1

−∫ ξ

0Pν

(η1 + 2ξ2

ξ(2η1 + 1

))g

(12, η1

)dη1 −

∫1/2

ξ

Pν

(ξ

ξ1

)g(ξ1, 0)dξ1.

(3.22)


Proof. Notice that the function

R(ξ1, η1; ξ, η

)= Pν

((ξ − η)(ξ1 − η1

)+ 2ξ1η1 + 2ξη

(ξ1 + η1

)(ξ + η

))

(3.23)

is a Riemann function for (3.19) (Copson [36]). Therefore, following Aldashev [10], wecan construct the function v(ξ, η) as a solution of a Goursat problem in D with boundaryconditions v(1/2, η) = 0 and v(ξ, 0) = τ(ξ) with some unknown function τ(ξ) ∈ C2(0, 1/2],which will be determined later:

v(ξ, η)= τ(ξ) +

∫1/2

ξ

τ(ξ1)∂

∂ξ1R(ξ1, 0; ξ, η

)dξ1

−∫1/2

ξ

∫η

0R(ξ1, η1; ξ, η

)g(ξ1, η1

)dη1dξ1.

(3.24)

Now, the boundary condition

(∂

∂ξ− ∂

∂η

)v

∣∣∣∣η=ξ

= 0. (3.25)

gives an integral equation for τ(ξ). For that reason, let us define the function G(ξ):

G(ξ) :=(∂

∂ξ− ∂

∂η

)∫1/2

ξ

(∫η

0R(ξ1, η1; ξ, η

)g(ξ1, η1

)dη1

)dξ1

∣∣∣∣∣η=ξ

=∫1/2

ξ

(∫ ξ

0P ′ν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))

ξ1 − η1ξ(ξ1 + η1

)g(ξ1, η1

)dη1

)dξ1

−∫ ξ

0g(ξ, η1

)dη1 −

∫1/2

ξ

g(ξ1, ξ)dξ1 = −∫ ξ

0g(ξ, η1

)dη1 −

∫1/2

ξ

g(ξ1, ξ)dξ1

−∫1/2

ξ

(∫ ξ

0g(ξ1, η1

)( ∂

∂ξ1− ∂

∂η1

)Pν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))dη1

)dξ1

=∫1/2

ξ

(∫ ξ

0Pν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))(

∂

∂ξ1− ∂

∂η1

)g(ξ1, η1

)dη1

)dξ1

−∫ ξ

0Pν

(η1 + 2ξ2

ξ(2η1 + 1

))g

(12, η1

)dη1 −

∫1/2

ξ

Pν

(ξ

ξ1

)g(ξ1, 0)dξ1.

(3.26)

Obviously, G ∈ C2[0, 1/2]. The condition (3.25) leads us to the following equation:

τ ′(ξ) − 1ξτ(ξ)P ′

ν(1) −∫1/2

ξ

τ(ξ1)ξ21

P ′′ν

(ξ

ξ1

)dξ1 = G(ξ). (3.27)


Then, using τ(1/2) = v(1/2, 0) = 0, we have

∫ ξ

1/2

d

dξ1

{ξ21τ

′(ξ1)}Pν

(ξ

ξ1

)dξ1 = ξ2G(ξ) − τ ′(1/2)

4Pν(2ξ). (3.28)

A necessary solvability condition for the unknown function τ ∈ C2(0, 1/2] is: τ ′(1/2) =G(1/2). One could solve this Volterra integral equation of the first kind, using Lemma 3.2.The result is

ξ2τ ′(ξ) − 14τ ′(12

)= ξ2G(ξ) − 1

4τ ′(12

)Pν(2ξ) +

∫1/2

ξ

P ′ν

(ξ1ξ

)4ξ21G(ξ1) − τ ′(1/2)Pν(2ξ1)4ξ1

dξ1.

(3.29)

Integrate, we find

τ(ξ) =∫ ξ

1/2

(G(z) +

1z2

∫1/2

z

P ′ν

(ξ1z

)ξ1G(ξ1)dξ1

)dz

+τ ′(1/2)

4

∫ ξ

1/2

(1z2

− Pν(2z)z2

− 1z2

∫z

1/2P ′ν

(ξ1z

)Pν(2ξ1)ξ−11 dξ1

)dz.

(3.30)

Now, using Lemma 3.3 and the equality

∫ ξ

1/2

1z2

(∫1/2

z

P ′ν

(ξ1z

)F(ξ1)dξ1

)dz =

∫ ξ

1/2

(Pν

(ξ1ξ

)− 1)F(ξ1)ξ1

dξ1, (3.31)

for F(ξ) = Pν(2ξ)ξ−1 one finds that the coefficient of τ ′(1/2) in (3.30) is zero. Using again(3.31) for F(ξ) = ξG(ξ), τ is simply

τ(ξ) =∫ ξ

1/2Pν

(ξ1ξ

)G(ξ1)dξ1. (3.32)

Obviously, τ ∈ C2(0, 1/2] and τ(1/2) = 0, τ ′(1/2) = G(1/2). Finally, the solution of Problem(P2.2) is given by the formulae (3.20), (3.21), and (3.22).

4. Proofs of Main Results

In order to study the behavior of the generalized solution of Problem (P2), in view of relations(3.18) and Lemma 3.4, we will examine the function v(ξ, η) defined by the formulae (3.20),(3.21), and (3.22). It is not hard to see that the part “responsible” for the singularity is theintegral in (3.21) for the function τ(ξ). In fact, τ(ξ) blows up at ξ = 0, since the argument ξ1/ξand thus the values of the Legendre function Pν in (3.21) go to infinity when ξ → 0. Actually,


Pν(z) grows like |z|ν at infinity. In the next lemma we find the dependance of the exact orderof singularity of τ(ξ) on the function G(ξ). It is governed by the constants

γk :=∫1/2

0ξν−2kG(ξ) dξ for k = 0, . . . ,

[ν + 12

]. (4.1)

Actually, these numbers are closely connected to the constants βnk,i from Theorem 2.3. We willclarify this relation later in Lemma 4.1 and the proof of Theorem 2.3.

Lemma 4.1. Let ν = n − 1/2, where n ∈ N, n ≥ 2, and let the function G(ξ) ∈ C1[0, 1/2]. Then thefunction

τ(ξ) =∫ ξ

1/2Pν

(ξ1ξ

)G(ξ1)dξ1 (4.2)

belongs to C2(0, 1/2] and satisfies the representation

τ(ξ) =[(ν+1)/2]∑

k=0

Cνkγkξ

−(ν−2k) + ψ(ξ), ξ ∈(0,

12

), (4.3)

where the function ψ(ξ) ∈ C2(0, 1/2], |ψ(ξ)| ≤ Cξmax{|G(ξ)| : ξ ∈ 0, 1/2]} and the nonzeroconstants Cν

kand C are independent of G(ξ).

Proof. The argument of the Legendre function Pν in (4.2) satisfies the inequality ξ1/ξ ≥ 1,which allows us to apply the representation (3.1):

τ(ξ) =1π

1ξν

∫ ξ

1/2

∫π

0

(ξ1 +

√ξ21 − ξ2 cos t

)ν

G(ξ1)dt dξ1. (4.4)

We will study the expansion at ξ = 0 of the function

F(ξ) :=∫1/2

ξ

∫π

0

(ξ1 +


)ν

G(ξ1)dt dξ1. (4.5)

Let us define the functions

Mνk(ξ1, ξ) := (−1)k (ν − 2k + 1)2k

2k(1/2)k

∫π

0

(ξ1 +


)ν−2ksin2kt dt, (4.6)

for ξ ≤ ξ1 ≤ 1/2. Then, obviously

F(ξ) =∫1/2

ξ

Mν0(ξ1, ξ)G(ξ1)dξ1. (4.7)


First, we will examine the properties of the functions Mνk(ξ1, ξ) and their derivatives with

respect to ξ. We start with the equality

Mνk(ξ, ξ) = a

νkξ

ν−2k, aνk /= 0. (4.8)

Further, the index k will be less than ν. Notice that for k < ν + 1/2 the integrals∫π0 (1 ±

cos t)ν−2ksin2ktdt are convergent. Then, for ξ ≤ ξ1 we have the equality

Mνk(ξ1, 0) = b

νkξ

ν−2k1 , bνk /= 0, (4.9)

and the inequality

∣∣Mνk(ξ1, ξ)

∣∣ ≤ cνkξν−2k1 . (4.10)

Differentiating with respect to ξ one finds

(−1)k+1 ∂∂ξMν

k(ξ1, ξ)

=(ν − 2k + 1)2k(ν − 2k)

2k(1/2)k

∫π

0

(ξ1 +


)ν−2k−1 ξ√ξ21 − ξ2

sin2kt cos tdt

=(ν − 2k)2k+1(ν − 2k − 1)

2k(1/2)k(2k + 1)

∫π

0ξ

(ξ1 +


)ν−2k−2sin2k+2tdt

= (−1)k+1ξMνk+1(ξ1, ξ).

(4.11)

Therefore, for the derivatives ofMν0 we find by induction

∂kMν0

∂ξk(ξ1, ξ) =

[k/2]∑

i=0

Cki ξ

k−2iMνk−i(ξ1, ξ), (4.12)

where the coefficients Cki are positive constants. We want to evaluate these derivatives ofMν

0at ξ = 0. Let us estimate the terms in the last sum for k < ν:

(i) when i is such that ν − 2(k − i) < 0 the inequality (4.10) gives the estimate

∣∣∣ξk−2iMνk−i(ξ1, ξ)

∣∣∣ ≤ ξk−2icνkξν−2(k−i)1 ≤ cνkξν−k; (4.13)

(ii) when ν − 2(k − i) ≥ 0 and k/2 > i, we have

∣∣∣ξk−2iMνk−i(ξ1, ξ)

∣∣∣ ≤ cνkξk−2i. (4.14)


Hence ξk−2iMνk−i(ξ1, ξ)|ξ=0 = 0 for 2i < k. Therefore, at the point ξ = 0 the only one nonzero

term in the sum (4.12) is for 2i = k, that is,

∂kMν0

∂ξk(ξ1, 0) =

{0, if k is oddCkk/2b

νk/2ξ

ν−k1 , if k is even.

(4.15)

The last observation is that (4.8) and (4.12) imply

∂kMν0

∂ξk

∣∣∣∣∣ξ1=ξ

= dνkξν−k, (4.16)

where dνk=∑[k/2]

i=0 Cki a

νk−i are constants.

Now, we go back to the function F(ξ). We want to differentiate [ν] times and evaluateat ξ = 0. Differentiating (4.7)we find the following:

F ′(ξ) = −aν0G(ξ)ξν +∫1/2

ξ

∂

∂ξMν

0(ξ1, ξ)G(ξ1)dξ1. (4.17)

Next, since the assertion for G(ξ) is only G(ξ) ∈ C1[0, 1/2], instead of F ′(ξ) we willdifferentiate the function

F1(ξ) := F ′(ξ) + aν0G(ξ)ξν =

∫1/2

ξ

∂

∂ξMν

0(ξ1, ξ)G(ξ1)dξ1. (4.18)

Notice that it belongs to C[0, 1/2] ∩ C1(0, 1/2] and the derivative is

F ′1(ξ) = −dν1G(ξ)ξν−1 +

∫1/2

ξ

∂2

∂ξ2Mν

0(ξ1, ξ)G(ξ1)dξ1. (4.19)

In the same way, after denoting F0(ξ) ≡ F(ξ), define for j = 1, . . . , [ν] the functions

Fj(ξ) := F ′j−1(ξ) + d

νj−1G(ξ)ξ

ν−j+1 (4.20)

with the constants dνj from (4.16). Then, using (4.16), it follows by induction that Fj iscontinuous in [0, 1/2] and

Fj(ξ) =∫1/2

ξ

∂j

∂ξjMν

0(ξ1, ξ)G(ξ1)dξ1, j = 0, . . . , [ν]. (4.21)

On the other hand,

Fj(0) =∫1/2

0

∂j

∂ξjMν

0(ξ1, 0)G(ξ1)dξ1, j = 0, . . . , [ν] − 1. (4.22)


Hence, according to (4.15), for j ≤ [ν] − 1,

Fj(0) =

{0, if j is oddγ ′i , if j is even.

(4.23)

The next step is to evaluate the integral F[ν]. Using (4.12), one could rewrite it in theform

F[ν](ξ) =∫1/2

ξ

∂[ν]

∂ξ[ν]Mν

0(ξ1, ξ)G(ξ1)dξ1 =[[ν]/2]∑

i=0

C[ν]i

∫1/2

ξ

ξ[ν]−2iMν[ν]−i(ξ1, ξ)G(ξ1)dξ1. (4.24)

For all the terms in the last sum, except one, the estimate is straightforward.(1) When i is such that [ν] − 2i ≥ 2, for the corresponding terms we have

∣∣∣ξ[ν]−2iMν[ν]−i(ξ1, ξ)

∣∣∣ ≤ ξ2ξ[ν]−2i−21 cν[ν]−i ξν−2([ν]−i)1 ≤ cν[ν]−i ξ2ξ−3/21 , (4.25)

and, therefore,

∣∣∣∣∣

∫1/2

ξ

ξ[ν]−2iMν[ν]−i(ξ1, ξ)G(ξ1)dξ1

∣∣∣∣∣ ≤ cν[ν]−iAξ

2∫1/2

ξ

ξ−3/21 dξ1 ≤ CAξ3/2, (4.26)

where A := max{|G(ξ)| : ξ ∈ 0, 1/2]}.(2) For the last term in (4.24)with i = [[ν]/2] there are two cases:

(2a) when [ν] = 2m is an even number this is the integral

∫1/2

ξ

Mνm(ξ1, ξ)G(ξ1)dξ1; (4.27)

(2b) when [ν] = 2m − 1 is an odd number, the integral is

∫1/2

ξ

ξMνm(ξ1, ξ)G(ξ1)dξ1. (4.28)

For simplicity let us define some constants γ ′i

γ ′i :=∫1/2

0Mν

i (ξ1, 0)G(ξ1)dξ1, (4.29)


related to the constants γi given by (4.1). Indeed, due to (4.9) the equality γ ′i = bνi γiholds. Let us begin with the following case.

(2a): [ν] = 2m, that is, ν = 2m + 1/2. We will evaluate the difference:

∣∣∣∣∣

∫1/2

ξ

Mνm(ξ1, ξ)G(ξ1)dξ1 − γ ′m

∣∣∣∣∣ =

∣∣∣∣∣

∫1/2

ξ

Mνm(ξ1, ξ)G(ξ1)dξ1 −

∫1/2

0Mν

m(ξ1, 0)G(ξ1)dξ1

∣∣∣∣∣

≤∣∣∣∣∣

∫1/2

ξ

{Mνm(ξ1, ξ) −Mν

m(ξ1, 0)}G(ξ1)dξ1∣∣∣∣∣

+

∣∣∣∣∣

∫ ξ

0Mν

m(ξ1, 0)G(ξ1)dξ1

∣∣∣∣∣.

(4.30)

For the first integral, using the estimate (4.10), we calculate

|Mνm(ξ1, ξ) −Mν

m(ξ1, 0)| =∣∣∣∣∣

∫ ξ

0

∂

∂ξ2Mν

m(ξ1, ξ2)dξ2

∣∣∣∣∣

=

∣∣∣∣∣

∫ ξ

0ξ2M

νm+1(ξ1, ξ2)dξ2

∣∣∣∣∣ ≤ cνm+1ξ

2ξν−2m−21 = cνm+1ξ

2ξ−3/21 ,

(4.31)

and, therefore,

∣∣∣∣∣

∫1/2

ξ


m(ξ1, 0)}G(ξ1)dξ1∣∣∣∣∣ ≤ C1Aξ

3/2. (4.32)

For the second integral

∣∣∣∣∣

∫ ξ

0Mν

m(ξ1, 0)G(ξ1)dξ1

∣∣∣∣∣ ≤ cνmA

∫ ξ

0ξν−2m1 dξ1 = CA

∫ ξ

0ξ1/21 dξ1 = C2Aξ

3/2. (4.33)

From the last two inequalities we get the estimate

∣∣∣∣∣

∫1/2

ξ

Mνm(ξ1, ξ)G(ξ1)dξ1 − γ ′m

∣∣∣∣∣ ≤ CAξ3/2. (4.34)

Therefore, in the case [ν] = 2m,m ∈ N,

F[ν](ξ) = γ ′m + ψ[ν](ξ), (4.35)

where |ψ[ν](ξ)| ≤ CAξ3/2.


(2b) When [ν] = 2m − 1, that is, ν = 2m − 1/2, we have to study the integral (4.28).Obviously,

∣∣∣∣∣

∫1/2

ξ

ξMνm(ξ1, ξ)G(ξ1)dξ1 − γ ′mξ

∣∣∣∣∣

≤ ξ∣∣∣∣∣

∫1/2

ξ


m(ξ1, 0)}G(ξ1)dξ1∣∣∣∣∣ + ξ

∣∣∣∣∣

∫ ξ

0Mν

m(ξ1, 0)G(ξ1)dξ1

∣∣∣∣∣.

(4.36)

For the last integral we have

∣∣∣∣∣

∫ ξ

0Mν

m(ξ1, 0)G(ξ1)dξ1

∣∣∣∣∣ ≤ cνmA

∫ ξ

0ξν−2m1 dξ1 = 2CAξ1/2. (4.37)

Now, to estimate the first term in the right-hand side of (4.36), there are two cases:

(i) when m ≥ 2, we have ν > [ν] = 2m − 1 ≥ m + 1 and similarly to the previouscase (2a) we can apply inequality (4.31). Thus, we estimate the difference:

|Mνm(ξ1, ξ) −Mν

m(ξ1, 0)| =∣∣∣∣∣

∫ ξ

0

∂

∂ξ2Mν

m(ξ1, ξ2)dξ2

∣∣∣∣∣ =

∣∣∣∣∣

∫ ξ

0ξ2M

νm+1(ξ1, ξ2)dξ2

∣∣∣∣∣

≤∫ ξ

0cνm+1ξ2ξ

ν−2m−21 dξ2 ≤ cνm+1ξ

2ξ−5/21 ,

(4.38)

(ii) when m = 1, denote for simplicity p :=√1 − ξ2/ξ21, then p ∈ [0, 1] and directly

from the definition (4.6) of the functionsMνm, we get

∣∣∣M3/21 (ξ1, ξ) −M3/2

1 (ξ1, 0)∣∣∣

= Cξ−1/21

∣∣∣∣∣

∫1

−1

{(1 − pz)−1/2 − (1 − z)−1/2

}√1 − z2dz

∣∣∣∣∣

≤ Cξ−1/21

∫1

−1

(1 − p)|z|

√1 + z

√1 − pz

(√1 − pz +√

1 − z)dz

≤ C1ξ−1/21

(1 − p) + C1ξ

−1/21

(1 − p)

∫1

0

1√(

1 − pz)(1 − z)dz

≤ C1ξ−1/21

(1 − p)

⎡⎢⎣1 +

∫1

0

1√(

1 − p + pτ)τdτ

⎤⎥⎦

≤ C2ξ−1/21

√1 − p ≤ C2ξ

−1/21

√1 − p2 = Cξξ−3/21 .

(4.39)


Thus form ≥ 1, both cases lead to

∣∣∣∣∣

∫1/2

ξ


m(ξ1, 0)}G(ξ1)dξ1∣∣∣∣∣ ≤ CAξ

1/2. (4.40)

Finally, substituting (4.37) and (4.40) in (4.36), we find the estimate

∣∣∣∣∣

∫1/2

ξ

Mνm(ξ1, ξ)G(ξ1)dξ1 − γ ′mξ

∣∣∣∣∣ ≤ CAξ3/2. (4.41)

Summarizing, in the case (2b) of odd [ν] = 2m − 1, we have

F[ν](ξ) = γ ′mξ + ψ[ν](ξ), (4.42)

where |ψν](ξ)| ≤ CAξ3/2.Now, we are ready to go backwards from F[ν](ξ) to F0(ξ). Integrating (4.20) we find

Fj(ξ) = Fj(0) +∫ ξ

0Fj+1(ξ1)dξ1 + dνj

∫ ξ

0ξν−j1 G(ξ1)dξ1, (4.43)

for j = 0, . . . , [ν] − 1. Using (4.43) and (4.23), one can find the relation between the functionsF(ξ) ≡ F0(ξ) and F[ν](ξ). Starting from (4.43)with j = 0 one expresses F1(ξ) by applying again(4.43) in the right-hand side but for j = 1:

F(ξ) ≡ F0(ξ) = F0(0) +∫ ξ

0F1(ξ1)dξ1 + dν0

∫ ξ

0ξν1G(ξ1)dξ1 = F0(0) + F1(0)ξ

+∫ ξ

0

(∫ t

0F2(ξ1)dξ1

)dt +

∫ ξ

0

{dν1

∫ t

0ξν−11 G(ξ1)dξ1 + dν0 t

νG(t)

}dt.

(4.44)

Similarly, (4.43) gives a representation of F2(ξ) through F3(ξ) and so on. Finally, we get thesum

F(ξ) = F0(0) +F1(0)1!

ξ +F2(0)2!

ξ2 + · · · + dν0∫ ξ

0G(ξ1)ξν1dξ1

+ dν1

∫ ξ

0G(ξ1)ξν−11 (ξ − ξ1)dξ1 +

dν22

∫ ξ

0G(ξ1)ξν−21 (ξ − ξ1)2dξ1 + · · · ,

(4.45)

by consequently substituting Fk+1(ξ) in the resulting expression at each step by applying thesame formula (4.43) for j = k + 1. Thus, for F(ξ) we find inductively

F(ξ) =∑

j

Fj(0)j!

ξj + · · · =∑

k

γ ′k(2k)!

ξ2k + · · · , (4.46)


since F2i+1(0) = 0 and F2i(0) = γ ′i , in view of (4.23). The process ends when F[ν], integrated[ν] times, appears and we can apply (4.35) or (4.42) instead of (4.43). Therefore, according to(4.35), when [ν] = 2m is an even number, the last γ ′i in this sum will be γ ′m, and its coefficientwill be (1/(2m)!)ξ2m, while when [ν] = 2m − 1 is an odd number, formula (4.42) shows thatthe last term will be also (γ ′m/(2m)!)ξ2m. In both cases m = [(ν + 1)/2] and the constantcoefficients are independent of G(ξ). Then, for the function F(ξ) we have the representation

F(ξ) =[(ν+1)/2]∑

k=0

γ ′k(2k)!

ξ2k + Ψ(ξ), (4.47)

where the function Ψ(ξ) is defined by

Ψ(ξ) :=[ν]−1∑

j=0

dνj

j!

∫ ξ

0G(ξ1)ξ

ν−j1 (ξ − ξ1)jdξ1 +

∫ ξ

0ψ[ν](ξ1)

(ξ − ξ1)[ν]−1([ν] − 1)!

dξ1. (4.48)

Therefore |Ψ(ξ)| ≤ CAξν+1, because |G(ξ)| ≤ A and |ψ[ν](ξ1)| ≤ CAξ3/21 .Finally, recall that γ ′k = bνkγk, with coefficients bνk /= 0 from (4.9) and τ(ξ) = −π−1ξ−νF(ξ)

from (4.4), and, therefore,

τ(ξ) =[(ν+1)/2]∑

k=0

Ckγkξ−(ν−2k) + ψ(ξ), (4.49)

where Ck = −π−1bνk/(2k)!/= 0 and |ψ(ξ)| ≤ CAξ.

This lemma, due to formula (3.20), helps us to examine the solution v(ξ, η) of Problem(P2.2) and therefore due to (3.18), the solution un(ρ, t) = ρ−1/2v of Problem (P2.1). First, fork = 0, . . . , [n/2], denote by αn

kthe parameters

αnk :=∫1/2

0

(∫1−t

t

Enk(ρ, t)fn(ρ, t)ρdρ

)dt. (4.50)

Theorem 4.2. Let fn ∈ C1(D1). Then the generalized solution un(ρ, t) of Problem (P2.1) belongs toC2(D1 \ (0, 0)) and has the following asymptotic expansion at the origin (0, 0):

un(ρ, t)=

[n/2]∑

k=0

ρ−1/2(ρ + t

)−(n−2k−1/2)αnkF

nk

(ρ, t)+ ρ1/2

(ln ρ

)Fn(ρ, t), (4.51)

where Fnk(ρ, t), Fn(ρ, t) ∈ C2(D1 \ (0, 0)),

∣∣Fnk(ρ, t)∣∣ ≤ C, ∣∣Fn

(ρ, t)∣∣ ≤ C

{maxD1

∣∣fn(ρ, t)∣∣ +max

D1

∣∣(fn)t

(ρ, t)∣∣}, (4.52)

with functions Fnkand a constant C independent of fn and limt→+0F

nk(t, t) = const /= 0.


First, let us shortly outline the proof. In order to find the behavior of un we applythe relation (3.18) and study the function v. Actually, Lemma 3.4 uses Lemmas 3.2 and 3.3to describe v by formulas (3.20)–(3.22) and the analysis passes to τ , G, and the Legendrefunction Pν. This way the base for the asymptotic expansion (4.51) is the expansion found inLemma 4.1 for τ given by (3.21).

Proof of Theorem 4.2. Denote

A := maxD1

∣∣fn(ρ, t)∣∣ +max

D1

∣∣(fn)t

(ρ, t)∣∣ (4.53)

and thus |g(ξ, η)| ≤ CA, |G(ξ)| ≤ CA with the constant C independent of fn. Our goal is toapply Lemma 4.1.

The key of this will be the equality

∫1/2

0ξν−2kG(ξ)dξ

=∫1/2

0

∫1/2

ξ

(∫ ξ

0ξν−2kPν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))(

∂

∂ξ1− ∂

∂η1

)g(ξ1, η1

)dη1

)dξ1dξ

−∫1/2

0

(∫ ξ

0ξν−2kPν

(η1 + 2ξ2

ξ(1 + 2η1

))g

(12, η1

)dη1

)dξ

−∫1/2

0

(∫1/2

ξ

ξν−2kPν

(ξ

ξ1

)g(ξ1, 0)dξ1

)dξ

=∫1/2

0

∫ ξ1

0

(∫ ξ1

η1

ξν−2kPν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))dξ

)(∂

∂ξ1− ∂

∂η1

)g(ξ1, η1

)dη1dξ1

−∫1/2

0

(∫1/2

η1

ξν−2kPν

(η1 + 2ξ2

ξ(1 + 2η1

))dξ

)g

(12, η1

)dη1

−∫1/2

0

(∫ ξ1

0ξν−2kPν

(ξ

ξ1

)dξ

)g(ξ1, 0)dξ1

= −∫1/2

0

∫ ξ1

0

(∂

∂ξ1− ∂

∂η1

)(∫ ξ1

η1

ξν−2kPν

(ξ1η1 + ξ2

ξ(ξ1 + η1

))dξ

)g(ξ1, η1

)dη1dξ1

= −∫1/2

0

∫ ξ1

0

(∂

∂ξ1− ∂

∂η1

){hνν−2k

(ξ1, η1

)}g(ξ1, η1

)dη1dξ1, k = 0, . . . ,

[n2

],

(4.54)

according to Definition (3.2) of functions hνkin Lemma 3.1.

On the other hand, Lemma 3.1 gives

(∂

∂ξ− ∂

∂η

)(ξ + η

)−1/2hνν−2k

(ξ, η)= ankE

nk

(ξ + η, ξ − η), (4.55)


and by definition f = (ξ + η)−1/2g. Therefore, we have the following relation between αnkdefined in (4.50) and the constants γk from Lemma 4.1:

αnk =∫1/2

0

(∫1−t

t

Enk(ρ, t)f(ρ, t)ρdρ

)dt

=2ank

∫1/2

0

(∫ ξ1

0

(∂

∂ξ1− ∂

∂η1

){(ξ1 + η1

)−1/2hνν−2k

(ξ1, η1

)}g(ξ1, η1

)(ξ1 + η1

)1/2dη1

)dξ1

= − 2ank

∫1/2

0ξν−2k1 G(ξ1)dξ1 = − 2

ank

γk,

(4.56)

with coefficients ank /= 0 from (3.3). Then Lemma 4.1 gives

τ(ξ) =[(ν+1)/2]∑

k=0

Cnkα

nkξ

−(ν−2k) + ψ(ξ), (4.57)

where the constants Cnk /= 0 are independent of f and |ψ(ξ)| ≤ CAξ. Hence, for the solution

v(ξ, η) of Problem (P2.1) we have

v(ξ, η)= τ(ξ) +

∫1/2

ξ

τ(ξ1)∂

∂ξ1Pν

((ξ − η)ξ1 + 2ξη

ξ1(ξ + η

))dξ1

−∫1/2

ξ

(∫η

0Pν

((ξ − η)(ξ1 − η1

)+ 2ξ1η1 + 2ξη

(ξ1 + η1

)(ξ + η

))g(ξ1, η1

)dη1

)dξ1

=[(ν+1)/2]∑

k=0

Cnkβ

nkG

nk

(ξ, η)ξ−(ν−2k) + Ψ1

(ξ, η),

(4.58)

where

Gnk

(ξ, η)= 1 + ξν−2k

∫1/2

ξ

ξ2k−ν1∂

∂ξ1Pν

((ξ − η)ξ1 + 2ξη

ξ1(ξ + η

))dξ1, (4.59)

Ψ1(ξ, η)= ψ(ξ) +

∫1/2

ξ

ψ(ξ1)∂

∂ξ1Pν

((ξ − η)ξ1 + 2ξη

ξ1(ξ + η

))dξ1

−∫1/2

ξ

∫η

0Pν

((ξ − η)(ξ1 − η1

)+ 2ξ1η1 + 2ξη

(ξ1 + η1

)(ξ + η

))g(ξ1, η1

)dξ1dη1.

(4.60)


Notice that the arguments of the Legendre’s functions Pν in (4.58), (4.59) and (4.60) vary inthe interval [0, 1]. Thus,

∣∣Gnk

(ξ, η)∣∣ ≤ 1 + C1

ηξν−2k+1

ξ + η

∫1/2

ξ

ξ2k−ν−21 dξ1 ≤ 1 +C1

ν + 1 − 2k= C. (4.61)

Therefore, the functions

Fnk(ρ, t):= 2ν−2kCn

kGnk

(ρ + t2

,ρ − t2

)(4.62)

are also bounded. On the other hand, v(ξ, 0) = τ(ξ) and therefore

Fnk (t, t) = 2ν−2kCnkG

nk(t, 0) = 2ν−2kCn

k (4.63)

with coefficients Cnk /= 0 from (4.57).

Let us now evaluate the function Ψ1 defined in (4.60). We have |ψ(ξ)| ≤ CAξ,∣∣∣∣∣

∫1/2

ξ

ψ(ξ1)∂

∂ξ1Pν

((ξ − η)ξ1 + 2ξη

ξ1(ξ + η

))dξ1

∣∣∣∣∣ ≤ CAξη

ξ + η

∣∣∣∣∣

∫1/2

ξ

ξ−11 dξ1

∣∣∣∣∣ ≤ CAξ|ln ξ|∣∣∣∣∣

∫1/2

ξ

∫η

0Pν

((ξ − η)(ξ1 − η1

)+ 2ξ1η1 + 2ξη

(ξ1 + η1

)(ξ + η

))g(ξ1, η1

)dξ1dη1

∣∣∣∣∣ ≤ CAξ.(4.64)

Finally, let us return to (ρ, t) coordinates using (3.17) and (3.18). The representation (4.58)gives (4.51), where the function

Fn(ρ, t):= ρ−1/2

(ln ρ

)−1Ψ1

(ρ + t2

,ρ − t2

)(4.65)

is continuous in D1 and the estimate |Fn(ρ, t)| ≤ C1Aρ1/2, holds with C1 = const.

Finally, we are ready to prove our main result.

Proof of Theorem 2.3. The uniqueness and the existence of the generalized solution when f ∈C1(Ω) is a trigonometric polynomial, follows from the results in [6]. Now the right-hand sidefunction satisfies (2.5), and thus it can be written in the form

f(x1, x2, t) =l∑

n=2

{f1n(|x|, t) cosnϕ + f2

n(|x|, t) sinnϕ}. (4.66)

According to [6] the unique generalized solution u(x1, x2, t) also has the form

u(x1, x2, t) =l∑

n=2

{u1n(ρ, t)cosnϕ + u2n

(ρ, t)sinnϕ

}, (4.67)


where the functions uin(ρ, t) are solutions of Problem (P2.1) with right-hand side functionfin ∈ C1(G) and are described in Theorem 4.2. Then, for the constants αn

kfrom Theorem 4.2

and βnk,i

from (2.6), we have the following relation:

αnk =∫1/2

0

(∫1−t

t

Enk(ρ, t)fin(ρ, t)ρdρ

)dt = π−1

∫

ΩWn

k,i(x, t)f(x, t)dx dt = π−1βnk,i. (4.68)

Therefore, from Theorem 4.2 it follows that

uin(ρ, t)= π−1

[n/2]∑

k=0

ρ−1/2(ρ + t

)−(n−2k−1/2)βnk,iF

n,ik

(ρ, t)+ ρ1/2

(ln ρ

)Fn,i

(ρ, t), (4.69)

where the functions Fn,ik

are independent of f , |Fn,ik(ρ, t)| ≤ C and

∣∣∣Fn,i(ρ, t)∣∣∣ ≤ C

(maxD1

∣∣∣fin∣∣∣ +max

D1

∣∣∣(fin

)

t

∣∣∣)

≤ C1

(maxΩ

∣∣f(x, t)∣∣ +max

Ω

∣∣ft(x, t)∣∣). (4.70)

Summing over n and i in (4.67) we get the expansion

u(x, t) =l∑

n=2

2∑

i=1

[n/2]∑

k=0

(|x|2 + t2

)k−n/2βnk,iF

nk,i(x, t)

+(|x|2 + t2

)1/4F(x, t) ln

(|x|2 + t2

),

(4.71)

where

Fnk,1(x, t) = π−1ρ−1/2

(ρ2 + t2

)k−n/2(ρ + t

)2k−n+1/2Fn,1k

(ρ, t)cosnϕ,

Fnk,2(x, t) = π−1ρ−1/2

(ρ2 + t2

)k−n/2(ρ + t

)2k−n+1/2Fn,2k

(ρ, t)sinnϕ,

F(x, t) =|x|1/2 ln|x|

(|x|2 + t2

)1/4ln(|x|2 + t2

)l∑

n=2

(Fn,1

(ρ, t)cosnϕ + Fn,2

(ρ, t)sinnϕ

).

(4.72)

Obviously the functions Fnk,i(x, t) are bounded and independent of f . Also, we have

|F(x, t)| ≤ C(maxΩ

∣∣f(x, t)∣∣ +max

Ω

∣∣ft(x, t)∣∣). (4.73)


Notice that the singularity (|x|2+t2)−m/2 for fixedm appears in the sum for the solutionu(x, t) when n = m,m + 2, m + 4, . . ., and the corresponding coefficients are βm0,iF

m0,i; β

m1,iF

m+21,i ;

βm2,iFm+42,i ; an so on, until n ≤ l. Therefore, (4.71) is equivalent to

u(x, t) =l∑

m=0

(|x|2 + t2

)−m/2 [(l−m)/2]∑

k=0

2∑

i=1

βm+2kk,i Fm+2k

k,i (x, t)

+(|x|2 + t2

)1/4F(x, t) ln

(|x|2 + t2

).

(4.74)

Thus, the properties (i), (ii), and (v) are proved.Finally, let us prove the properties (iii) and (iv). For a fixed direction (α1, α2, 1)t =

(cos γ, sin γ, 1)t ∈ S2, 0 < t < 1/2, γ ∈ [0, 2π) we have the expressions

Fnk,1(α1t, α2t, t) = π−122k−n+1/2Fn,1

k (t, t) cosnγ,

Fnk,2(α1t, α2t, t) = π−122k−n+1/2Fn,2

k (t, t) sinnγ,(4.75)

with the functions Fn,ik (ρ, t) from (4.69) and Fnk,i(x1, x2, t) from (2.8). Therefore, according to(2.8) and (4.74),

limt→+0

Fm(α1t, α2t, t)

=[(l−m)/2]∑

k=0

{Cmk,1β

m+2kk,1 cos(m + 2k)γ + Cm

k,2βm+2kk,2 sin(m + 2k)γ

} (4.76)

with some constants Cmk,i /= 0. Thus, this expression is zero for all γ ∈ [0, 2π] if and only if

all the constants βm+2kk,i involved are zero, because the trigonometric functions are linearly

independent in [0, 2π]. Thus, if at least one βm+2kk,i /= 0, one could choose γ , that is, a direction

(α1, α2, 1), such that limt→+0Fm(α1t, α2t, t) = Cm = const/= 0, which proves (iii).

For (iv) in the casem = 0 we have

limt→+0

F0(α1t, α2t, t) =[l/2]∑

k=1

{C0k,1β

2kk,1 cos 2kγ + C

0k,2β

2kk,2 sin 2kγ

}, (4.77)

and the sum starts at k = 1 since β00,1 = β00,2 = 0 according to Definition (2.6) and the specialform (2.5) of f . Now, F0 is continuous at (0, 0, 0) only when the expression in the right-handside of (4.77) is a constant. However, the constant 1 and the trigonometric functions involvedin (4.77) are linearly independent. Therefore, if at least one β2k

k,iis not zero, then F0 is not

continuous at the origin.

Acknowledgments

This research was supported in part by the research grants of the Priority R&D Groupfor Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik


University College. The research of N. Popivanov and T. Popov was partially supported bythe Bulgarian NSF under Grants DO 02-115/2008 and DO 02-75/2008. The authors wouldlike to thank the anonymous referees for the useful comments and suggestions.

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Research ArticleA Note on the Second Order of AccuracyStable Difference Schemes for the NonlocalBoundary Value Hyperbolic Problem

Allaberen Ashyralyev1 and Ozgur Yildirim2

1 Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey2 Department of Mathematics, Yildiz Technical University, Esenler 34210, Istanbul, Turkey

Correspondence should be addressed to Ozgur Yildirim, [email protected]



Copyright q 2012 A. Ashyralyev and O. Yildirim. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The second order of accuracy absolutely stable difference schemes are presented for the nonlocalboundary value hyperbolic problem for the differential equations in a Hilbert space H with theself-adjoint positive definite operator A. The stability estimates for the solutions of these differenceschemes are established. In practice, one-dimensional hyperbolic equationwith nonlocal boundaryconditions and multidimensional hyperbolic equation with Dirichlet conditions are considered.The stability estimates for the solutions of these difference schemes for the nonlocal boundaryvalue hyperbolic problem are established. Finally, a numerical method proposed and numericalexperiments, analysis of the errors, and related execution times are presented in order to verifytheoretical statements.

1. Introduction

Hyperbolic partial differential equations play an important role in many branches of scienceand engineering and can be used to describe a wide variety of phenomena such as acoustics,electromagnetics, hydrodynamics, elasticity, fluid mechanics, and other areas of physics (see[1–5] and the references given therein).

While applying mathematical modelling to several phenomena of physics, biology,and ecology, there often arise problems with nonclassical boundary conditions, which thevalues of unknown function on the boundary are connected with inside of the given domain.Such type of boundary conditions are called nonlocal boundary conditions. Over the lastdecades, boundary value problems with nonlocal boundary conditions have become arapidly growing area of research (see, e.g., [6–16] and the references given therein).


In the present work, we consider the nonlocal boundary value problem

d2u(t)dt2

+Au(t) = f(t) (0 ≤ t ≤ 1),

u(0) =n∑

j=1

αju(λj)+ ϕ, ut(0) =

n∑

j=1

βjut(λj)+ ψ,

0 < λ1 < λ2 < · · · < λn ≤ 1,

(1.1)

where A is a self-adjoint positive definite operator in a Hilbert spaceH.A function u(t) is called a solution of the problem (1.1), if the following conditions are

satisfied:

(i) u(t) is twice continuously differentiable on the segment [0, 1]. The derivatives at theendpoints of the segment are understood as the appropriate unilateral derivatives.

(ii) The element u(t) belongs toD(A), independent of t, and dense inH for all t ∈ [0, 1]and the function Au(t) is continuous on the segment [0, 1].

(iii) u(t) satisfies the equation and nonlocal boundary conditions (1.1).

In the paper of [8], the following theorem on the stability estimates for the solution ofthe nonlocal boundary value problem (1.1) was proved.

Theorem 1.1. Suppose that ϕ ∈ D(A), ψ ∈ D(A1/2), and f(t) is a continuously differentiablefunction on [0, 1] and the assumption

n∑

k=1

∣∣αk + βk∣∣ +

n∑

m=1

|αm|n∑

k=1k /=m

∣∣βk∣∣ <

∣∣∣∣∣1 +n∑

k=1

αkβk

∣∣∣∣∣ (1.2)

holds. Then, there is a unique solution of problem (1.1) and the stability inequalities

max0≤t≤1

‖u(t)‖H ≤M[∥∥ϕ∥∥H +∥∥∥A−1/2ψ

∥∥∥H+max

0≤t≤1

∥∥∥A−1/2f(t)∥∥∥H

],

max0≤t≤1

∥∥∥A1/2u(t)∥∥∥H

≤M[∥∥∥A1/2ϕ

∥∥∥H+∥∥ψ∥∥H +max

0≤t≤1

∥∥f(t)∥∥H

],

max0≤t≤1

∥∥∥∥∥d2u(t)dt2

∥∥∥∥∥H

+max0≤t≤1

‖Au(t)‖H ≤M[∥∥Aϕ

∥∥H +∥∥∥A1/2ψ

∥∥∥H+∥∥f(0)

∥∥H +∫1

0

∥∥f ′(t)∥∥Hdt

]

(1.3)

hold, whereM does not depend on ϕ, ψ, and f(t), t ∈ [0, 1].

Moreover, the first order of accuracy difference scheme

τ−2(uk+1 − 2uk + uk−1) +Auk+1 = fk, fk = f(tk),

tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,


u0 =n∑

r=1

αru[λr/τ] + ϕ,

τ−1(u1 − u0) =n∑

r=1

βr(u[λr/τ]+1 − u[λr/τ]

)1τ+ ψ,

(1.4)

for the approximate solution of problem (1.1) was presented. The stability estimates for thesolution of this difference scheme, under the assumption

n∑

k=1

|αk| +n∑

k=1

∣∣βk∣∣ +

n∑

k=1

|αk|n∑

k=1

∣∣βk∣∣ < 1, (1.5)

were established.In the development of numerical techniques for solving PDEs, the stability has been

an important research topic (see [6–31]). A large cycle of works on difference schemes forhyperbolic partial differential equations, in which stability was established under theassumption that the magnitude of the grid steps τ and h with respect to the time and spacevariables, are connected. In abstract terms, this particularly means that τ‖Ah‖ → 0 whenτ → 0.

We are interested in studying the high order of accuracy difference schemes forhyperbolic PDEs, in which stability is established without any assumption with respect to thegrid steps τ and h. Particularly, a convenient model for analyzing the stability is provided bya proper unconditionally absolutely stable difference scheme with an unbounded operator.

In the present paper, the second order of accuracy unconditionally stable differenceschemes for approximately solving boundary value problem (1.1) is presented. The stabilityestimates for the solutions of these difference schemes and their first and second orderdifference derivatives are established. This operator approach permits one to obtain the sta-bility estimates for the solutions of difference schemes of nonlocal boundary value problems,for one-dimensional hyperbolic equation with nonlocal boundary conditions in space var-iable and multidimensional hyperbolic equation with Dirichlet condition in space variables.

Some results of this paper without proof were presented in [7].Note that nonlocal boundary value problems for parabolic equations, elliptic equa-

tions, and equations of mixed types have been studied extensively by many scientists (see,e.g., [11–16, 20–24, 32–38] and the references therein).

2. The Second Order of Accuracy Difference Scheme Generated by A2

Throughout this paper for simplicity λ1 > 2τ and λn < 1 will be considered. Let us associateboundary value problem (1.1) with the second order of accuracy difference scheme

τ−2(uk+1 − 2uk + uk−1) +Auk +τ2

4A2uk+1 = fk,

fk = f(tk), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,


u0 =n∑

m=1

αm

{u[λm/τ] + τ

−1(u[λm/τ] − u[λm/τ]−1) ×(λm −

[λmτ

]τ

)}+ ϕ,

(I +

τ2A

2

)τ−1(u1 − u0) − τ

2(f0 −Au0

)

=n∑

k=1

βk

{τ−1(u[λk/τ] − u[λk/τ]−1

)+(τ

2+(λk −

[λkτ

]τ

))× (f[λk/τ] −Au[λk/τ]

)}+ ψ,

f0 = f(0).

(2.1)

A study of discretization, over time only, of the nonlocal boundary value problem alsopermits one to include general difference schemes in applications, if the differential operatorin space variablesA is replaced by the difference operatorAh that act in the Hilbert space andare uniformly self-adjoint positive definite in h for 0 < h ≤ h0.

In general, we have not been able to obtain the stability estimates for the solutionof difference scheme (2.1) under assumption (1.5). Note that the stability of solution ofdifference scheme (2.1) will be obtained under the strong assumption

n∑

k=1

|αk|(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)+

n∑

k=1

∣∣βk∣∣(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)

+n∑

k=1

|αk|n∑

k=1

∣∣βk∣∣(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)

+n∑

k=1

|αk|∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣n∑

k=1

∣∣βk∣∣∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣ < 1.

(2.2)

Now, let us give some lemmas that will be needed in the sequel.


‖R‖H�→H ≤ 1,∥∥∥R∥∥∥H�→H

≤ 1,∥∥∥R−1R

∥∥∥H�→H

≤ 1,∥∥∥R−1R

∥∥∥H�→H

≤ 1,∥∥∥τA1/2R

∥∥∥H�→H

≤ 1,∥∥∥τA1/2R

∥∥∥H�→H

≤ 1.

(2.3)

Here,H is the Hilbert space, R = (I + iτA1/2 − (τ2/2)A)−1, and R = (I − iτA1/2 − (τ2/2)A)−1.

Lemma 2.2. Suppose that assumption (2.2) holds. Denote

Bτ =n∑

m=1

αm

(I +

τ2A

2

)−1[R[λm/τ]−1(I − iτA1/2) + R[λm/τ]−1(I + iτA1/2)

2

]

+n∑

k=1

βk

(I +

τ2A

2

)−1[R[λk/τ]−1R−1 + R[λk/τ]−1R−1

2

]


− 12

n∑

m=1

n∑

k=1

αmβk

(I +

τ2A

2

)−1[R[λm/τ]−1R[λk/τ]−1 + R[λm/τ]−1R[λk/τ]−1

]

+n∑

m=1

αm

(λm −

[λmτ

]τ

)(iA1/2

)(I +

τ2A

2

)−1

×[R[λm/τ]−1(I − (iτ/2)A1/2)(I + iτA1/2) − R[λm/τ]−1(I + (iτ/2)A1/2)(I − iτA1/2)

2

]

+n∑

k=1

βk

(λk −

[λkτ

]τ

)(iA1/2

)(I +

τ2A

2

)−1[R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

]

+14

n∑

m=1

n∑

k=1

αmβk

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−2

× iA1/2[R[λm/τ]−1R[λk/τ]−1

(I +

iτ

2A1/2

)− R[λm/τ]−1R[λk/τ]−1

(I − iτ

2A1/2

)]

− 12

n∑

m=1

n∑

k=1

αmβk

(λk −

[λkτ

]τ

)(iA1/2

)(I +

τ2A

2

)−1

×[R[λm/τ]−1R[λk/τ]−1 − R[λm/τ]−1R[λk/τ]−1

]

− 12

n∑

m=1

n∑

k=1

αmβk

(λm −

[λmτ

]τ

)(λk −

[λkτ

]τ

)A

(I +

τ2A

2

)−1

×[R[λm/τ]−1R[λk/τ]−1

(I − iτ

2A1/2

)+ R[λm/τ]−1R[λk/τ]−1

(I +

iτ

2A1/2

)].

(2.4)

Then, the operator I − Bτ has an inverse

Tτ = (I − Bτ)−1, (2.5)

and the following estimate holds:

‖Tτ‖H�→H ≤M. (2.6)

Proof. The proof of estimate (2.6) is based on the estimate

‖I − Bτ‖H�→H ≥ 1 −n∑

k=1

|αk|(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)−

n∑

k=1

∣∣βk∣∣(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)

−n∑

k=1

|αk|n∑

k=1

∣∣βk∣∣(1 +∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣)

−n∑

k=1

|αk|∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣n∑

k=1

∣∣βk∣∣∣∣∣∣λk −

[λkτ

]τ

∣∣∣∣.

(2.7)

Estimate (2.7) follows from the triangle inequality and estimate (2.3). Lemma 2.2 is proved.


Now, we will obtain the formula for the solution of problem (2.1). It is easy to showthat (see, e.g., [18]) there is unique solution of the problem

τ−2(uk+1 − 2uk + uk−1) +Auk +τ2

4A2uk+1 = fk,

fk = f(tk), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,(I +

τ2A

2

)τ−1(u1 − u0) − τ

2(f0 −Au0

)= ω, f0 = f(0), u0 = μ,

(2.8)

and the following formula holds:

u0 = μ, u1 =

(I +

τ2A

2

)−1(μ + τω +

τ2

2f0

],

uk =

(I +

τ2A

2

)−1(Rk−1(I − iτA1/2) + Rk−1(I + iτA1/2)

2

)μ

+

(I +

τ2A

2

)−1(iA1/2

)−1(Rk−1R−1 − Rk−1R−1

2

)(ω +

τ

2f0)

−k−1∑

s=1

τ

2iA−1/2

(Rk−s − Rk−s

)fs, 2 ≤ k ≤N.

(2.9)

Applying formula (2.9) and the nonlocal boundary conditions in problem (2.1), we get

μ = Tτ

{[n∑

m=1

αmτ

2f0

×⎛

⎝(I +

τ2A

2

)−1(iA1/2

)−1 × R[λm/τ]−1R−1 − R[λm/τ]−1R−1

2

+n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1

×(R[λm/τ]−1R−1(I − (iτ/2)A1/2) + R[λm/τ]−1R−1(I + (iτ/2)A1/2)

2

)⎞

⎠

−(

n∑

m=1

αm

[λm/τ]−1∑

s=1

τ

2iA−1/2

(R[λm/τ]−s − R[λm/τ]−s

)fs −

n∑

m=1

αm

(λm −

[λmτ

]τ

)

×[λm/τ]−2∑

s=1

τ

2iA1/2

(iA−1/2

)(R[λm/τ]−s

(I − iτ

2A1/2)+ R[λm/τ]−s

(I − iτ

2A1/2

))fs

+n∑

m=1

αmτ

(λm −

[λmτ

]τ

)RRf[λm/τ]−1 − ϕ

)]


×⎡

⎣I +n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A

(I +

τ2A

2

)−1(iA1/2

)−1

×(R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

)−

n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1


2

)]

+

⎡

⎣n∑

m=1

αm

(I +

τ2A

2

)−1(iA1/2

)−1 R[λm/τ]−1R−1 − R[λm/τ]−1R−1

2

+n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1


2

)]

×⎡

⎣τ2f0

⎛

⎝n∑

k=1

βk

(I +

τ2A

2

)−1

×(R[λk/τ]−1R−1(I − (iτ/2)A1/2)

2+R[λk/τ]−1R−1(I + (iτ/2)A1/2)

2

)

−n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A ×(I +

τ2A

2

)−1(iA1/2

)−1

× R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

⎞

⎠ +n∑

k=1

βk

[λk/τ]−2∑

s=1

τ

2i

×A−1/2(iA1/2

)(R[λk/τ]−s

(I +

iτ

2A1/2

)+ R[λk/τ]−s

(I − iτ

2A1/2

))fs

+n∑

k=1

βkτRRf[λk/τ]−1 +n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A

[λk/τ]−1∑

s=1

τ

2iA−1/2

×(R[λk/τ]−s − R[λk/τ]−s

)fs +

n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))f[λk/τ]−1 + ψ

]},

ω = Tτ

⎧⎨

⎩

⎡

⎣I −n∑

m=1

αm

(I +

τ2A

2

)−1

×(R[λm/τ]−1(I − iτA1/2)

2+R[λm/τ]−1(I + iτA1/2)

2

)

−n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1(iA1/2

)


× R[λm/τ]−1(I − (iτ/2)A1/2)(I + iτA1/2) − R[λm/τ]−1(I + (iτ/2)A1/2)(I − iτA1/2)

2

]

×⎡

⎣τ2f0

⎛

⎝n∑

k=1

βk

(I +

τ2A

2

)−1

×(R[λk/τ]−1R−1(I − (iτ/2)A1/2)

2+R[λk/τ]−1R−1(I + (iτ/2)A1/2)

2

)

−n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A ×(I +

τ2A

2

)−1(iA1/2

)−1

× R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

⎞

⎠ +n∑

k=1

βk

[λk/τ]−2∑

s=1

τ

2i

×A−1/2(iA1/2

)(R[λk/τ]−s

(I +

iτ

2A1/2

)+ R[λk/τ]−s

(I − iτ

2A1/2

))fs

+n∑

k=1

βkRRf[λk/τ]−1 +n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A

[λk/τ]−1∑

s=1

τ

2i

×A−1/2(R[λk/τ]−s − R[λk/τ]−s

)fs +(τ

2+(λk −

[λkτ

]τ

))f[λk/τ]−1 + ψ

⎤

⎦

−⎡

⎣n∑

k=1

βkτ

2A

(I +

τ2A

2

)−1R[λk/τ]−1(I − iτA1/2) + R[λk/τ]−1(I + iτA1/2)

2

− 12

n∑

k=1

βk

(I +

τ2A

2

)−1(R[λk/τ]−1

(I − iτ

2A1/2

)(I + iτA1/2

)

−R[λk/τ]−1(I +

iτ

2A1/2

)(I − iτA1/2

))⎤

⎦

×⎡

⎣τ2f0

⎛

⎝n∑

m=1

αm

(I +

τ2A

2

)−1(iA1/2

)−1 R[λm/τ]−1R−1 − R[λm/τ]−1R−1

2

+n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1


2

)⎞

⎠

−(

n∑

m=1

αm

[λm/τ]−1∑

s=1

τ

2iA−1/2

(R[λm/τ]−s − R[λm/τ]−s

)fs +

n∑

m=1

αm

(λm −

[λmτ

]τ

)

×[λm/τ]−2∑

s=1

τ

2iA−1/2

(iA1/2

)[R[λm/τ]−s

(I+

iτ

2A1/2

)+R[λm/τ]−s

(I − iτ

2A1/2

)]fs


+n∑

m=1

αmτ

(λm −

[λmτ

]τ

)RRf[λm/τ]−1 − ϕ

)]}.

(2.10)

Thus, formulas (2.9) and (2.10) give a solution of problem (2.1).

Theorem 2.3. Suppose that assumption (2.2) holds and ϕ ∈ D(A), ψ ∈ D(A1/2). Then, for thesolution of difference scheme (2.1) the stability inequalities

max0≤k≤N

‖uk‖H ≤M{N−1∑

k=0

∥∥∥A−1/2fk∥∥∥Hτ +∥∥∥A−1/2ψ

∥∥∥H+∥∥ϕ∥∥H

}, (2.11)

max0≤k≤N

∥∥∥A1/2uk∥∥∥H

≤M{N−1∑

k=0

∥∥fk∥∥Hτ +

∥∥∥A1/2ϕ∥∥∥H+∥∥ψ∥∥H

}, (2.12)

max1≤k≤N−1

∥∥∥τ−2(uk+1 − 2uk + uk−1)∥∥∥H+ max

0≤k≤N−1

∥∥∥∥∥Auk +τ2

4A2uk+1

∥∥∥∥∥H

≤M{N−1∑

k=1

∥∥fk − fk−1∥∥H +∥∥f0∥∥H +∥∥∥A1/2ψ

∥∥∥H+∥∥Aϕ

∥∥H

} (2.13)

hold, whereM does not depend on τ , ϕ, ψ, and fk, 0 ≤ k ≤N − 1.

Proof. By [18], the following estimates

max0≤k≤N

‖uk‖H ≤M{N−1∑

k=0

∥∥∥A−1/2fk∥∥∥Hτ +∥∥∥A−1/2ω

∥∥∥H+∥∥μ∥∥H

}, (2.14)

max0≤k≤N

∥∥∥A1/2uk∥∥∥H

≤M{N−1∑

k=0

∥∥fk∥∥Hτ +

∥∥∥A1/2μ∥∥∥H+ ‖ω‖H

}, (2.15)

max1≤k≤N−1

∥∥∥τ−2(uk+1 − 2uk + uk−1)∥∥∥H+ max

0≤k≤N−1

∥∥∥∥∥Auk +τ2

4A2uk+1

∥∥∥∥∥H

≤M{N−1∑

k=1

∥∥fk − fk−1∥∥H +∥∥f0∥∥H +∥∥∥A1/2ω

∥∥∥H+∥∥Aμ

∥∥H

} (2.16)

hold for the solution of (2.8). Using formulas of μ,ω, and (2.3) and (2.6) the following estimat-es obtained

∥∥μ∥∥H ≤M

{N−1∑

s=0

∥∥∥A−1/2fs∥∥∥Hτ +∥∥∥A−1/2ψ

∥∥∥H+∥∥ϕ∥∥H

}, (2.17)

∥∥∥A−1/2ω∥∥∥H

≤M{N−1∑

s=0

∥∥∥A−1/2fs∥∥∥Hτ +∥∥∥A−1/2ψ

∥∥∥H+∥∥ϕ∥∥H

}. (2.18)


Estimate (2.11) follows from (2.14), (2.17), and (2.18). In a similar manner, we obtain

max0≤k≤N

∥∥∥A1/2uk∥∥∥H

≤M{N−1∑

k=0

∥∥fk∥∥Hτ +

∥∥∥A1/2ϕ∥∥∥H+∥∥ψ∥∥H

}. (2.19)

Now, we obtain the estimates for ‖Aμ‖H and ‖A1/2ω‖H . First, applyingA to the formula of μand using Abel’s formula, we can write

Aμ = Tτ

⎧⎨

⎩

⎡

⎣τ2f0

⎛

⎝n∑

m=1

αm

(I +

τ2A

2

)−1A1/2 R

[λm/τ]−1R−1 − R[λm/τ]−1R−1

2i

+n∑

m=1

αm

(λm −

[λmτ

]τ

)A

(I +

τ2A

2

)−1

× R[λm/τ]−1R−1(I − (iτ/2)A1/2) + R[λm/τ]−1R−1(I + (iτ/2)A1/2)

2

⎞

⎠

−(

12

n∑

m=1

αm

[λm/τ]−1∑

s=2

(R[λm/τ]−s

(I +

iτ

2A1/2

)−1+ R[λm/τ]−s

(I − iτ

2A1/2

)−1)

×(fs−fs−1)+

n∑

m=1

αm12

(R[λm/τ]−1

(I+

iτ

2A1/2

)−1+R[λm/τ]−1

(I− iτ

2A1/2

)−1)f1

−n∑

m=1

αm

(I +

τ2A

4

)−1f[λm/τ]−1 +

n∑

m=1

αm

(λm −

[λmτ

]τ

)

×[[λm/τ]−2∑

s=2

12iA1/2 ×

(R[λm/τ]−s + R[λm/τ]−s

)(fs − fs−1

)

+12iA1/2

(R[λm/τ]−1 + R[λm/τ]−1

)f1 + iA1/2f[λm/τ]−2

]

+n∑

m=1

αmτ

(λm −

[λmτ

]τ

)ARRf[λm/τ]−1 −Aϕ

)⎤

⎦

×⎡

⎣I +n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A

(I +

τ2A

2

)−1(iA1/2

)−1

×(R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

)−

n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1


2

⎤

⎦


+

⎡

⎣n∑

m=1

αm

(I +

τ2A

2

)−1R[λm/τ]−1R−1 − R[λm/τ]−1R−1

2i+

n∑

m=1

αm

(λm −

[λmτ

]τ

)

×A1/2

(I +

τ2A

2

)−1


2

⎤

⎦

×⎡

⎣τ2f0

⎛

⎝n∑

k=1

βkA1/2

(I +

τ2A

2

)−1

×(R[λk/τ]−1R−1(I − (iτ/2)A1/2)

2+R[λk/τ]−1R−1(I + (iτ/2)A1/2)

2

)

+n∑

k=1

βki

(τ

2+(λk −

[λkτ

]τ

))A×(I +

τ2A

2

)−1

× R[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

⎞

⎠

−n∑

k=1

βk

[λk/τ]−2∑

s=2

12i

(R[λk/τ]−s + R[λk/τ]−s

)(fs − fs−1

)

−n∑

k=1

βk12i

(R[λk/τ]−1 + R[λk/τ]−1

)f1 −

n∑

k=1

βkif[λm/τ]−2

+n∑

k=1

βkA1/2τRRf[λk/τ]−1 +

n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))

×[12A−1/2

[λk/τ]−1∑

s=2

(R[λk/τ]−s

(I +

iτ

2A1/2

)−1+ R[λk/τ]−s

(I − iτ

2A1/2

)−1)

×(fs−fs−1)+12A−1/2

(R[λk/τ]−1

(I+

iτ

2A1/2

)−1+R[λk/τ]−1

(I− iτ

2A1/2

)−1)f1

−A−1/2(I +

τ2A

4

)−1f[λk/τ]−1

⎤

⎦

+n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A1/2f[λk/τ]−1 +A

1/2ψ

⎤

⎦

⎫⎬

⎭.

(2.20)


Second, applying A1/2 to the formula of ω and using Abel’s formula, we can write

A1/2ω = Tτ

⎧⎨

⎩

⎡

⎣I −n∑

m=1

αm

(I +

τ2A

2

)−1

×(R[λm/τ]−1(I − iτA1/2)

2+R[λm/τ]−1(I + iτA1/2)

2

)

−n∑

m=1

αm

(λm −

[λmτ

]τ

)(I +

τ2A

2

)−1(iA1/2

)

× R[λm/τ]−1(I−(iτ/2)A1/2)(I+iτA1/2)−R[λm/τ]−1(I+(iτ/2)A1/2)(I−iτA1/2)

2

⎤

⎦

×⎡

⎣12f0

⎛

⎝n∑

k=1

βkA1/2τ

(I +

τ2A

2

)−1

×(R[λk/τ]−1R−1(I − (iτ/2)A1/2)

2+R[λk/τ]−1R−1(I + (iτ/2)A1/2)

2

)

+n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))(I +

τ2A

2

)−1

×A1/2iR[λk/τ]−1R−1 − R[λk/τ]−1R−1

2

⎞

⎠

−n∑

k=1

βk

[λk/τ]−2∑

s=2

12i

(R[λk/τ]−s + R[λk/τ]−s

)

× (fs − fs−1) −

n∑

k=1

βk12i

(R[λk/τ]−1 + R[λk/τ]−1

)f1 −

n∑

k=1

βkif[λk/τ]−2

+n∑

k=1

βkA1/2τRRf[λk/τ]−1 +

n∑

k=1

βk

(τ

2+(λk −

[λkτ

]τ

))A−1/2

×⎡

⎣12

[λk/τ]−1∑

s=2

(R[λk/τ]−s

(I +

iτ

2A1/2

)−1+ R[λk/τ]−s

(I − iτ

2A1/2

)−1)

×(fs−fs−1)+12

(R[λk/τ]−1

(I+

iτ

2A1/2

)−1+R[λk/τ]−1

(I− iτ

2A1/2

)−1)f1

−(I +

τ2A

4

)−1f[λk/τ]−1

⎤

⎦ +A1/2ψ

⎤

⎦


−⎡

⎣n∑

k=1

βkA1/2 τ

2

(I +

τ2A

2

)−1×(R[λk/τ]−1(I − iτA1/2) + R[λk/τ]−1(I + iτA1/2)

2

)

−n∑

k=1

βkA−1/2(I +

τ2A

2

)−1

×(R[λk/τ]−1(I − (iτ/2)A1/2)(I + iτA1/2)

2

−R[λk/τ]−1(I + (iτ/2)A1/2)(I − iτA1/2)

2

)]

×⎡

⎣12f0

⎛

⎝n∑

m=1

αm

(I +

τ2A

2

)−1× τA1/2 R

[λm/τ]−1R−1 − R[λm/τ]−1R−1

2i

+n∑

m=1

αm

(λm −

[λmτ

]τ

)A

(I +

τ2A

2

)−1


2

))

−(

12

n∑

m=1

αmA−1/2

×[λm/τ]−1∑

s=2

(R[λm/τ]−s

(I +

iτ

2A1/2

)−1+ R[λm/τ]−s

(I − iτ

2A1/2

)−1)

× (fs − fs−1)+12

n∑

m=1

αmA−1/2

×(R[λm/τ]−1

(I +

iτ

2A1/2

)−1+ R[λm/τ]−1

(I − iτ

2A1/2

)−1)f1

−n∑

m=1

αmA−1/2(I +

τ2A

4

)−1f[λm/τ]−1 −

n∑

m=1

αm

(λm −

[λmτ

]τ

)

×[[λm/τ]−2∑

s=2

12iA1/2 ×

(R[λm/τ]−s + R[λm/τ]−s

)(fs − fs−1

)

+12iA1/2

(R[λm/τ]−1 + R[λm/τ]−1

)f1 + iA1/2f[λm/τ]−2

]

+n∑

m=1

αmτ

(λm −

[λmτ

]τ

)ARRf[λm/τ]−1 −Aϕ

)⎤

⎦

⎫⎬

⎭.

(2.21)

The following estimates


∥∥Aμ∥∥H ≤M

{N−1∑

s=1

∥∥fs − fs−1∥∥H +∥∥f0∥∥H +∥∥∥A1/2ψ

∥∥∥H+∥∥Aϕ

∥∥H

},

∥∥∥A1/2ω∥∥∥H

≤M{N−1∑

s=1

∥∥fs − fs−1∥∥H +∥∥f0∥∥H +∥∥∥A1/2ψ

∥∥∥H+∥∥Aϕ

∥∥H

} (2.22)

are obtained by using formulas (2.20), (2.21), (2.3), and (2.6).Estimate (2.13) follows from (2.16), and (2.22). Theorem 2.3 is proved.Now, let us consider the applications of Theorem 2.3. First, the nonlocal the mixed

boundary value problem for hyperbolic equation

utt − (a(x)ux)x + δu = f(t, x), 0 < t < 1, 0 < x < 1,

u(0, x) =n∑

m=1

αmu(λm, x) + ϕ(x), 0 ≤ x ≤ 1,

ut(0, x) =n∑

k=1

βkut(λk, x) + ψ(x), 0 ≤ x ≤ 1,

u(t, 0) = u(t, 1), ux(t, 0) = ux(t, 1), 0 ≤ t ≤ 1,

(2.23)

under assumption (2.2), is considered. Here ar(x), (x ∈ (0, 1)), ϕ(x), ψ(x) (x ∈ [0, 1]) andf(t, x) (t ∈ (0, 1), x ∈ (0, 1)) are given smooth functions and ar(x) ≥ a > 0, δ > 0. Thediscretization of problem (2.23) is carried out in two steps.

In the first step, the grid space is defined as follows:

[0, 1]h = {x : xr = rh, 0 ≤ r ≤ K,Kh = 1}. (2.24)

We introduce the Hilbert space L2h = L2([0, 1]h), W12h = W1

2h([0, 1]h) and W22h = W2

2h([0, 1]h)of the grid functions ϕh(x) = {ϕr}K−1

1 defined on [0, 1]h, equipped with the norms

∥∥∥ϕh∥∥∥L2h

=

(K−1∑

r=1

∣∣∣ϕh(x)∣∣∣2h

)1/2

,

∥∥∥ϕh∥∥∥W1

2h

=∥∥∥ϕh∥∥∥L2h

+

(K−1∑

r=1

∣∣∣∣(ϕh)

x,j

∣∣∣∣2

h

)1/2

,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh∥∥∥L2h

+

(K−1∑

r=1

∣∣∣∣(ϕh)

x,j

∣∣∣∣2

h

)1/2

+

(K−1∑

r=1

∣∣∣∣(ϕh)

xx,j

∣∣∣∣2

h

)1/2

,

(2.25)

respectively. To the differential operator A generated by problem (2.23), we assign the differ-ence operator Ax

h by the formula

Axhϕ

h(x) ={−(a(x)ϕx)x,r + δϕr

}K−11 , (2.26)

acting in the space of grid functions ϕh(x) = {ϕr}K0 satisfying the conditions ϕ0 = ϕK, ϕ1−ϕ0 =ϕK − ϕK−1. With the help of Ax

h, we arrive at the nonlocal boundary value problem


d2vh(t, x)dt2

+Axhv

h(t, x) = fh(t, x), 0 ≤ t ≤ 1, x ∈ [0, 1]h,

vh(0, x) =n∑

j=1

αjvh(λj , x

)+ ϕh(x), x ∈ [0, 1]h,

vht (0, x) =n∑

j=1

βjvht

(λj , x)+ ψh(x), x ∈ [0, 1]h,

(2.27)

for an infinite system of ordinary differential equations.In the second step, we replace problem (2.27) by difference scheme (2.28)

uhk+1(x) − 2uhk(x) + uhk−1(x)

τ2+Ax

huhk(x) +

τ2

4(Axh

)2uhk+1(x) = f

hk (x), x ∈ [0, 1]h,

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

uh0(x)=n∑

j=1

αj

{uh[λj/τ](x)+τ

−1(uh[λj/τ](x)−u

h[λj/τ]−1(x)

)(λj−[λj

τ

]τ

)}+ϕh(x), x ∈ [0, 1]h,

(I +

τ2Axh

2

)uh1(x) − uh0(x)

τ− τ

2

(fh0 (x) −Ax

huh0(x))

=n∑

j=1

βj

⎧⎨

⎩

uh[λj/τ](x) − uh

[λj/τ]−1(x)

τ+

(τ

2+

(λj −[λj

τ

]τ

))×(fh[λj/τ](x) −A

xhu

h[λj/τ]

(x))⎫⎬

⎭

+ ψh(x),

fh0 (x) = fh(0, x), x ∈ [0, 1]h.

(2.28)

Theorem 2.4. Let τ and h be sufficiently small positive numbers. Suppose that assumption (2.2)holds. Then, the solution of difference scheme (2.28) satisfies the following stability estimates:

max0≤k≤N


+ max0≤k≤N

∥∥∥(uhk

)

x

∥∥∥L2h

≤M1

[max

0≤k≤N−1


+∥∥∥ψh∥∥∥L2h

+∥∥∥ϕhx∥∥∥L2h

],

max1≤k≤N−1

∥∥∥τ−2(uhk+1 − 2uhk + u

hk−1)∥∥∥

L2h+ max

0≤k≤N

∥∥∥(uhk

)

xx

∥∥∥L2h

≤M1

[∥∥∥fh0∥∥∥L2h

+ max1≤k≤N−1

∥∥∥τ−1(fhk − fhk−1

)∥∥∥L2h

+∥∥∥ψhx∥∥∥L2h

+∥∥∥(ϕhx

)

x

∥∥∥L2h

].

(2.29)

Here,M1 does not depend on τ , h, ϕh(x), ψh(x) and fhk, 0 ≤ k < N.

The proof of Theorem 2.4 is based on abstract Theorem 2.3 and symmetry propertiesof the operator Ax

hdefined by (2.26).


Second, for them-dimensional hyperbolic equation under assumption (2.2) is consider-ed. Let Ω be the unit open cube in the m-dimensional Euclidean space R

m{x = (x1, . . . , xm) :0 < xj < 1, 1 ≤ j ≤ m} with boundary S, Ω = Ω ∪ S. In [0, 1] ×Ω, the mixed boundary valueproblem for the multidimensional hyperbolic equation

∂2u(t, x)∂t2

−m∑

r=1

(ar(x)uxr )xr = f(t, x),

x = (x1, . . . , xm) ∈ Ω, 0 < t < 1,

u(0, x) =n∑

j=1

αju(λj , x)+ ϕ(x), x ∈ Ω;

ut(0, x) =n∑

k=1

βkut(λk, x) + ψ(x), x ∈ Ω;

u(t, x) = 0, x ∈ S(2.30)

is considered.Here, ar(x), (x ∈ Ω), ϕ(x), ψ(x) (x ∈ Ω) and f(t, x) (t ∈ (0, 1), x ∈ Ω) are given

smooth functions and ar(x) ≥ a > 0. The discretization of problem (2.30) is carried out in twosteps. In the first step, let us define the grid sets

Ωh ={x = xr = (h1r1, . . . , hmrm), r = (r1, . . . , rm), 0 ≤ rj ≤Nj, hjNj = 1, j = 1, . . . , m

},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.(2.31)

We introduce the Banach space L2h = L2(Ωh), W12h =W1

2h(Ωh) andW22h =W2

2h(Ωh) of the gridfunctions ϕh(x) = {ϕ(h1r1, . . . , hmrm)} defined on Ωh, equipped with the norms

∥∥∥ϕh∥∥∥L2(Ωh)

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hm

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W1

2h

=∥∥∥ϕh∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

m∑

r=1

∣∣∣∣(ϕh)

xr ,jr

∣∣∣∣2

h1 · · ·hm⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

m∑

r=1

∣∣∣∣(ϕh)

xr

∣∣∣∣2

h1 · · ·hm⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

m∑

r=1

∣∣∣∣(ϕh)

xrxr ,jr

∣∣∣∣2

h1 · · ·hm⎞

⎠1/2

,

(2.32)


respectively. To the differential operator A generated by problem (2.27), we assign the differ-ence operator Ax

hby the formula

Axhu

h = −m∑

r=1

(ar(x)uhxr

)

xr ,jr, (2.33)

acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ Sh.Note that Ax

his a self-adjoint positive definite operator in L2(Ωh). With the help of Ax

hwe

arrive at the nonlocal boundary value problem

d2vh(t, x)dt2

+Axhv

h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,

vh(0, x) =n∑

l=1

αlvh(λl, x) + ϕh(x), x ∈ Ωh,

dvh(0, x)dt

=n∑

l=1

βlvht (λl, x) + ψ

h(x), x ∈ Ω,

(2.34)

for an infinite system of ordinary differential equations.In the second step, we replace problem (2.34) by difference scheme (2.35)

uhk+1(x) − 2uh

k(x) + uhk−1(x)

τ2+Ax

huhk(x) +

τ2

4(Axh

)2uhk+1(x) = f

hk (x), x ∈ Ωh,

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

uh0(x) =n∑

l=1

αl{uh[λl/τ](x) + τ

−1(uh[λl/τ](x) − u

h[λl/τ]−1(x)

)(λl − [λl/τ]τ)

}+ ϕh(x), x ∈ Ωh,

(I +

τ2Axh

2

)uh1(x) − uh0(x)

τ− τ

2

(fh0 (x) −Ax

huh0(x))

=n∑

l=1

βl

⎧⎨

⎩uh[λl/τ](x) − u

h[λl/τ]−1(x)

τ+(τ

2+(λl −[λlτ

]τ

))×(fh[λl/τ](x) −A

xhu

h[λl/τ]

(x))⎫⎬

⎭

+ ψh(x),

fh0 (x) = fh(0, x), x ∈ Ωh.

(2.35)


Theorem 2.5. Let τ and h be sufficiently small positive numbers. Suppose that assumption (2.2)holds. Then, the solution of difference scheme (2.35) satisfies the following stability estimates:

max0≤k≤N


+ max0≤k≤N

m∑

r=1

∥∥∥∥(uhk

)

xr ,jr

∥∥∥∥L2h

≤M1

[max

0≤k≤N−1


+∥∥∥ψh∥∥∥L2h

+m∑

r=1

∥∥∥ϕhxr ,jr∥∥∥L2h

],

max1≤k≤N−1

∥∥∥τ−2(uhk+1 − 2uhk + u

hk−1)∥∥∥

L2h+ max

0≤k≤N

m∑

r=1

∥∥∥∥(uhk

)

xrxr ,jr

∥∥∥∥L2h

≤M1

[∥∥∥fh0∥∥∥L2h

+ max1≤k≤N−1

∥∥∥τ−1(fhk − fhk−1

)∥∥∥L2h

+m∑

r=1

∥∥∥ψhxr ,jr∥∥∥L2h

+m∑

r=1

∥∥∥ϕhxrxr ,jr∥∥∥L2h

].

(2.36)

Here,M1 does not depend on τ , h, ϕh(x), ψh(x) and fhk, 0 ≤ k < N.

The proof of Theorem 2.5 is based on abstract Theorem 2.3, symmetry properties of theoperatorAx

hdefined by formula (2.33), and the following theorem on the coercivity inequality

for the solution of the elliptic difference problem in L2h.

Theorem 2.6. For the solutions of the elliptic difference problem

Axhu

h(x) = ωh(x), x ∈ Ωh,

uh(x) = 0, x ∈ Sh,(2.37)


m∑

r=1

∥∥∥uhxrxr ,jr∥∥∥L2h

≤M∥∥∥ωh∥∥∥L2h. (2.38)

3. The Second Order of Accuracy Difference Scheme Generated by A

Note that the difference scheme (2.1) generated by A2 is based on the second order approxi-mation formula for the differential equation u′′(t) +Au(t) = f(t) and second order of approx-imation formulas for nonlocal boundary conditions. Using the differential equation above,


one can construct the second order of approximation formula for this differential equation.Thus, we obtain the following difference scheme:

τ−2(uk+1 − 2uk + uk−1) +12Auk +

14A(uk+1 + uk−1) = fk,

fk = f(tk), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

u0 =n∑

m=1

αm

{u[λm/τ] + τ

−1(u[λm/τ] − u[λm/τ]−1) ×(λm −

[λmτ

]τ

)}+ ϕ,

(I +

τ2A

4

)[(I +

τ2A

4

)τ−1(u1 − u0) − τ

2(f0 −Au0

)]

=n∑

k=1

βk

{τ−1(u[λk/τ] − u[λk/τ]−1

)+(τ

2+(λk −

[λkτ

]τ

))× (f[λk/τ] −Au[λk/τ]

)}+ ψ,

f0 = f(0).(3.1)

In exactly the same manner using the method of Section 2 one proves

Theorem 3.1. Suppose that assumption (2.2) holds and ϕ ∈ D(A), ψ ∈ D(A1/2). Then, for thesolution of difference scheme (3.1) the following stability inequalities

max0≤k≤N

‖uk‖H ≤M{N−1∑

k=0

∥∥∥A−1/2fk∥∥∥Hτ +∥∥∥A−1/2ψ

∥∥∥H+∥∥ϕ∥∥H

}, (3.2)

max0≤k≤N

∥∥∥A1/2uk∥∥∥H

≤M{N−1∑

k=0

∥∥fk∥∥Hτ +

∥∥∥A1/2ϕ∥∥∥H+∥∥ψ∥∥H

}, (3.3)

max1≤k≤N−1

∥∥∥τ−2(uk+1 − 2uk + uk−1)∥∥∥H+ max

1≤k≤N−1

∥∥∥∥12Auk +

14A(uk+1 + uk−1)

∥∥∥∥H

≤M{N−1∑

k=1

∥∥fk − fk−1∥∥H +∥∥f0∥∥H +∥∥∥A1/2ψ

∥∥∥H+∥∥Aϕ

∥∥H

} (3.4)

hold, whereM does not depend on τ , ϕ, ψ and fk, 0 ≤ k ≤N − 1.

Moreover, we can construct the difference schemes of a second order of accuracy withrespect to one variable for approximate solution of nonlocal boundary value problems (2.30)and (2.34). Therefore, abstract Theorem 3.1 permits us to obtain the stability estimates for thesolution of these difference schemes.


In this section, several numerical examples are presented for the solution of problem (4.1),demonstrating the accuracy of the difference schemes. In general, we have not been able to


determine a sharp estimate for the constants figuring in the stability inequalities. However,the numerical examples are presented for the following nonlocal boundary value problem

∂2u(t, x)∂t2

− ∂2u(t, x)∂x2

+ u(t, x) =(6t + 2t3

)sinx, 0 < t < 1, 0 < x < π,

u(0, x) =14u(1, x) − 1

4u

(12, x

)− 732

sinx, 0 ≤ x ≤ π,

ut(0, x) =14ut(1, x) − 1

4ut

(12, x

)− 916

sinx, 0 ≤ x ≤ π,

u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1.

(4.1)

The exact solution of this problem is u(t, x) = t3 sinx.We consider the set [0, 1]τ × [0, π]h of a family of grid points depending on the small

parameters τ and h: [0, 1]τ × [0, π]h = {(tk, xn) : tk = kτ, 0 ≤ k ≤ N,Nτ = 1, xn =nh, 0 ≤ n ≤ M,Mh = π}. For the approximate solution of problem (4.1), we have appliedthe first order and the two different types of second orders of accuracy difference schemes,respectively. Solving these difference equations we have applied a procedure of modifiedGauss elimination method with respect to nwith matrix coefficients.

First let us consider the first order of accuracy in time difference scheme (1.4) for theapproximate solution of the Cauchy problem (4.1). Using difference scheme (1.4), we obtain

uk+1n − 2ukn + uk−1n

τ2− uk+1n+1 − 2uk+1n + uk+1n−1

h2+ uk+1n = f(tk+1, xn), tk = kτ,

1 ≤ k ≤N − 1, Nτ = 1, xn = nh, 1 ≤ n ≤M − 1, Mh = π,

u0n =14uNn − 1

4u(N/2)+1n − 7

32sinx, 1 ≤ n ≤M − 1,

τ−1(u1n − u0n

)=14τ−1(uNn − uN−1

n

)− 14τ−1(u(N/2)+1n −uN/2n

)− 916

sinx,

xn = nh, 1 ≤ n ≤M − 1, uk0 = ukM = 0, 0 ≤ k ≤N,

(4.2)

for the approximate solution of Cauchy problem (4.1). We have (N + 1) × (N + 1) system oflinear equation in (4.2) and we can write them in the matrix form as

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = 0, UM = 0,(4.3)

where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 0 00 0 a 0 · 0 0 00 0 0 a · 0 0 0· · · · · · · ·0 0 0 0 · 0 a 00 0 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,


B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 · 14

0 · 0 0 0 −14

b c d 0 0 · 0 0 · 0 0 0 00 b c d 0 · 0 0 · 0 0 0 00 0 b c d · 0 0 · 0 0 0 0· · · · · · · · · · · · ·0 0 0 0 0 · 0 0 · 0 0 0 00 0 0 0 0 · 0 0 · b c d 00 0 0 0 0 · 0 0 · 0 b c d

−1 1 0 0 0 · −14

14

· 0 014

−14

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

C = A, D =

⎡⎢⎢⎣

1 0 · 00 1 · 0· · · ·0 0 · 1

⎤⎥⎥⎦

(N+1)×(N+1)

,

ϕkn =

⎡⎢⎢⎢⎣

ϕ0n

ϕ1n...ϕNn

⎤⎥⎥⎥⎦

(N+1)×1

, 0 ≤ k ≤N,

ϕ0n = − 7

32sin(xn), ϕNn = − 9

16sin(xn),

ϕkn =

⎧⎨

⎩

(6(tk+1) + 2(tk+1)3

)sin(xn),

f(tk+1, xn), 1 ≤ k ≤N − 1,

Uks =

⎡⎢⎢⎢⎣

u0su1s...uNs

⎤⎥⎥⎥⎦

(N+1)×1

, 0 ≤ k ≤N, s = n ± 1, n.

(4.4)

Here,

a = − 1h2, b =

1τ2, c =

−2τ2, d =

1τ2

+2h2

+ 1. (4.5)

For the solution of difference equation (4.2), we have applied the modified Gauss eliminationmethod. Therefore, we seek a solution of the matrix equation by using the following iterationformula:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 2, 1, 0, (4.6)


where αj , βj(j = 1, . . . ,M) are (N + 1)× (N + 1) square matrices and γj are (N + 1)× 1 columnmatrices. Now, we obtain formula of αn+1, βn+1,

αn+1 = −(B + Cαn)−1A,


), n = 1, 2, 3, . . .M − 1.

(4.7)

Note that

α1 =

⎡

⎣0 · 0· · ·0 · 0

⎤

⎦

(N+1)×(N+1)

, β1 =

⎡

⎣0 · 0· · ·0 · 0

⎤

⎦

(N+1)×(N+1)

, (4.8)

andUM = 0.Thus, using the formulas and matrices above we obtain the difference scheme first

order of accuracy in t and second order of accuracy in x for approximate solution of nonlocalboundary value problem (4.1).

Second, let us consider the second order of accuracy in time implicit difference scheme(3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain

Uk+1n − 2Uk

n +Uk−1n

τ2− Uk

n+1 − 2Ukn +U

kn−1

2h2− Uk+1

n+1 − 2Uk+1n +Uk+1

n−14h2

− Uk−1n+1 − 2Uk−1

n +Uk−1n−1

4h2

=(6(tk) + 2(tk)3

)sin(xn), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

xn = nh, 1 ≤ n ≤M − 1, Mh = π,

u0n =14uN − 1

4u(N/2)+1 − 7

32sinx, 1 ≤ n ≤M − 1,

(I +

τ2Axh

4

)[(I +

τ2Axh

4

)(uh1(xn) − uh0(xn)

)+τ

2

(fh(0, xn) −Ax

huh0(xn)

)]

=14(2τ)−1

(−uhN−2(xn) + 4uhN−1(xn) − 3uhN(xn)

)

− 14(2τ)−1

(−3uh(N/2)+1(xn)+4uh(N/2)+1(xn)−uh(N/2)+3(xn)

)− 916

sin(xn), 1 ≤ n ≤M − 1,

uk0 = ukM = 0, 0 ≤ k ≤N,

(4.9)


for the approximate solution of problem (4.1). We have again the same linear system (4.3);therefore, we use exactly the same method that we have used for the solution of differenceequation (4.2), but matrices are different as follows:

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 0 0z w z 0 · 0 0 00 z w z · 0 0 0· · · · · · · ·0 0 0 0 · z w z0 0 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · 14

0 0 · 0 0 −14

b c d 0 · 0 0 0 · 0 0 00 b c d · 0 0 0 · 0 0 0· · · · · · · · · · · ·0 0 0 0 · 0 0 0 · 0 0 00 0 0 0 · 0 0 0 · c d 00 0 0 0 · 0 0 0 · b c d

−32τ

42τ −1

2τ 0 · −3

8τ

48τ −1

8τ · −1

8τ

48τ −3

8τ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

C = A, D =

⎡⎢⎢⎣

1 0 · 00 1 · 0· · · ·0 0 · 1

⎤⎥⎥⎦

(N+1)×(N+1)

.

(4.10)

Here,

x =1τ2

+12h2

+14, y = − 2

τ2+

1h2

+12, z =

−14h2

, w = − 12h2

. (4.11)

Thus, using the above matrices we obtain the difference scheme second order of accuracy int and x for approximate solution of nonlocal boundary value problem (4.1).

Next, let us consider the second order of accuracy in time implicit difference scheme(3.1) for the approximate solution of problem (4.1). Using difference scheme (3.1), we obtain


τ2− ukn+1 − 2ukn + u

kn−1

h2+ ukn

+τ2

4

(uk+1n+2 − 4uk+1n+1 + 6uk+1n − 4uk+1n−1 + u

k+1n−2

h4− 2

uk+1n+1 − 2uk+1n + uk+1n−1h2

)

=(6(tk) + 2(tk)3

)sin(xn),


tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, xn = nh, 1 ≤ n ≤M − 1, Mh = π,

u0n =14uN − 1

4u(N/2)+1 − 7

32sinx, 1 ≤ n ≤M − 1,

(I +

τ2Axh

4

)[(I +

τ2Axh

4

)(uh1(xn) − uh0(xn)

)+τ

2

(fh(0, xn) −Ax

huh0(xn)

)]

=14(2τ)−1

(−uhN−2(xn) + 4uhN−1(xn) − 3uhN(xn)

)

− 14(2τ)−1

(−3uh(N/2)+1(xn) + 4uh(N/2)+1(xn) − uh(N/2)+3(xn)

)

− 916

sin(xn), 1 ≤ n ≤M − 1,

uk0 = ukM = 0, 0 ≤ k ≤N.

uk1 =45uk2 −

15uk3 , ukM−1 =

45ukM−2 −

15ukM−3, 0 ≤ k ≤N,

(4.12)

for the approximate solution of problem (4.1). We have (N + 1) × (N + 1) system of linearequation in (4.12) and we can write in the matrix form as

AUn+2 + BUn+1 + CUn +DUn−1 + EUn−2 = Rϕn, 2 ≤ n ≤M − 2,

U0 = UM = 0, Uk3 = 4Uk

2 − 5Uk1 , Uk

M−3 = 4UkM−2 − 5Uk

M−1, 0 ≤ k ≤N,(4.13)

where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 0 00 0 x 0 · 0 0 00 0 0 x · 0 0 0· · · · · · · ·0 0 0 0 · 0 x 00 0 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · 0 0 00 w y 0 · 0 0 00 0 w y · 0 0 0· · · · · · · ·0 0 0 0 · w y 00 0 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,


C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · 14

0 0 · 0 0 −14

v t z 0 · 0 0 0 · 0 0 00 v t z · 0 0 0 · 0 0 0· · · · · · · · · · · ·0 0 0 0 · 0 0 0 · 0 0 00 0 0 0 · 0 0 0 · t z 00 0 0 0 · 0 0 0 · v t z

−32τ

42τ −1

2τ 0 · −3

8τ

48τ −1

8τ · −1

8τ

48τ −3

8τ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

D = B, E = A, R =

⎡⎢⎢⎣

1 0 · 00 1 · 0· · · ·0 0 · 1

⎤⎥⎥⎦

(N+1)×(N+1)

,

Uks =

⎡⎢⎢⎢⎢⎣

u0su1s...uNs

⎤⎥⎥⎥⎥⎦

(N+1)×1

, 0 ≤ k ≤N, s = n ± 2, n ± 1, n.

(4.14)

Here,

x =τ2

4h4, y = −τ

2

h4, z =

1τ2

+3τ2

2h4, t =

−2τ2,

d =1τ2

+2h2

+ 1, v =1τ2, w = − 1

h2.

(4.15)

For the solution of the linear system (4.13), we use the modified variant Gauss eliminationmethod and seek a solution of the matrix equation by the following form:

un = αn+1un+1 + βn+1un+2 + γn+1, n =M − 2, . . . , 2, 1, 0, (4.16)

where αj , βj (j = 1 :M−1) are (N+1)×(N+1) square matrices and γj-s are (N+1)×1 columnmatrices. We obtain the following formulas of αn+1, βn+1, γn+1 from linear system (4.13) byusing formula (4.16):

αn+1 =[C +Dαn + Eαn−1αn + Eβn−1

]−1

× [−B −Dβn − Eαn−1βn],

βn+1 = −[C +Dαn + Eαn−1αn + Eβn−1]−1

A,

γn+1 =[C +Dαn + Eαn−1αn + Eβn−1

]−1

× [Rϕn −Dγn + Eαn−1γn + Eγn−1].

(4.17)


We have α1, β1, γ1, α2, β2, γ2 :

α1 =

⎡

⎣0 · 0· · ·0 · 0

⎤

⎦

(N+1)×(N+1)

, β1 =

⎡

⎣0 · 0· · ·0 · 0

⎤

⎦

(N+1)×(N+1)

,

γ1 =

⎡⎢⎢⎢⎣

00...0

⎤⎥⎥⎥⎦

(N+1)×(1)

, γ2 =

⎡⎢⎢⎢⎣

00...0

⎤⎥⎥⎥⎦

(N+1)×(1)

,

α2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

45

0 · 0

045

· 0

· · · ·0 0 · 4

5

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, β2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

−15

0 · 0

0 −15

· 0

· · · ·0 0 · −1

5

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

UM = 0, UM−1 =[(βM−2 + 5I

) − (4I − αM−2)αM−1]−1[(4I − αM−2)γM−1 − γM−2

].

(4.18)

Thus, using the above matrices we obtain the difference scheme second order of accuracy int and x for approximate solution of nonlocal boundary value problem (4.1).

Matlab is a programming language for numeric scientific computation. One of itscharacteristic features is the use of matrices as the only data type. Therefore, the implementa-tions of numerical examples are carried out by Matlab.

Now, wewill give some numerical results for the solutions of (4.2), (4.9), and (4.12) fordifferent M = N values where N and M are the step numbers for the time and spacevariables, respectively. Note that the grid step numbersN andM in the given examples arechosen equal for clarity and this is not necessary for the stability and solutions of the differ-ence schemes.

The errors are computed by the following formula:

ENM = max1≤k≤M−1

(N−1∑

k=1

∣∣∣u(tk, xn) − ukn∣∣∣2h

)1/2

. (4.19)

Here, u(tk, xn) represents the exact solution and ukn represents the approximate solution at(tk, xn). The errors and the related CPU times are presented in Tables 1 and 2, respectively, forN = M = 20, 30, 40, 80, 300, and 500. The implementations are carried out by MATLAB 7.1software package and obtained by a PC System 64 bit, Pentium (R) Core(TM) i5 CPU, 3.206Hz, 3.19Hz, 4000Mb of RAM.

Let us denote the first order of accuracy difference scheme (4.2) as F, the second orderof accuracy difference scheme generated by three points (4.9) as S1, and the second order ofaccuracy difference scheme generated by five points (4.12) as S2.

In order to get accurate results, CPU times are recorded by running each program 100times for small valuesN =M = 20, 30, 40, 80 and taking the average of the elapsed time.


Table 1: Comparison of errors for approximate solutions.

Difference Scheme 20 30 40 80 300 500F 0.0494 0.0324 0.0241 0.0119 0.0032 0.0019S1 0.0020 0.0008 0.0004918 0.0001230 0.0000087 0.0000031S2 0.0031 0.0015 0.0007875 0.0001971 0.0000140

Table 2: CPU times.

Difference Scheme 20 30 40 80 300 500F 0.0080 0.0261 0.0542 0.4518 69.384 545.411S1 0.0086 0.0267 0.0572 0.4631 69.933 532.852S2 0.0112 0.0441 0.1042 0.9842 169.880

The following conclusions can be noted for the comparison of the numerical resultspresented in the tables above.

(i) In the tables, it is noted that almost the same accuracy is achieved by S1 with dataerror = 0.0020,N = 20 and by F with data error = 0.0019,N = 500 in different CPUtimes; 545 s and 0.0086 s, respectively. This means the use of the difference schemeS1 accelerates the computation with a ratio of more than 545/0.0086 ∼= 63372 times,that is, S1 is considerably faster than F.

(ii) It is also noted that almost the same accuracy is achieved by the difference schemeS2 with data error = 0.0031,N = 20, and by the difference scheme F with dataerror = 0.0032,N = 300 in different CPU times; 0.0112 s and 69 s, respectively. Thus,the use of the difference scheme S2 accelerates the computation with a ratio of morethan 69/0.0112 ∼= 6160 times, that is, S2 is considerably faster than F.

(iii) When we consider almost the same CPU times for F and S1 as 0.0080 s ∼=0.0086 s, F computes the solution with an error = 0.0494, for N = 20, where thedifference scheme S1 computes the solution with an error = 0.0020, which is almost25 times smaller error than the computation error of F. Namely, S1 yields 25 timesmore accurate results than F. For S2, this ratio reduces to 0.0112/0.0080 ∼= 1.4 times.

(iv) While both types of the second order difference schemes reach approximately thesame accuracy, the CPU time for the difference scheme S2 is always greater thanS1.

(v) The CPU times of difference schemes F, S1, and S2 recorder for N ≥ 100 exceedones. Matlab gives “out of memory” error for S2 with valuesN = 500 which meansthe memory of the computer is not enough. It is not necessary to have results for S2when N = 500, since it is obvious that CPU time of S2 is more than that of F andS1.

(vi) It is observed from the tables that for largerN values the numerical results becomeapproximately the same for each difference scheme in the reliable range of the CPUtimes. This indicates that the approximation made for the solution of problem (4.1)is valid.

In conclusion, the second order difference schemes are much more accurate than thefirst order difference scheme, and the second order difference scheme generated by threepoints is more convenient than the second order difference scheme generated by five points


when considering the CPU times and the error levels. Comparing with many other numericalmethods, our method is not based on the relationship between the grid step sizes of time andspace variables (see [25–30] and the references therein).

Acknowledgment

The authors would like to thank Professor P. E. Sobolevskii for his helpful suggestions to theimprovement of this paper.

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Research ArticleClassification of Exact Solutions for SomeNonlinear Partial Differential Equations withGeneralized Evolution

Yusuf Pandir,1 Yusuf Gurefe,1, 2

Ugur Kadak,1, 3 and Emine Misirli2

1 Department of Mathematics, Faculty of Science, Bozok University, 66100 Yozgat, Turkey2 Department of Mathematics, Faculty of Science, Ege University, 35100 Bornova-Izmir, Turkey3 Department of Mathematics, Faculty of Science, Gazi University, 06500 Teknikokullar-Ankara, Turkey

Correspondence should be addressed to Yusuf Gurefe, [email protected]

Received 13 March 2012; Accepted 17 May 2012


Copyright q 2012 Yusuf Pandir et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We obtain the classification of exact solutions, including soliton, rational, and elliptic solutions,to the one-dimensional general improved Camassa Holm KP equation and KdV equation by thecomplete discrimination system for polynomial method. In discussion, we propose a more generaltrial equation method for nonlinear partial differential equations with generalized evolution.

1. Introduction

To construct exact solutions to nonlinear partial differential equations, some importantmethods have been defined such as Hirota method, tanh-coth method, the exponentialfunction method, (G′/G)-expansion method, the trial equation method, [1–15]. There area lot of nonlinear evolution equations that are integrated using the various mathematicalmethods. Soliton solutions, compactons, singular solitons, and other solutions have beenfound by using these approaches. These types of solutions are very important and appearin various areas of applied mathematics.

In Section 2, we give a new trial equation method for nonlinear evolution equationswith higher-order nonlinearity. In Section 3, as applications, we obtain some exact solutionsto two nonlinear partial diffeential equations such as the one-dimensional general improvedCamassa Holm KP equation [16]:

(ut + 2kux − uxxt + aun(un)x)x + uyy = 0, (1.1)


the dimensionless form of the generalized KdV equation [17]:

ut + aunux + bu2nux + δuxxx = 0. (1.2)

In discussion, we propose a more general trial equation method.

2. The Extended Trial Equation Method

Step 1. For a given nonlinear partial differential equation,

P(u, ut, ux, uxx, . . .) = 0, (2.1)

take the general wave transformation:

u(x1, x2, . . . , xN, t) = u(η), η = λ

⎛

⎝N∑

j=1

xj − ct⎞

⎠, (2.2)

where λ/= 0 and c /= 0. Substituting (2.2) into (2.1) yields a nonlinear ordinary differentialequation:

N(u, u′, u′′, . . .

)= 0. (2.3)

Step 2. Take the finite series and trial equation as follows:

u =δ∑

i=0

τiΓi, (2.4)

where

(Γ′)2 = Λ(Γ) =

Φ(Γ)Ψ(Γ)

=ξθΓθ + · · · + ξ1Γ + ξ0ζεΓε + · · · + ζ1Γ + ζ0

. (2.5)

Using (2.4) and (2.5), we can write

(u′)2 =

Φ(Γ)Ψ(Γ)

(δ∑

i=0

iτiΓi−1)2

,

u′′ =Φ′(Γ)Ψ(Γ) −Φ(Γ)Ψ′(Γ)

2Ψ2(Γ)

(δ∑

i=0

iτiΓi−1)

+Φ(Γ)Ψ(Γ)

(δ∑

i=0

i(i − 1)τiΓi−2),

(2.6)


where Φ(Γ) and Ψ(Γ) are polynomials. Substituting these relations into (2.3) yields anequation of polynomial Ω(Γ) of Γ:

Ω(Γ) = sΓs + · · · + 1Γ + 0 = 0. (2.7)

According to the balance principle, we can find a relation of θ, ε, and δ. We can computesome values of θ, ε, and δ.

Step 3. Let the coefficients of Ω(Γ) all be zero will yield an algebraic equations system:

i = 0, i = 0, . . . , s. (2.8)

Solving this system, we will determine the values of ξ0, . . . , ξθ, ζ0, . . . , ζε, and τ0, . . . , τδ.

Step 4. Reduce (2.5) to the elementary integral form:

±(η − η0)=∫

dΓ√Λ(Γ)

=∫ √

Ψ(Γ)Φ(Γ)

dΓ. (2.9)

Using a complete discrimination system for polynomial to classify the roots ofΦ(Γ), we solve(2.9) and obtain the exact solutions to (2.3). Furthermore, we can write the exact travelingwave solutions to (2.1), respectively.

3. Applications

Example 3.1 (Application to the Camassa Holm KP equation). In order to look for travellingwave solutions of (1.1), wemake the transformation u(x, y, t) = u(η), η = m(x+y−ct), wherem and c are arbitrary constants. Then, integrating this equation with respect to η twice andsetting the integration constant to zero, we obtain

(2k + 1 − c)u +a

2u2n +m2cu′′ = 0. (3.1)

We use the following transformation:

u = v1/(2n−1). (3.2)

Equation (3.1) turns into the equation

m2c(2n − 1)vv′′ + 2m2c(1 − n)v′2 + (2k + 1 − c)(2n − 1)2v2 +a

2(2n − 1)2v3 = 0. (3.3)


Substituting (2.6) into (3.3) and using balance principle yield

θ = ε + δ + 2. (3.4)

After this solution procedure, we obtain the results as follows.

Case 1. If we take ε = 0, δ = 1, and θ = 3, then

(v′)2 =

(τ1)2(ξ3Γ3 + ξ2Γ2 + ξ1Γ + ξ0

)

ζ0, (3.5)

where ξ3 /= 0, ζ0 /= 0. Respectively, solving the algebraic equation system (2.8) yields

ξ1 =−a(1 − 2n)2ζ0τ30 +m2ξ0τ

21 (2 + 4k + 4n + 8kn + 3aτ0)

m2τ0τ1(q + aτ0

) ,

ξ2 =−2a(1 − 2n)2ζ0τ30 +m2τ21

(q + 3aτ0

)

m2τ20(q + aτ0

) ,

ξ3 =aτ1(−(1 − 2n)2ζ0τ20 +m2ξ0τ

21

)

m2τ20(q + aτ0

) , ξ0 = ξ0, ζ0 = ζ0, τ0 = τ0, τ1 = τ1,

c =(1 − 2n)2ζ0τ20

(q + aτ0

)

(1 + 2n)((1 − 2n)2ζ0τ20 −mξ0τ21

) ,

(3.6)

where q = (1 + 2k)(1 + 2n). Substituting these results into (2.5) and (2.9), we have

± (η − η0)

=m√A

∫dΓ

√√√√√Γ3+−2a(1−2n)2ζ0τ30 +m2τ21

(q + 3aτ0

)

aτ1(−(1 − 2n)2ζ0τ20 +m2ξ0τ

21

) Γ2+AΓ+m2ξ0τ

20

(q + aτ0

)

aτ1(−(1−2n)2ζ0τ20 +m2ξ0τ

21

)

,

(3.7)


whereA denotes by ((−a(1 − 2n)2ζ0τ40+m2ξ0τ0τ

21 (2+4k+4n+8kn+3aτ0))/aτ

21 (−(1 − 2n)2ζ0τ20+

m2ξ0τ21 ))), and A = ζ0τ

20 (q + aτ0)/aτ1(−(1 − 2n)2ζ0τ20 + m2ξ0τ

21 ). Integrating (3.7), we obtain

the solutions to the (1.1) as follows:

±(η − η0)= −2m

√A

1√Γ − α1

, (3.8)

±(η − η0)= 2m

√A

α2 − α1 arctan√

Γ − α2α2 − α1 , α2 > α1, (3.9)

±(η − η0)= m

√A

α1 − α2 ln∣∣∣∣∣

√Γ − α2 −

√α1 − α2√

Γ − α2 +√α1 − α2

∣∣∣∣∣, α1 > α2, (3.10)

±(η − η0)= 2m

√A

α1 − α3F(ϕ, l), α1 > α2 > α3, (3.11)

where

A =ζ0τ

20

(q + aτ0

)

aτ1(−(1 − 2n)2ζ0τ20 +m2ξ0τ

21

) , F(ϕ, l)=∫ϕ

0

dψ√1 − l2sin2ψ

, (3.12)

ϕ = arcsin

√Γ − α3α2 − α3 , l2 =

α2 − α3α1 − α3 .

(3.13)

Also α1, α2, and α3 are the roots of the polynomial equation

Γ3 +ξ2ξ3Γ2 +

ξ1ξ3Γ +

ξ0ξ3

= 0. (3.14)

Substituting the solutions (3.8)–(3.10) into (2.4) and (3.2), we have

u(x, y, t

)=

[τ0 + τ1α1 +

4τ1A(x + y − Bt − η0/m

)2

]1/(2n−1),

u(x, y, t

)=

[τ0 + τ1α2 + τ1(α1 − α2)tanh2

(12

√α1 − α2A

(x + y − Bt + η0

m

))]1/(2n−1),

u(x, y, t

)=

[τ0 + τ1α1 + τ1(α1 − α2)cosech2

(12

√α1 − α2A

(x + y − Bt)

)]1/(2n−1),

(3.15)

where B denote by ((1 − 2n)2ζ0τ20 (q + aτ0)/(1 + 2n)((1 − 2n)2ζ0τ20 −mξ0τ21 )).


If we take τ0 = −τ1α1 and η0 = 0, then the solutions (3.15) can reduce to rationalfunction solution:

u(x, y, t

)=

⎡⎢⎣

2√A

x + y −((1 − 2n)2α21

(q − aτ1α1

)/(1 + 2n)

((1 − 2n)2ζ0α21 −mξ0

))t

⎤⎥⎦

2/(2n−1)

,

(3.16)

1-soliton solution:

u(x, y, t

)=

A1

cosh2/(2n−1)[B(x + y − vt)]

, (3.17)

and singular soliton solution:

u(x, y, t

)=

A2

sinh2/(2n−1)[B(x + y − vt)]

, (3.18)

where A = Aτ1, A1 = (τ1(α2 − α1))1/(2n−1), A2 = (τ1(α1 − α2))1/(2n−1), B = (1/2)√(α1 − α2)/A,

and v = (1 − 2n)2ζ0α21(q − aτ1α1)/(1 + 2n)((1 − 2n)2ζ0α21 − mξ0). Here, A1 and A2 are theamplitudes of the solitons, while v is the velocity and B is the inverse width of the solitons.Thus, we can say that the solitons exist for τ1 > 0.

Case 2. If we take ε = 0, δ = 2 and θ = 4, then

(v′)2 =

(τ1 + 2τ2Γ)2(ξ4Γ4 + ξ3Γ3 + ξ2Γ2 + ξ1Γ + ξ0

)

ζ0, (3.19)

where ξ4 /= 0, ζ0 /= 0. Respectively, solving the algebraic equation system (2.8) yields as follows.Subcase 2.1. It holds that

ξ0 = ξ0, ξ1 =ξ4τ

31

4τ32+4ξ0τ2τ1

, ξ2 =5ξ4τ214τ22

+4ξ0τ22τ21

, ξ3 =2ξ4τ1τ2

, ξ4 = ξ4, τ1 = τ1, τ2 = τ2,

ζ0 = −m2(−16aξ0τ42 + ξ4τ21

(aτ21 + 4τ2q

))

a(1 − 2n)2τ21 τ22

, τ0 =τ214τ2

,

c = 1 + 2k +a(ξ4τ

41 − 16ξ0τ42

)

4(1 + 2n)ξ4τ21 τ2,

(3.20)


where q = (1 + 2k)(1 + 2n). Substituting these results into (2.5) and (2.9), we get

± (η − η0)

= m

√√√√(16aξ0τ42 − ξ4τ41

(aτ21 + 4qτ2

))

a(1 − 2n)2ξ4τ21 τ22

×∫

dΓ√Γ4 + (2τ1/τ2)Γ3 +

((5τ21/4τ

22

)+(4ξ0τ22/τ

21

))Γ2 +

((τ31/4τ

32

)+ (4ξ0τ2/τ1)

)Γ + (ξ0/ξ4)

.

(3.21)

Integrating (3.21), we obtain the solutions to (1.1) as follows:

±(η − η0)= − mB

Γ − α1 ,(3.22)

±(η − η0)=

2mBα1 − α2

√Γ − α2Γ − α1 , α1 > α2, (3.23)

±(η − η0)=

mB

α1 − α2 ln∣∣∣∣Γ − α1Γ − α2

∣∣∣∣, (3.24)

± (η − η0)

=mB

√(α1 − α2)(α1 − α3)

ln

∣∣∣∣∣

√(Γ − α2)(α1 − α3) −

√(Γ − α3)(α1 − α2)√

(Γ − α2)(α1 − α3) +√(Γ − α3)(α1 − α2)

∣∣∣∣∣,

α1 > α2 > α3,

(3.25)

±(η − η0)=

2mB√(α1 − α3)(α2 − α4)

F(ϕ, l), α1 > α2 > α3 > α4, (3.26)

where

B =

√√√√(16aξ0τ42 − ξ4τ41

(aτ21 + 4qτ2

))

a(1 − 2n)2ξ4τ21 τ22

, F(ϕ, l)=∫ϕ

0


, (3.27)

ϕ = arcsin

√(Γ − α1)(α2 − α4)(Γ − α2)(α1 − α4) , l2 =

(α2 − α3)(α1 − α4)(α1 − α3)(α2 − α4) .

(3.28)

Also α1, α2, α3, and α4 are the roots of the polynomial equation:

Γ4 +ξ3ξ4Γ3 +

ξ2ξ4Γ2 +

ξ1ξ4Γ +

ξ0ξ4

= 0. (3.29)


Substituting the solutions (3.22)–(3.25) into (2.4) and (3.2), we have

u(x, y, t

)=

⎡

⎣τ0 + τ1α1 ± τ1B

C − (η0/m) + τ2

(α1 ± B

C − (η0/m))2⎤

⎦1/(2n−1)

,

u(x, y, t

)=

{τ0 + τ1α1 +

4B2(α2 − α1)τ14B2 − [(α1 − α2)

(C − η0)]2

+τ2

(α1 +

4B2(α2 − α1)4B2 − [(α1 − α2)

(C − η0)]2

)2⎫⎬

⎭

1/(2n−1)

,

u(x, y, t

)=

{τ0 + τ1α2 +

(α2 − α1)τ1exp

[((α1 − α2)/B)

(C − η0)] − 1

+τ2

(α2 +

(α2 − α1)exp

[((α1 − α2)/B)

(C − η0)] − 1

)2⎫⎬

⎭

1/(2n−1)

,

u(x, y, t

)=

{τ0 + τ1α1 +

(α1 − α2)τ1exp

[((α1 − α2)/B)

(C − η0)] − 1

+τ2

(α1 +

(α1 − α2)exp

[((α1 − α2)/B)

(C − η0)] − 1

)2⎫⎬

⎭

1/(2n−1)

,

u(x, y, t

)=

⎧⎨

⎩τ0 + τ1α1 − 2(α1 − α2)(α1 − α3)τ12α1 − α2 − α3 + (α3 − α2) cosh

[(√(α1 − α2)(α1 − α3)/B

)(C)

]

+τ2

⎛⎜⎝α1− 2(α1 − α2)(α1 − α3)

2α1−α2−α3+(α3−α2) cosh[(√

(α1−α2)(α1−α3)/B)(C)

]

⎞⎟⎠

2⎫⎪⎬

⎪⎭

1/(2n−1)

,

(3.30)

where C denotes by x + y − (1 + 2k + (a(ξ4τ41 − 16ξ0τ42 )/4(1 + 2n)ξ4τ21 τ2))t.For simplicity, we can write the solutions (3.30) as follows:

u(x, y, t

)=

⎡

⎣2∑

i=0

τi

(α1 ± B

C − (η0/m))i⎤

⎦1/(2n−1)

,


u(x, y, t

)=

⎡

⎣2∑

i=0

τi

(α1 +

4B2(α2 − α1)4B2 − [(α1 − α2)

(C − η0)]2

)i⎤

⎦1/(2n−1)

,

u(x, y, t

)=

⎡

⎣2∑

i=0

τi

(α2 +

(α2 − α1)exp

[((α1 − α2)/B)

(C − η0)] − 1

)i⎤

⎦1/(2n−1)

,

u(x, y, t

)=

⎡

⎣2∑

i=0

τi

(α1 +

(α1 − α2)exp

[((α1 − α2)/B)

(C − η0)] − 1

)i⎤

⎦1/(2n−1)

,

u(x, y, t

)=

⎡⎢⎣

2∑

i=0

τi

⎛⎜⎝α1− 2(α1 − α2)(α1 − α3)

2α1−α2−α3+(α3−α2) cosh[(√

(α1−α2)(α1−α3)/B)(C)

]

⎞⎟⎠

i⎤⎥⎦

1/(2n−1)

,

(3.31)

Subcase 2.2. It holds that

ξ0 = ξ0, ξ1 = ξ3 = 0, ξ2 = 2√ξ0ξ4, ξ4 = ξ4, τ1 = 0, τ2 = τ2,

ζ0 = −4(qm2ξ4 + am2

√ξ0ξ4τ2

)

a(1 − 2n)2τ2, τ0 =

√ξ0τ2√ξ4

, c = 1 + 2k +a√ξ0τ2

(1 + 2n)√ξ4,

(3.32)

where q = (1 + 2k)(1 + 2n). Substituting these results into (2.5) and (2.9), we get

±(η − η0)= m

√√√√−4qξ4 − a√ξ0ξ4τ2

a(1 − 2n)2ξ4τ2

∫dΓ

√Γ4 +

(2√ξ0ξ4/ξ4

)Γ2 + (ξ0/ξ4)

. (3.33)

Integrating (3.33), we obtain the solutions to the (1.1) as follows.If we denote

F(Γ) = Γ4 +2√ξ0ξ4ξ4

Γ2 +ξ0ξ4

= R2 + d1R + d0, (3.34)

where Γ2 = R, F(Γ) = G(R), then we can write complete discrimination system of G(R) asfollows:

Δ = d21 − 4d0. (3.35)

Correspondingly, there are the following two cases to be discussed.


(1) IfΔ > 0,thenwe have F(Γ) = (Γ−√α1)(Γ+√α1)(Γ−√α2)(Γ+√α2), α1 /=α2. Therefore,the solution is given by

±(η − η0)= mC

√α1F

(ϕ, l), (3.36)

where

C =

√√√√−4qξ4 − a√ξ0ξ4τ2

a(1 − 2n)2ξ4τ2, F

(ϕ, l)=∫ϕ

0


, (3.37)

ϕ = arcsin(

Γ√α1

), l2 =

α1α2, α2 > α1. (3.38)

(2) If Δ = 0, then we have F(Γ) = (Γ −√α1)

2(Γ +√α1)

2. From here, the solutions can befound as

±(η − η0)=mC√α1

arctanh(

Γ√α1

), (3.39)

u(x, y, t

)=

{τ0 + τ1

√α1 tanh

[±√α1C

(x + y −

(1 + 2k +

a√ξ0τ2

(1 + 2n)√ξ4

)t − η0

m

)]

+τ2α1tanh2

[±√α1C

(x + y −

(1 + 2k +

a√ξ0τ2

(1 + 2n)√ξ4

)t − η0

m

)]}1/(2n−1).

(3.40)

For simplicity, we can write (3.40) as follows:

u(x, y, t

)

=

⎧⎨

⎩

2∑

i=0

τi

{√α1 tanh

[±√α1C

(x + y −

(1 + 2k +

a√ξ0τ2

(1 + 2n)√ξ4

)t− η0m

)]}i⎫⎬

⎭

1/(2n−1)

.(3.41)

Example 3.2 (Application to the generalized KdV equation). Using a complex variation ηdefined as η = kx +wt, we can convert (1.2) into ordinary different equation, which reads

wu′ + akunu′ + bku2nu′ + δk3u′′′ = 0, (3.42)

where the prime denotes the derivative with respect to η. Integrating (3.42), and setting theconstant of integration to be zero, we obtain

wu +ak

n + 1un+1 +

bk

2n + 1u2n+1 + δk3u′′ = 0. (3.43)


By the using of the transformation u = v1/n, (3.43) reduces to

δk3n(n + 1)(2n + 1)vv′′ + δk3(1 − n2

)(2n + 1)

(v′)2 + bkn2(n + 1)v4 + akn2(2n + 1)v3

+ n2(n + 1)(2n + 1)wv2 = 0.(3.44)

Substituting (2.6) into (3.44) and using balance principle yield

θ = ε + 2δ + 2. (3.45)

If we take θ = 4, ε = 0, and δ = 1, then

(v′)2 =

τ21(ξ4Γ4 + ξ3Γ3 + ξ2Γ2 + ξ1Γ + ξ0

)

ζ0, (3.46)

where ξ4 /= 0, ζ0 /= 0. Respectively, solving the algebraic equation system (2.8) yields

ξ0 =D2τ31

, ξ1 =Fτ31,

ξ2 = ξ2, ξ3 = ξ3, ξ4 =b(2 + n)ξ34τ1

2(a + 2an + 2b(2 + n)τ0), ζ0 = − k2(1 + n)(2 + n)(1 + 2n)δξ3

2n2(a + 2an + 2b(n + 2)τ0)τ1,

τ0 = τ0, τ1 = τ1, w =−2k[3ξ3τ0(a + 2an + b(2 + n)τ0) − ξ2τ1(a + 2an + b(2 + n)τ0)]

ξ3(2 + 7n + 7n2 + 2n3).

(3.47)

where D denotes by τ20 (−(ξ3τ0(4a(1 + 2n) + 5b(2 + n)τ0)/(a + 2an + 2b(2 + n)τ0)) + 2ξ2τ1) andF denote by τ0(−(ξ3τ0(3a(1 + 2n) + 4b(2 + n)τ0)/(a + 2an + 2b(2 + n)τ0)) + 2ξ2τ1). Substitutingthese results into (2.5) and (2.9), we can write

± (η − η0)=k

τ1

√−(1 + n)(1 + 2n)δ

4n2b

×∫

dΓ√Γ4 +

N2b(2 + n)τ1

Γ3 +ξ2(N)

2b(2 + n)ξ2τ1Γ2 +

F(N)2b(2 + n)ξ3τ41

Γ +D(N)

4b(2 + n)ξ3τ41

,(3.48)


where N denotes by a + 2an + b(2 + n)τ0. Integrating (3.48), we obtain the solutions to (1.2)as follows:

±(η − η0)= − kB

Γ − α1 ,(3.49)

±(η − η0)=

2kBα1 − α2

√Γ − α2Γ − α1 , α2 > α1, (3.50)

±(η − η0)=

kB

α1 − α2 ln∣∣∣∣Γ − α1Γ − α2

∣∣∣∣, (3.51)

±(η − η0)=

kB√(α1 − α2)(α1 − α3)

ln

∣∣∣∣∣

√(Γ − α2)(α1 − α3) −

√(Γ − α3)(α1 − α2)√

(Γ − α2)(α1 − α3) +√(Γ − α3)(α1 − α2)

∣∣∣∣∣, α1 > α2 > α3,

(3.52)

±(η − η0)=

2kB√(α1 − α3)(α2 − α4)

F(ϕ, l), α1 > α2 > α3 > α4, (3.53)

where

B =1τ1

√−(1 + n)(1 + 2n)δ

4n2b, F

(ϕ, l)=∫ϕ

0


, (3.54)

ϕ = arcsin

√(Γ − α1)(α2 − α4)(Γ − α2)(α1 − α4) , l2 =

(α2 − α3)(α1 − α4)(α1 − α3)(α2 − α4) .

(3.55)

Also α1, α2, α3, and α4 are the roots of the polynomial equation:

Γ4 +ξ3ξ4Γ3 +

ξ2ξ4Γ2 +

ξ1ξ4Γ +

ξ0ξ4

= 0. (3.56)

Substituting the solutions (3.49)–(3.52) into (2.4) and (3.2), we find

u(x, t) =

[τ0 + τ1α1 ± τ1B

M− (η0/k)]1/n

,


u(x, t) =

{τ0 + τ1α1 +

4B2(α2 − α1)τ14B2 − [(α1 − α2)

(M− (η0/k))]2

}1/n

,

u(x, t) =

{τ0 + τ1α2 +

(α2 − α1)τ1exp

[((α1 − α2)/B)

(M− (η0/k))] − 1

}1/n

,

u(x, t) =

{τ0 + τ1α1 +

(α1 − α2)τ1exp

[((α1 − α2)/B)

(M− (η0/k))] − 1

}1/n

,

u(x, t) =

⎧⎨

⎩τ0 + τ1α1 − 2(α1 − α2)(α1 − α3)τ12α1 − α2 − α3 + (α3 − α2) cosh

[(√(α1 − α2)(α1 − α3)/B

)M]

⎫⎬

⎭

1/n

,

(3.57)

whereM denotes by x + ((−2[3ξ3τ0(a + 2an + b(2 + n)τ0) − ξ2τ1(a + 2an + b(2 + n)τ0)])/ξ3(2 +7n + 7n2 + 2n3))t.

If we take τ0 = −τ1α1 and η0 = 0, then the solutions (3.57) can reduce to rationalfunction solutions:

u(x, t) =

[± B

x + (2τ1(a + 2an − b(2 + n)τ1α1)(3ξ3α1 + ξ2)/ξ3(2 + 7n + 7n2 + 2n3))t

]1/n,

u(x, t) =

⎧⎨

⎩4B2(α2 − α1)

τ1[4B2−((α1−α2)(x+(2τ1(a+2an−b(2+n)τ1α1)(3ξ3α1+ξ2)/ξ3(2+7n+7n2+2n3))t))2

]

⎫⎬

⎭

1/n

,

(3.58)

traveling wave solutions:

u(x, t)={(α2−α1)τ1

2

{1∓coth

[(α1 − α2)

2B

(x+

2τ1(a+2an−b(2+n)τ1α1)(3ξ3α1 + ξ2)ξ3(2 + 7n + 7n2 + 2n3)

t

)]}}1/n

,

(3.59)

and soliton solution:

u(x, t) =A3

(D + cosh[B1(x − vt)])1/n, (3.60)

where B = Bτ1, A3 = (2(α1 − α2)(α1 − α3)τ1/(α3 − α2))1/n, B1 =√(α1 − α2)(α1 − α3)/B, D =

(2α1 −α2 −α3)/(α3 −α2), and v = −2τ1(a+2an−b(2+n)τ1α1)(3ξ3α1 + ξ2)/ξ3(2+7n+7n2 +2n3).Here, A3 is the amplitude of the soliton, while v is the velocity and B1 is the inverse width ofthe soliton. Thus, we can say that the solitons exist for τ1 < 0.


4. Discussion

Thus we give a more general extended trial equation method for nonlinear partial differentialequations as follows.

Step 1. The extended trial equation (2.4) can be reduced to the following more general form:

u =A(Γ)B(Γ)

=∑δ

i=0 τiΓi

∑μ

j=0ωjΓj, (4.1)

where

(Γ′)2 = Λ(Γ) =

Φ(Γ)Ψ(Γ)

=ξθΓθ + · · · + ξ1Γ + ξ0ζεΓε + · · · + ζ1Γ + ζ0

. (4.2)

Here, τi (i = 0, . . . , δ), ωj (j = 0, . . . , μ), ξς (ς = 0, . . . , θ), and ζσ (σ = 0, . . . , ε) are theconstants to be determined.

Step 2. Taking trial equations (4.1) and (4.2), we derive the following equations:

(u′)2 =

Φ(Γ)Ψ(Γ)

(A′(Γ)B(Γ) −A(Γ)B′(Γ))2

B4(Γ),

u′′ =(A′(Γ)B(Γ) −A(Γ)B′(Γ)){(Φ′(Γ)Ψ(Γ) −Φ(Γ)Ψ′(Γ))B(Γ) − 4Φ(Γ)Ψ(Γ)B′(Γ)}

2B3(Γ)Ψ2(Γ)

+2Φ(Γ)Ψ(Γ)B(Γ)(A′′(Γ)B(Γ) −A(Γ)B′′(Γ))

2B3(Γ)Ψ2(Γ),

(4.3)

and other derivation terms such as u′′′, and so on.

Step 3. Substituting u′, u′′ and other derivation terms into (2.3) yields the following equation:

Ω(Γ) = sΓs + · · · + 1Γ + 0 = 0. (4.4)

According to the balance principle, we can determine a relation of θ, ε, δ and μ.

Step 4. Letting the coefficients of Ω(Γ) all be zero will yield an algebraic equations systemi = 0 (i = 0, . . . , s). Solving this equations system, we will determine the values τ0, . . . τδ;ω0, . . . , ωμ; ξ0 . . . , ξθ, and ζ0, . . . , ζε.

Step 5. Substituting the results obtained in Step 4 into (4.2) and integrating equation (4.2), wecan find the exact solutions of (2.1).

5. Conclusions and Remarks

In this study, we proposed a new trial equationmethod and used it to obtain some soliton andelliptic function solutions to the CamassaHolmKP equation and the one-dimensional general


improved KdV equation. Otherwise, we discussed a more general trial equation method. Wethink that the proposed method can also be applied to other nonlinear differential equationswith nonlinear evolution.

Also, the convergence analysis of obtained elliptic solutions is given as follows:

F(φ, l)=∫φ

0

dφ√(

1 − l2 sin2φ) , (5.1)

where

sinφ =

√Γ − α3α2 − α3 , l =

√α2 − α3α1 − α3 .

(5.2)

Especially, φ = π/2, we have

F(π2, l)=∫π/2

0

dφ√(

1 − l2 sin2φ) = φ +

12k2v2 + · · · + 1.3 · · · (2n − 1)

2.4 · · · (2n) k2nv2n + · · · , (5.3)

where v2n =∫φ0 sin2nφ dφ. Taking the value φ = π/2, we have v2n = ((1.3 · · · (2n −

1))/(2.4 · · · (2n)))(π/2). Therefore, if we take Γ(t) = α2 in (3.13), Γ(t) = α4 in (3.28) and (3.55),Γ(t) =

√α1 in (3.38), for each t, then we have

F(π2, l)=π

2

∞∑

n=0

((2n)!

22n(n!)2

)2

l2n. (5.4)

By the using radius of convergence of power series:

R =1

limn→∞

(an+1/an), (5.5)

where an = ((2n)!/22n(n!)2)2. We have the radius of convergence of power series R = 1. We

can say that power series converges for each 0 < l < 1, diverges for each l > 1. Consequently,the inequalities in (3.11), (3.26)–(3.53), and (3.38) are obtained by using 0 < l < 1, respectively.

Acknowledgment

The research has been supported by the Scientific and Technological Research Council ofTurkey (TUBITAK) and Yozgat University Foundation.


References

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[8] Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of the trial equation method for solving somenonlinear evolution equations arising inmathematical physics,” Pramana, vol. 77, pp. 1023–1029, 2011.

[9] Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of an irrational trial equation method to high-dimensional nonlinear evolution equations,” Journal of Advanced Mathematical Studies, vol. 5, pp. 41–47, 2012.

[10] C. S. Liu, “Trial equation method and its applications to nonlinear evolution equations,” Acta PhysicaSinica, vol. 54, no. 6, pp. 2505–2509, 2005.

[11] C. S. Liu, “Trial equation method for nonlinear evolution equations with rank inhomogeneous:mathematical discussions and applications,” Communications in Theoretical Physics, vol. 45, pp. 219–223, 2006.

[12] C. S. Liu, “A new trial equation method and its applications,” Communications in Theoretical Physics,vol. 45, pp. 395–397, 2006.

[13] C. Y. Jun, “Classification of traveling wave solutions to the Vakhnenko equations,” Computers &Mathematics with Applications, vol. 62, no. 10, pp. 3987–3996, 2011.

[14] C. Y. Jun, “Classification of traveling wave solutions to the modified form of the Degasperis-Procesiequation,” Mathematical and Computer Modelling, vol. 56, pp. 43–48, 2012.

[15] C.-S. Liu, “Applications of complete discrimination system for polynomial for classifications oftraveling wave solutions to nonlinear differential equations,” Computer Physics Communications, vol.181, no. 2, pp. 317–324, 2010.

[16] M. M. Kabir, “Analytic solutions for a nonlinear variant of the (2+1) dimensional Camassa Holm KPequation,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 12, pp. 1566–1577, 2011.

[17] W. Zhang, Q. Chang, and B. Jiang, “Explicit exact solitary-wave solutions for compound KdV-typeand compound KdV-Burgers-type equations with nonlinear terms of any order,” Chaos, Solitons andFractals, vol. 13, no. 2, pp. 311–319, 2002.


Research ArticleNumerical Solution of StochasticHyperbolic Equations

Necmettin Aggez1 and Maral Ashyralyyewa2

1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey2 Department of Applied Mathematics and Informatics, Magtymguly Turkmen State University,Ashgabat, Turkmenistan

Correspondence should be addressed to Necmettin Aggez, [email protected]

Received 30 March 2012; Revised 25 May 2012; Accepted 27 May 2012


Copyright q 2012 N. Aggez and M. Ashyralyyewa. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A two-step difference scheme for the numerical solution of the initial-boundary value problemfor stochastic hyperbolic equations is presented. The convergence estimate for the solution of thedifference scheme is established. In applications, the convergence estimates for the solution ofthe difference scheme are obtained for different initialboundary value problems. The theoreticalstatements for the solution of this difference scheme are supported by numerical examples.

1. Introduction

Stochastic partial differential equations have been studied extensively by many researchers.For example, the method of operators as a tool for investigation of the solution to stochasticequations in Hilbert and Banach spaces have been used systematically by several authors(see, [1–7] and the references therein). Numerical methods and theory of solutions of initialboundary value problem for stochastic partial differential equations have been studied in [8–16]. Moreover, the authors of [17] presented a two-step difference scheme for the numericalsolution of the following initial value problem:

dv(t) = −Av(t)dt + f(t)dwt, 0 < t < T,

v(0) = 0, v(0) = 0,(1.1)

for stochastic hyperbolic differential equations. We have the following.(i) wt is a standard Wiener process given on the probability space (Ω, F, P).(ii) For any z ∈ [0, T], f(z) is an element of the space M2

w([0, T],H1), where H1 is asubspace ofH.


Here, M2w([0, T],H) [18] denote the space of H-valued measurable processes which

satisfy

(a) φ(t) is Ft measurable, a.e. in t,

(b) E∫T0 ‖φ(t)‖Hdt <∞.

The convergence estimates for the solution of the difference scheme are established.In the present work, we consider the following initial value problem:

dv(t) +Av(t)dt = f(t)dwt, 0 < t < T,

v(0) = ϕ, v(0) = ψ,(1.2)

for stochastic hyperbolic equation in a Hilbert space H with a self-adjoint positive definiteoperator Awith A ≥ δI, where δ > δ0 > 0. In addition to (i) and (ii), we put the following.

(iii) ϕ and ψ are elements of the space M2w([0, T],H2) of H2-valued measurable

processes, whereH2 is a subspace ofH.By the solutions provided in [19] (page 423, (0.4)) and in [20] (page 1005, (2.9)), under

the assumptions (i), (ii), and (iii), the initial value problem (1.2) has a unique mild solutiongiven by the following formula:

v(t) = c(t)ϕ + s(t)ψ +∫ t

0s(t − z)f(z)dwz. (1.3)

For the theory of cosine and sine operator-function we refer to [21, 22].Our interest in this study is to construct and investigate the difference scheme for

the initial value problem (1.2). The convergence estimate for the solution of the differencescheme is proved. In applications, the theorems on convergence estimates for the solutionof difference schemes for the numerical solution of initial-boundary value problems forhyperbolic equations are established. The theoretical statements for the solution of thisdifference scheme are supported by the result of the numerical experiments.

2. The Exact Difference Scheme

We consider the following uniform grid space:

[0, T]τ = {tk = kτ, k = 0, 1, . . . ,N, Nτ = T}, (2.1)

with step τ > 0. Here,N is a fixed positive integer.


Theorem 2.1. Let v(tk) be the solution of the initial value problem (1.2) at the grid points t = tk.Then, {v(tk)}N0 is the solution of the initial value problem for the following difference equation:

1τ2

(v(tk+1) − 2v(tk) + v(tk−1)) +2τ2

(I − c(τ))v(tk) = 1τ

(f1,k+1 + s(τ)f2,k − c(τ)f1,k

),

f1,k =1τ

∫ tk

tk−1s(tk − z)f(z)dwz, f2,k =

1τ

∫ tk

tk−1c(tk − z)f(z)dwz, 1 ≤ k ≤N − 1,

v(0) = ϕ, v(τ) = c(τ)ϕ + s(τ)ψ + τf1,1.

(2.2)

Proof. Putting t = tk into the formula (1.3), we can write

v(tk) = c(tk)ϕ + s(tk)ψ +∫ tk

0s(tk − z)f(z)dwz. (2.3)

Using (2.3), the definition of the sine and cosine operator function, we obtain

v(tk) = c(tk)ϕ + s(tk)ψ +k∑

j=1

∫ tj

tj−1s(tk − tj + tj − z

)f(z)dwz

= c(tk)ϕ + s(tk)ψ + τk∑

j=1

(s(tk − tj

)f2,j + c

(tk − tj

)f1,j).

(2.4)

It follows that

v(tk+1) + v(tk−1) = [c(tk+1) + c(tk−1)]ϕ + [s(tk+1) + s(tk−1)]ψ

+ 2c(τ)τk∑

j=1

(s(tk − tj

)f2,j + c

(tk − tj

)f1,j)

+ τ(f1,k+1 + s(τ)f2,k − c(τ)f1,k

).

(2.5)

Hence, we get the relation between v(tk) and v(tk±1) as

v(tk+1) + v(tk−1) − 2c(τ)v(tk) = τ(f1,k+1 + s(τ)f2,k − c(τ)f1,k

). (2.6)

This relation and equality (2.2) are equivalent. Theorem 2.1 is proved.


3. Convergence of the Difference Scheme

For the approximate solution of problem (1.2), we need to approximate the following expres-sions:

f1,k =1τ

∫ tk

tk−1s(tk − z)f(z)dwz, f2,k =

1τ

∫ tk

tk−1c(tk − z)f(z)dwz,

exp(±iτA1/2

).

(3.1)

Using Taylor’s formula and Pade approximation of the function exp(−z) at z = 0, we get

exp(±iτA1/2

)≈(I ± iτA1/2

2

)(I ∓ iτA1/2

2

)−1= R(±iτA1/2

),

f1,k ≈ − 1τ

∫ tk

tk−1(z − tk)f(z)dwz = f1,k, f2,k ≈ 1

τ

∫ tk

tk−1f(z)dwz = f2,k.

(3.2)

Applying the difference scheme (2.2) and formula (3.2), we can construct the followingdifference scheme:

1τ2

(uk+1 − 2uk + uk−1) +2τ2

(I − cτ(τ))uk =1τ

(f1,k+1 + sτ(τ)f2,k − cτ(τ)f1,k

), (3.3)

cτ(τ) =R(iτA1/2) + R

(−iτA1/2)

2,

sτ(τ) = A−1/2R(iτA1/2) − R(−iτA1/2)

2i, 1 ≤ k ≤N − 1,

u0 = ϕ, u1 = cτ(τ)ϕ + sτ(τ)ψ + τf1,1,

(3.4)

for the approximate solution of the initial value problem (1.2). Using the definition of cτ(τ)and sτ(τ), we can write (3.3) in the following equivalent form:

1τ2

(uk+1 − 2uk + uk−1) +12Auk +

14Auk+1 +Auk−1

=1τ

((I +

14τ2A

)f1,k+1 + τf2,k −

(I − 1

4τ2A

)f1,k

),

tk = kτ, 1 ≤ k ≤N − 1,

u0 = ϕ, u1 = cτ(τ)ϕ + sτ(τ)ψ + τf1,1.

(3.5)

Now, let us give the lemma we need in the sequel from papers [23, 24].



‖c(t)‖H→H ≤ 1,∥∥∥A1/2s(t)

∥∥∥H→H

≤ 1 (t ≥ 0), (3.6)

‖cτ(kτ)‖H→H ≤ 1,∥∥∥A1/2sτ(kτ)

∥∥∥H→H

≤ 1 (k ≥ 0), (3.7)

∥∥∥A−(1+α)(cτ(kτ) − c(tk))∥∥∥H→H

≤ Cτ (3/2+α), 0 ≤ α ≤ 12, (3.8)

∥∥∥A−(1/2+α)(sτ(kτ) − s(tk))∥∥∥H→H

≤ Cτ (3/2+α), (k ≥ 0), (3.9)

where

cτ(kτ) =Rk(iτA1/2) + Rk

(−iτA1/2)

2,

sτ(kτ) = A−1/2Rk(iτA1/2) − Rk

(−iτA1/2)

2i.

(3.10)

The following Theorem on convergence of difference scheme (3.5) is established.


E(∥∥Aϕ

∥∥2H

)≤ C, E

(∥∥∥(A1/2ψ

)∥∥∥2

H

)≤ C, E

∫T

0

∥∥Af(t)∥∥2Hdt ≤ C, (3.11)

then the estimate of convergence

(N∑

k=1

E‖v(tk) − uk‖2H)1/2

≤ C1(δ)τ (3.12)

holds. Here, C1(δ) does not depend on τ .

Proof. Using the formula for the solution of second order difference equation and thedefinition of cτ(kτ) and sτ(kτ), we can write

uk = cτ(kτ)ϕ + sτ(kτ)ψ + τk∑

j=1

(sτ((k − j)τ)f2,j + cτ

((k − j)τ)f1,j

), 1 ≤ k ≤N. (3.13)


Using (2.4) and (3.13), we obtain

v(tk) − uk = [c(kτ) − cτ(kτ)]ϕ + [s(kτ) − sτ(kτ)]ψ

+ τk∑

j=1

(s(tk − tj

)f2,j + c

(tk − tj

)f1,j)

− τk∑

j=1

(sτ((k − j)τ)f2,j + cτ

((k − j)τ)f1,j

)

= [c(kτ) − cτ(kτ)]ϕ + [s(kτ) − sτ(kτ)]ψ

+ τk−1∑

j=1

sτ((k − j)τ)

(f2,j − f2,j

)

+ τk−1∑

j=1

(s(tk − tj

) − sτ((k − j)τ))f2,j

+ τk∑

j=1

cτ((k − j)τ)

(f1,j − f1,j

)

+ τk∑

j=1

(c(tk − tj

) − cτ((k − j)τ))f1,j

= J1,k + J2,k + J3,k + J4,k + J5,k + J6,k, 1 ≤ k ≤N,

(3.14)

where

J1,k = [c(kτ) − cτ(kτ)]ϕ, J2,k = [s(kτ) − sτ(kτ)]ψ,

J3,k = τk−1∑

j=1

sτ((k − j)τ)

(f2,j − f2,j

),

J4,k = τk−1∑

j=1

(s(tk − tj

) − sτ((k − j)τ))f2,j ,

J5,k = τk∑

j=1

cτ((k − j)τ)

(f1,j − f1,j

),

J6,k = τk∑

j=1

(c(tk − tj

) − cτ((k − j)τ))f1,j .

(3.15)


Let us estimate the expected value of Jm,k for all m = 1, . . . , 6, separately. We start with J1,kand J2,k. Using (3.6), (3.7), and (3.8), we obtain

(N∑

k=1

E‖J1,k‖2H)1/2

=

(N∑

k=1

E∥∥∥A−1[c(kτ) − cτ(kτ)]Aϕ

∥∥∥2

H

)1/2

≤ C(

N∑

k=1

τ3E∥∥Aϕ

∥∥2H

)1/2

≤ τC(E∥∥Aϕ

∥∥2H

)1/2,

(N∑

k=1

E‖J2,k‖2H)1/2

=

(N∑

k=1

E∥∥∥A−1/2[s(kτ) − sτ(kτ)]A1/2ψ

∥∥∥2

H

)1/2

≤ C(

N∑

k=1

τ3E∥∥∥A1/2ψ

∥∥∥2

H

)1/2

≤ τC(E∥∥∥A1/2ψ

∥∥∥2

H

)1/2

.

(3.16)

Estimates for the expected value of Jm,k for all m = 3, . . . , 6, separately, were also used inpaper [17]. Combining these estimates, we obtain (3.12). Theorem 3.2 is proved.

4. Applications

First, letΛ be the unit open cube in the n-dimensional Euclidean space Rn = {x = (x1, . . . , xn) :

0 < xi < 1, i = 1, . . . , n} with boundary S, Λ = Λ ∪ S. In [0, T] ×Λ, the initial-boundary valueproblem for the following multidimensional hyperbolic equation:

du(t, x) −n∑

r=1

(ar(x)uxr )xrdt = f(t, x)dwt, 0 < t < T, x = (x1, . . . , xn) ∈ Λ,

u(0, x) = ϕ(x), u(0, x) = ψ(x), x ∈ Λ; u(t, x) = 0, x ∈ S, 0 ≤ t ≤ T(4.1)

with the Dirichlet condition is considered. Here, ar(x), (x ∈ Λ), δ ≥ 0 and f(t, x) (t ∈(0, 1), x ∈ Λ) are given smooth functions with respect to x and ar(x) ≥ a > 0.

The discretization of (4.1) is carried out in two steps. In the first step, define the gridspace Λh = {x = xm = (h1m1, . . . , hnmn);m = (m1, . . . , mn), 0 ≤ mr ≤ Nr, hrNr = 1, r =1, . . . , n}, Λh = Λh ∩Λ, Sh = Λh ∩ S.

Let L2h denote the Hilbert space as

L2h = L2

(Λh

)=

⎧⎪⎨

⎪⎩ϕh(x) :

⎛

⎝∑

x∈Λh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

<∞

⎫⎪⎬

⎪⎭. (4.2)

The differential operator A in (4.1) is replaced with

Axhu

h(x) = −n∑

r=1

(ar(x)uhxr

)

xr ,jr+ δuh(x), (4.3)


where the difference operator Axh is defined on these grid functions uh(x) = 0, for all x ∈ Sh.

As it is proved in [25], Axhis a self-adjoint positive definite operator in L2h. Using (4.1) and

(4.3), we get

duh(t, x) +Axhu

h(t, x)dt = fh(t, x)dwt, 0 < t < T, x ∈ Λh,

uh(0, x) = ϕh(x), uh(0, x) = ψh(x), x ∈ Λh.(4.4)

In the second step, we replace (4.4) with the difference scheme (3.5) as

1τ2

(uhk+1(x) − 2uhk(x) + u

hk−1(x)

)+12Axhu

hk(x)

+14

(Axhu

hk+1(x) +A

xhu

hk−1(x)

)= ϕhk(x),

ϕhk(x) =1τ

[(I +

14τ2Ax

h

)ϕh1,k+1(x) + τϕ

h2,k(x) −

(I − 1

4τ2Ax

h

)ϕh1,k(x)

],

ϕh1,k(x) = − 1τ

∫ tk

tk−1(z − tk)fh(z, x)dwz, ϕh2,k(x) =

1τ

∫ tk

tk−1fh(z, x)dwz,

tk = kτ, 1 ≤ k ≤N − 1, Nτ = T, x ∈ Λh,

uh1(x) = cτ(τ)ϕh(x) + sτ(τ)ψh(x) −

∫ τ

0(z − τ)fh(z, x)dwz, x ∈ Λh,

uh0(x) = ϕh(x).

(4.5)

Theorem 4.1. Let τ and |h| =√h21 + · · · + h2n be sufficiently small numbers. Then, the solution of

difference scheme (4.5) satisfies the convergence estimate as

(N∑

k=1

E∥∥∥uh(tk) − uhk

∥∥∥2

L2h

)1/2

≤ C(δ)(τ + |h|2

), (4.6)

where C(δ) does not depend on τ and |h|.


hdefined by (4.3).

Second, in [0, T] × Λ, the initial-boundary value problem for the following multidi-mensional hyperbolic equation:

du(t, x) −n∑

r=1

(ar(x)uxr )xrdt + δu(t, x)dt = f(t, x)dwt,

0 < t < T, x = (x1, . . . , xn) ∈ Λ,

u(0, x) = ϕ(x), u(0, x) = ψ(x), x ∈ Λ,∂u(t, x)∂n

= 0, x ∈ S, 0 ≤ t ≤ T

(4.7)


with the Neumann condition is considered. Here, n is the normal vector to Λ, δ > 0, ar(x),(x ∈ Λ), and f(t, x) (t ∈ (0, 1), x ∈ Λ) are given smooth functions with respect to x andar(x) ≥ a > 0.

The discretization of (4.7) is carried out in two steps. In the first step, the differentialoperator A in (4.7) is replaced with

Axhu

h(x) = −n∑

r=1

(ar(x)uhxr

)

xr ,jr+ δuh(x), (4.8)

where the difference operator Axh is defined on those grid functions Dhuh(x) = 0, for all

x ∈ Sh, where Dhuh(x) = 0 is the second order of approximation of ∂u(t, x)/∂n. As it isproved in [25], Ax

his a self-adjoint positive definite operator in L2h. Using (4.7) and (4.8), we

get

duh(t, x) +Axhu


uh(0, x) = ϕh(x), uh(0, x) = ψh(x), x ∈ Λh.(4.9)

In the second step, we replace (4.9) with the difference scheme (3.5) as

1τ2

(uhk+1(x) − 2uhk(x) + u

hk−1(x)

)+12Axhu

hk(x) +

14

(Axhu

hk+1(x) +A

xhu

hk−1(x)

)= ϕhk(x),

ϕhk(x) =1τ

((I +

14τ2Ax

h

)ϕh1,k+1(x) + τϕ

h2,k(x) −

(I − 1

4τ2Ax

h

)ϕh1,k(x)

),

ϕh1,k(x) = − 1τ

∫ tk

tk−1(z − tk)fh(z, x)dwz, ϕh2,k(x) =

1τ

∫ tk

tk−1fh(z, x)dwz,

tk = kτ, 1 ≤ k ≤N − 1, Nτ = T, x ∈ Λh,


∫ τ

0(z − τ)fh(z, x)dwz,

uh0(x) = ϕh(x), x ∈ Λh.

(4.10)

Theorem 4.2. Let τ and |h| =√h21 + · · · + h2n be sufficiently small numbers. Then, the solution of

difference scheme (4.10) satisfies the convergence estimate as

(N∑

k=1


∥∥∥2

L2h

)1/2

≤ C(δ)(τ + |h|2

), (4.11)



hdefined by (4.8).


Third, in [0, T] ×Λ, the mixed boundary value problem for the following multidimen-sional hyperbolic equation:

du(t, x) −n∑

r=1

(ar(x)uxr )xrdt + δu(t, x)dt = f(t, x)dwt,

0 < t < T, x = (x1, . . . , xn) ∈ Λ,

u(0, x) = ϕ(x), u(0, x) = ψ(x), x ∈ Λ,

∂u(t, x)∂n

= 0, x ∈ S2, 0 ≤ t ≤ T, S1 ∪ S2 = S,

u(t, x) = 0, x ∈ S1

(4.12)

with the Dirichlet-Neumann condition is considered. Here, n is the normal vector to Λ, δ >0, ar(x), (x ∈ Λ), and f(t, x) (t ∈ (0, 1), x ∈ Λ) are given smooth functions with respect to xand ar(x) ≥ a > 0.

The discretization of (4.12) is carried out in two steps. In the first step, the differentialoperator A in (4.12) is replaced with

Axhu

h(x) = −n∑

r=1

(ar(x)uhxr

)

xr ,jr+ δuh(x), (4.13)

where the difference operator Axhis defined on those grid functions uh(x) = 0, for all x ∈ S1

h

and Dhuh(x) = 0, for all x ∈ S2h, S

1h ∪ S2

h = Sh, where Dhuh(x) = 0 is the second orderof approximation of ∂u(t, x)/∂n. By [25], we can conclude that Ax

h is a self-adjoint positivedefinite operator in L2h. Using (4.12) and (4.13), we get

duh(t, x) +Axhu


uh(0, x) = ϕ(x), uh(0, x) = ψ(x), x ∈ Λh.(4.14)

In the second step, we replace (4.14)with the difference scheme (3.5) as

1τ2

(uhk+1(x) − 2uhk(x) + u

hk−1(x)

)+12Axhu

hk(x) +

14

(Axhu

hk+1(x) +A

xhu

hk−1(x)

)= ϕhk(x),

ϕhk(x) =1τ

((I +

14τ2Ax

h

)ϕh1,k+1(x) + τϕ

h2,k(x) −

(I − 1

4τ2Ax

h

)ϕh1,k(x)

),


ϕh1,k(x) = − 1τ

∫ tk

tk−1(z − tk)fh(z, x)dwz, ϕ

h2,k(x) =

1τ

∫ tk

tk−1fh(z, x)dwz,

tk = kτ, 1 ≤ k ≤N − 1, Nτ = T, x ∈ Λh,


∫ τ

0(z − τ)fh(z, x)dwz,

uh0(x) = ϕh(x), x ∈ Λh.

(4.15)

Theorem 4.3. Let τ and |h| =√h21 + · · · + h2n be sufficiently small positive numbers. Then, the

solution of difference scheme (4.15) satisfies the convergence estimate as

(N∑

k=1


∥∥∥2

L2h

)1/2

≤ C(δ)(τ + |h|2

), (4.16)



hdefined by (4.13).

5. Numerical Examples

In this section, we apply finite difference scheme (2.2) to four examples which are stochastichyperbolic equation with Neumann, Dirichlet, Dirichlet-Neumann, and Neumann-Dirichletconditions.

Example 5.1. The following initial-boundary value problem:

du(t, x) − ∂2u(t, x)∂x2

dt + u(t, x)dt = f(t, x)dwt,

f(t, x) =√2 cosx, wt =

√tξ, 0 < t < 1, 0 < x < π,

u(0, x) = cosx, u(0, x) = 0, 0 ≤ x ≤ π,ux(t, 0) = ux(t, π) = 0, 0 ≤ t ≤ 1

(5.1)

for a stochastic hyperbolic equation is considered. The exact solution of this problem is

u(t, x) =∫ t

0sin(√

2(t − s))cosxdws + cos

(√2t)cosx. (5.2)


For the approximate solution of the (5.1), we apply the finite difference scheme (2.2) and weget


τ+τ

2

[−u

kn+1 − 2ukn + u

kn−1

h2+ ukn

]

+τ

4

[−u

k+1n+1 − 2uk+1n + uk+1n−1

h2+ uk+1n − uk−1n+1 − 2uk−1n + uk−1n−1

h2+ uk−1n

]= fkn ,

fkn =√2ξ cosxn

[√tk+1 −

√tk−1 +

τ2 − 22τ

[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k−√t3k−1

)]],

Nτ = 1, xn = nh, 1 ≤ n ≤M − 1, Mh = π, 1 ≤ k ≤N − 1, tk = kτ,

u0n = cosxn, u1n − u0n =4 + τ2

6

√2τ3ξ cosxn, 1 ≤ n ≤M − 1,

uk0 = uk1 , ukM = ukM−1, 1 ≤ k ≤N.

(5.3)

The system can be written in the following matrix form:

Aun+1 + Bun + Cun−1 = Dϕn, 1 ≤ n ≤M − 1,

u0 = u1, uM = uM−1.(5.4)

Here,

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0 0 0a 2a a 0 · · · 0 0 0 00 a 2a a · · · 0 0 0 0...

......

.... . .

......

......

0 0 0 0 · · · a 2a a 00 0 0 0 · · · 0 a 2a asa −sa 0 0 · · · 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · · · 0 0 0 0b c b 0 · · · 0 0 0 00 b c b · · · 0 0 0 0...

......

.... . .

......

......

0 0 0 0 · · · b c b 00 0 0 0 · · · 0 b c bsb −sb 0 0 · · · 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

(5.5)


the matrix is C = A,

a =−τ4h2

, b =1τ+

τ

2h2+τ

4, c =

−2τ

+τ

h2+τ

2,

sa =τ2

4h2, sb = 1 +

τ2

2h2+τ2

4,

(5.6)

fn =

⎡⎢⎢⎢⎢⎢⎢⎣

f0n

f1n

f2n...fNn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

,

fkn =√2ξ cosxn

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]], 1 ≤ k ≤N − 1,

f0n = cosxn, 0 ≤ n ≤M,

fNn =4 + τ2

6

√2τ3ξ cosxn, 0 ≤ n ≤M,

(5.7)

and D = IN+1 is the identity matrix,

Us =

⎡⎢⎢⎢⎢⎢⎢⎣

u0su1su2s...

uNs

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, s = n − 1, n, n + 1. (5.8)

This type of system was used by [26] for difference equations. For the solution of matrixequation (5.4), we will use modified Gauss elimination method. We seek a solution of thematrix equation by the following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 2, 1, (5.9)

where uM = (I − αM)−1βM, αj (j = 1, . . . ,M − 1) are (N + 1) × (N + 1) square matrices,βj (j = 1, . . . ,M − 1) are (N + 1) × 1 column matrices α1 is an identity and β1 is a zeromatrices, and

αn+1 = −(B + Cαn)−1A,


), n = 1, 2, 3, . . . ,M − 1.

(5.10)



du(t, x) − ∂2u(t, x)∂x2


f(t, x) =√2 sinx, wt =

√tξ, 0 < t < 1, 0 < x < π,

u(0, x) = sinx, u(0, x) = 0, 0 ≤ x ≤ π,u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1

(5.11)

for a stochastic hyperbolic equation is considered. We use the same procedure as in the firstexample. The exact solution of this problem is

u(t, x) =∫ t

0sin(√

2(t − s))sinxdws + cos

(√2t)sinx. (5.12)

For the approximate solution of the (5.11), we can construct the following difference scheme:


τ+τ

2

[−u

kn+1 − 2ukn + u

kn−1

h2+ ukn

]

× τ

4

[−u

k+1n+1 − 2uk+1n + uk+1n−1

h2+ uk+1n − uk−1n+1 − 2uk−1n + uk−1n−1

h2+ uk−1n

]= fkn ,

fkn =√2ξ sinxn

[√tk+1 −

√tk−1

+τ2 − 22τ

[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k−√t3k−1

)]],


u0n = sinxn, u1n − u0n =4 + τ2

6

√2τ3ξ sinxn, 1 ≤ n ≤M − 1,

uk0 = ukM = 0, 1 ≤ k ≤N,

(5.13)

and it can be written in the following matrix form:

Aun+1 + Bun + Cun−1 = Dfn, 1 ≤ n ≤M − 1,

u0 = uM = 0.(5.14)


Here, the matrices A, B, C, D are given in the previous example, and

fn =

⎡⎢⎢⎢⎢⎢⎢⎣

f0n

f1n

f2n...fNn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

,

fkn =√2ξ sinxn

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]], 1 ≤ k ≤N − 1,

f0n = sinxn, 0 ≤ n ≤M,

fNn =4 + τ2

6

√2τ3ξ sinxn, 0 ≤ n ≤M.

(5.15)

For the solution of matrix equation (5.14), we will use modified Gauss elimination method.We seek a solution of the matrix equation in the following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 2, 1, (5.16)

where uM = 0, αj (j = 1, . . . ,M−1) are (N+1)×(N+1) square matrices, βj (j = 1, . . . ,M−1)are (N + 1) × 1 column matrices. α1 and β1 are zero matrices, and

αn+1 = −(B + Cαn)−1A,


), n = 1, 2, 3, . . . ,M − 1.

(5.17)


du(t, x) − ∂2u(t, x)∂x2


f(t, x) =√52

sin(x2

), wt =

√tξ, 0 < t < 1, 0 < x < π,

u(0, x) = sin(x2

), u(0, x) = 0, 0 ≤ x ≤ π,

u(t, 0) = ux(t, π) = 0, 0 ≤ t ≤ 1

(5.18)



u(t, x) =

[∫ t

0sin

(√52

(t − s))dws + cos

(√52t

)]sin(x2

). (5.19)

We get the following difference scheme:


τ+τ

2

[−u

kn+1 − 2ukn + u

kn−1

h2+ ukn

]τ

4

[−u

k+1n+1 − 2uk+1n + uk+1n−1

h2

+ uk+1n − uk−1n+1 − 2uk−1n + uk−1n−1h2

+ uk−1n

]= fkn ,

fkn =√52ξ sin

xn2

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]],


u0n = sinxn2, u1n − u0n =

4 + τ2

12

√5τ3ξ sin

xn2, 1 ≤ n ≤M − 1,

uk0 = 0, ukM = ukM−1, 1 ≤ k ≤N,

(5.20)

for the approximate solutions of (5.18), and we obtain the following matrix equation:

Aun+1 + Bun + Cun−1 = Dfn, 1 ≤ n ≤M − 1,

uk0 = 0, ukM = ukM−1, 1 ≤ k ≤N.(5.21)

Here, the matrices A, B, C, D are same as in the first example, and

fn =

⎡⎢⎢⎢⎢⎢⎢⎣

f0n

f1n

f2n...fNn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

,

fkn =√52ξ sin

xn2

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]], 1 ≤ k ≤N − 1,


f0n = sin

xn2, 0 ≤ n ≤M,

fNn =4 + τ2

12

√5τ3ξ sin

xn2, 0 ≤ n ≤M.

(5.22)

For the solution of matrix equation (5.21), we use the same procedure as in the previousexamples. Moreover, uM = 0, α1 is an identity and β1 is a zero matrices, and

αn+1 = −(B + Cαn)−1A, βn+1 = (B + Cαn)−1(Dϕn − Cβn

), n = 1, 2, 3, . . . ,M − 1.

(5.23)

Example 5.4. The following initial boundary value problem:

du(t, x) − ∂2u(t, x)∂x2


f(t, x) =√52

cos(x2

), wt =

√tξ, 0 < t < 1, 0 < x < π,

u(0, x) = cos(x2

), u(0, x) = 0, 0 ≤ x ≤ π,

ux(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1

(5.24)


u(t, x) =

[∫ t

0sin

(√52

(t − s))dws + cos

(√52t

)]cos(x2

). (5.25)

The following difference scheme:


τ+τ

2

[−u

kn+1 − 2ukn + u

kn−1

h2+ ukn

]τ

4

[−u

k+1n+1 − 2uk+1n + uk+1n−1

h2

+ uk+1n − uk−1n+1 − 2uk−1n + uk−1n−1h2

+ uk−1n

]= fkn ,

fkn =√52ξ cos

xn2

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]],



u0n = cosxn2, u1n − u0n =

4 + τ2

12

√5τ3ξ cos

xn2, 1 ≤ n ≤M − 1,

uk0 = uk1 , ukM = 0, 1 ≤ k ≤N(5.26)

is obtained for the approximate solutions of (5.24), and we obtain the following matrixequation:

Aun+1 + Bun + Cun−1 = Dfn, 1 ≤ n ≤M − 1,

u1 = u2, uM = 0.(5.27)

Here, the matrices A, B, C, D are same as in the first example, and

fn =

⎡⎢⎢⎢⎢⎢⎢⎣

f0n

f1n

f2n...fNn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

,

fkn =√52ξ cos

xn2

[√tk+1 −

√tk−1 +

τ2 − 22τ

×[τ(√

tk+1 −√tk−1)− 23

(√t3k+1 −

√t3k −√t3k−1

)]], 1 ≤ k ≤N − 1,

f0n = cos

xn2, 0 ≤ n ≤M,

fNn =4 + τ2

12

√5τ3ξ cos

xn2, 0 ≤ n ≤M.

(5.28)

Using (5.27) that we get α1 is an identity and β1 is a zero matrices and uM = (I − αM)−1βM.The rest are the same as in Example 5.3.

For these examples, the errors of the numerical solution derived by difference scheme(2.2) computed by

ENM = max1≤k≤N−1,1≤n≤M−1

(N∑

k=1

∣∣∣u(tk, xn) − ukn∣∣∣2)1/2

(5.29)

and the results are given in Table 1.

The numerical solutions are recorded for different values of N = M, where u(tk, xn)represents the exact solution and ukn represents the numerical solution at (tk, xn). To obtain


Table 1

N =M = 10 N =M = 20 N =M = 40Example 5.1 (Neumann) 0.3028 0.1219 0.0554Example 5.2 (Dirichlet) 0.4004 0.2137 0.1342Example 5.3 (Dirichlet-Neumann) 0.3040 0.1145 0.0494Example 5.4 (Neumann-Dirichlet) 0.3439 0.1844 0.0957

the results, we simulated the 1000 sample paths of Brownian motion for each level of discret-ization.

Thus, results show that the error is stable and decreases in an exponential manner.

Acknowledgment

The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey)for the helpful suggestions to the improvement of this paper.

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Research ArticleSpectral Properties ofNon-Self-Adjoint Perturbations fora Spectral Problem with Involution

Asylzat A. Kopzhassarova,1 Alexey L. Lukashov,2, 3

and Abdizhakhan M. Sarsenbi1

1 Department of Mathematics, M. Auezov South Kazakhstan State University,Tauke Han av., 5, 160012 Shymkent, Kazakhstan

2 Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey3 Department of Mechanics and Mathematics, N.G. Chernyshevsky Saratov State University,Astrakhanskaya, 83, 410012 Saratov, Russia

Correspondence should be addressed to Alexey L. Lukashov, [email protected]

Received 15 May 2012; Accepted 8 July 2012


Copyright q 2012 Asylzat A. Kopzhassarova et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Full description of Riesz basis property for eigenfunctions of boundary value problems for firstorder differential equations with involutions is given.

1. Introduction

Differential equations with involutions were considered firstly in [1]. They are a particularcase of functional differential equations that appear in several applications (see, for instance,monographs [2, 3] and papers [4–6]). Different spectral problems for equations of this formwere considered in [7, 8].

In particular, main questions about the following spectral problem:

u′(x) = λu(−x), −1 < x < 1, (1.1)

u(−1) = γu(1), (1.2)

were solved in [7]. Namely, (1.1)-(1.2) is a Volterra operator if and only if γ2 = −1;furthermore, (1.1)-(1.2) is self-adjoint if and only if γ is a real number. For γ2 /= − 1, thesystem of eigenfunctions for (1.1)-(1.2) is a Riesz basis in L2(−1, 1). Observe also that forγ2 /= − 1, (1.1)-(1.2) has no associated functions, that is, all eigenvalues are simple. Note that


problem (1.1)-(1.2) is an example of a generalized spectral problem of the form Au = λSu,with A = d/dx and (Sf)(x) = f(−x). In general, they were considered in [9] when A and Sare operators in a Banach space. Equiconvergence questions for two different perturbations of(1.1)-(1.2)were deeply studied in [10]. The main goal of the paper is to study questions aboutRiesz basis property of eigenfunctions for the following non-self-adjoint spectral problem:

u′(−x) + αu′(x) = λu(x), −1 < x < 1, (1.3)

u(−1) = γu(1). (1.4)

Also note that problems similar to (1.3)-(1.4) appear when the Fourier methodis applied for solving boundary value problems for partial differential equations withinvolution (see, for example, [11] and the bibliography therein).

2. Results

Theorem 2.1. If α2 /= 1, γ /=α ±√α2 − 1, then the eigenfunctions system for (1.3)-(1.4) is a Riesz

basis in L2(−1, 1).

Proof. Before the proof, we need several facts about (1.3)-(1.4). First of all, it is easy to seethat the general solution of (1.3)-(1.4)with α2 /= 1 is given by the following formula:

u(x) = C(√

1 + α cosλx√1 − α2

+√1 − α sin

λx√1 − α2

). (2.1)

Next, we observe that for α2 /= 1, γ2 /= 1 eigenvalues are equal to

λk =√1 − α2

[kπ + arctan

1 − γ1 + γ

√1 + α√1 − α

], k = 0,±1, . . . (2.2)

The related eigenfunctions are given by the following formula:

uk(x) =√1 + α cos

[kπ + arctan

1 − γ1 + γ

√1 + α√1 − α

]x

+√1 − α sin

[kπ + arctan

1 − γ1 + γ

√1 + α√1 − α

]x, k = 0,±1, . . .

(2.3)

Observe also that for γ = 1, the eigenvalues are

λk =√1 − α2 kπ, k = 0,±1,±2, . . . (2.4)

The corresponding eigenfunctions are

uk(x) =√1 + α cos(kπx) +

√1 − α sin(kπx), k = 0,±1,±2, . . . ; (2.5)


and for γ = −1, the eigenvalues are

λk =√1 − α2

(kπ +

π

2

), k = 0,±1, . . . , (2.6)

and the corresponding eigenfunctions are

uk(x) =√1 + α cos

(kπ +

π

2

)x +

√1 − α sin

(kπ +

π

2

)x, k = 0,±1, . . . (2.7)

We introduce the differential operator L by

Lu = u′(−x) + αu′(x), −1 < x < 1, (2.8)

and by the boundary condition (1.4). Suppose that Lu belongs to the domain of L, Lu ∈ D(L).Then we consider

L2u = L(u′(−x) + αu′(x)) = −

(1 − α2

)u′′(x). (2.9)

From the boundary condition (1.4), for Lu we deduce that L2 is the second orderdifferential operator generated by the following relations:

L2u = −(1 − α2

)u′′(x), −1 < x < 1, (2.10)

(α − γ)u′(−1) + (

1 − αγ)u′(1) = 0,

u(−1) − γu(1) = 0.(2.11)

Spectral problem (2.10)-(2.11) is a typical spectral problem for an ordinary secondorder differential operator. These problems are studied very well and they have numerousapplications (see, for example, [12–14].) Recall [12, Chapter 2] that the following boundaryconditions:

a1u′(−1) + b1u′(1) + c1u(−1) + d1u(1) = 0,

c0u(−1) + d0u(1) = 0,(2.12)

for an ordinary second order differential operator are regular if b1c0 + a1d0 /= 0; and they arestrongly regular if additionally θ20 /= 4θ−1θ1, where

θ−1 = θ1 = b1c0 + a1d0,

θ0 = 2(a1c0 + b1d0).(2.13)

Since a1 = (α − γ), b1 = (1 − αγ), c0 = 1, d0 = −γ , we obtain θ−1 = θ1 = γ2 − 2αγ + 1, θ0 =2(α − 2γ + αγ2). It follows from α2 /= 1, γ2 /= 1 and γ /=α ±

√α2 − 1 that the boundary conditions


(2.11) are strongly regular. It is known [12, 13] that the eigenfunctions of an operator withstrongly regular boundary conditions constitute a Riesz basis in L2(−1, 1). By (2.2) numbers−λk cannot be eigenvalues of L, hence any eigenfunction of L2 which corresponds to λ2

kwill

be an eigenfunction Lwhich corresponds to λk as

(L2 − λ2kE

)uk = (L + λkE)(L − λkE)uk = 0. (2.14)

Finally, we deduce the assertion of Theorem 2.1 in the case γ2 /= 1.For the case γ2 = 1 the explicit representations of eigenfunctions (2.5) and (2.7) give

the Riesz basis property for these systems directly.

Remark 2.2. If α = 0, then (1.3)-(1.4) coincide with the unperturbed problem (1.1)-(1.2)whichis a Volterra operator for γ2 = −1, that is, γ = α ±

√α2 − 1. If α/= 0 and γ = α ±

√α2 − 1, then

the boundary conditions (2.7)–(2.10) are nonregular and hence the system of eigenfunctionsis incomplete [12, 13]. Finally, for α2 = 1, (1.3) has only trivial solution.

Now, we consider other types of non-selfadjoint perturbations of (1.1)-(1.2).

Theorem 2.3. If γ2 /= ± 1, then the eigenfunctions of the following spectral problem:

u′(−x) + αu(−x) = λu(x), −1 < x < 1,

u(−1) = γu(1),(2.15)

constitute a Riesz basis of L2(−1, 1).

Proof. The proof is analogous to the proof of Theorem 2.1. It uses the following spectralproblem:

−u′′(x) + α2u(x) = λu(x), −1 < x < 1, (2.16)

γu′(−1) − u′(1) + α(γ2 − 1

)u(1) = 0,

u(−1) − γu(1) = 0.(2.17)

Boundary conditions (2.17) are regular for γ2 = −1, and nonregular for γ2 = 1. Then,basis property for eigenfunctions of an ordinary differential second order operator withconstant coefficients gives the result for γ2 = 1.

For γ2 /= ± 1, boundary conditions (2.17) are strongly regular and the proof terminatesanalogously to the proof of Theorem 2.1.

Remark 2.4. The perturbation u′(−x)+αu(x) = λu(x) of (1.1)-(1.2) has the same form after thesubstitution λ−α = μ. Hence, the result of [7] gives full description of basis properties for thefollowing spectral problem:

u′(−x) + αu(x) = λu(x), −1 < x < 1,

u(−1) = λu(1).(2.18)


References

[1] C. Babbage, “An essay towards the calculus of calculus of functions, Part II,” Philosophical Transactionsof the Royal Society B, vol. 106, pp. 179–256, 1816.

[2] D. Przeworska-Rolewicz, Equations with Transformed Argument, An Algebraic Approach, Elsevier-PWN, Amsterdam, The Netherlands, 1973.

[3] J. Wiener, Generalized Solutions of Functional-Differential Equations, World Scientific Publishing,Singapore, 1993.

[4] W. Watkins, “Modified Wiener equations,” International Journal of Mathematics and MathematicalSciences, vol. 27, no. 6, pp. 347–356, 2001.

[5] M. Sh. Burlutskaya, V. P. Kurdyumov, A. S. Lukonina, and A. P. Khromov, “A functional-differentialoperator with involution,” Doklady Mathematics, vol. 75, pp. 399–402, 2007.

[6] W. T.Watkins, “Asymptotic properties of differential equations with involutions,” International Journalof Pure and Applied Mathematics, vol. 44, no. 4, pp. 485–492, 2008.

[7] M.A. Sadybekov andA.M. Sarsenbi, “Solution of fundamental spectral problems for all the boundaryvalue problems for a first-order differential equation with a deviating argument,” Uzbek MathematicalJournal, no. 3, pp. 88–94, 2007 (Russian).

[8] A. M. Sarsenbi, “Unconditional bases related to a nonclassical second-order differential operator,”Differential Equations, vol. 46, no. 4, pp. 506–511, 2010.

[9] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 1966.[10] M. Sh. Burlutskaya and A. P. Khromov, “On an equiconvergence theorem on the whole interval

for functional-differential operators,” Proceedings of Saratov University, vol. 9:4, no. 1, pp. 3–12, 2009(Russian).

[11] M. Sh. Burlutskaya and A. P. Khromov, “Classical solution of a mixed problem with involution,”Doklady Mathematics, vol. 82, pp. 865–868, 2010.

[12] M. A. Naimark, Linear Differential Operators. Part I: Elementary Theory of Linear Differential Operators,Frederick Ungar, New York, NY, USA, 1967.

[13] N. Dunford and J. T. Schwartz, Linear Operators. Part III, Spectral Operators, John Wiley & Sons, NewYork, NY, USA, 1988.

[14] V. A. Il’in and L. V. Kritskov, “Properties of spectral expansions corresponding to non-self-adjointdifferential operators,” Journal of Mathematical Sciences, vol. 116, no. 5, pp. 3489–3550, 2003.


Research ArticleThe Numerical Solution of the Bitsadze-SamarskiiNonlocal Boundary Value Problems withthe Dirichlet-Neumann Condition

Allaberen Ashyralyev1, 2 and Elif Ozturk3

1 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey2 Department of Mathematics, International Turkmen-Turkish University, 74400 Ashgabat, Turkmenistan3 Department of Mathematics, Uludag University, 16059 Bursa, Turkey

Correspondence should be addressed to Elif Ozturk, [email protected]



Copyright q 2012 A. Ashyralyev and E. Ozturk. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We are interested in studying the stable difference schemes for the numerical solution of thenonlocal boundary value problemwith the Dirichlet-Neumann condition for themultidimensionalelliptic equation. The first and second orders of accuracy difference schemes are presented. Aprocedure of modified Gauss elimination method is used for solving these difference schemesfor the two-dimensional elliptic differential equation. The method is illustrated by numericalexamples.

1. Introduction

Methods of solution of the Bitsadze-Samarskii nonlocal boundary value problems for ellipticdifferential equations have been studied extensively by many researchers (see [1–22] and thereferences given therein).

Let Ω be the unit open cube in Rm (x = (x1, . . . , xm) : 0 < xk < 1, 1 ≤ k ≤ m) withboundary S, Ω = Ω ∪ S. In [0, 1] × Ω, the Bitsadze-Samarskii-type nonlocal boundary valueproblem for the multidimensional elliptic equation

−utt −m∑

r=1

(ar(x)uxr )xr + ηu = f(t, x), 0 < t < 1, x = (x1, . . . , xm) ∈ Ω,

u(0, x) = ϕ(x), u(1, x) =J∑

j=1

αju(λj , x

)+ ψ(x), x ∈ Ω,


J∑

j=1

∣∣αj∣∣ ≤ 1, 0 < λ1 < · · · < λJ < 1,

u(t, x)|x∈S1 = 0,∂u(t, x)∂�n

∣∣∣∣x∈S2

= 0, S1 ∪ S2 = S

(1.1)

is considered. Here ar(x), (x ∈ Ω), ψ(x), ϕ(x) (x ∈ Ω), and f(t, x) (t ∈ (0, 1), x ∈ Ω) aregiven smooth functions, ar(x) ≥ a > 0, η is a positive number, and �n is the normal vector toΩ. We are interested in studying the stable difference schemes for the numerical solution ofthe nonlocal boundary value problem (1). The first and second orders of accuracy differenceschemes are presented. The stability and almost coercive stability of these difference schemesare established. A procedure of modified Gauss elimination method is used for solving thesedifference schemes in the case of two-dimensional elliptic partial differential equations.

2. Difference Schemes: The Stability and Coercive Stability Estimates

The discretization of problem (1) is carried out in two steps. In the first step, let us define thegrid sets

Ωh = {x = xm = (h1m1, . . . , hmmm), m = (m1, . . . , mm),

0 ≤ mr ≤Nr, hrNr = 1, r = 1, . . . , m},

Ωh = Ωh ∩Ω, Srh = Ωh ∩ Sr, r = 1, 2.

(2.1)

We introduce the Hilbert space L2h = L2(Ωh) andW22h =W2

2 (Ωh) of the grid functions ϕh(x) ={ϕ(h1m1, . . . , hmmm)} defined on Ωh, equipped with the norms

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

m∑

r=1

∣∣∣∣(ϕh)

xr

∣∣∣∣2

h1 · · ·hm⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

m∑

r=1

∣∣∣∣(ϕh)

xrxr ,mr

∣∣∣∣2

h1 · · ·hm⎞

⎠1/2

,

∥∥∥ϕh∥∥∥L2(Ωh)

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hm

⎞

⎠1/2

.

(2.2)

To the differential operatorA generated by problem (1), we assign the difference operatorAxh

by the formula

Axhu

h = −m∑

r=1

(ar(x)uhxr

)

xr ,jr+ ηuh(x), (2.3)


acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ S1h

andDhuh(x) = 0 for all x ∈ S2

h. Here,Dhu

h(x) is an approximation to ∂u/∂�n. It is known thatAxhis a self-adjoint positive definite operator in L2(Ωh). With the help of Ax

h, we arrive at the

nonlocal boundary value problem

−d2uh(t, x)dt2

+Axhu

h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,

uh(0, x) = ϕh(x), uh(1, x) =J∑

j=1

αjuh(λj , x

)+ ψh(x), x ∈ Ωh,

J∑

j=1

∣∣αj∣∣ ≤ 1, 0 < λ1 < · · · < λJ < 1,

(2.4)

for an infinite system of ordinary differential equations. In the second step, we replaceproblem (2.4) by the first and second orders of accuracy difference schemes

−uhk+1(x) − 2uhk(x) + u

hk−1(x)

τ2+Ax

huhk(x) = f

hk (x),


uh0(x) = ϕh(x), x ∈ Ωh,

uhN(x) =J∑

j=1

αjuh[λj/τ]

(x) + ψh(x), x ∈ Ωh,

(2.5)

−uhk+1(x) − 2uh

k(x) + uhk−1(x)

τ2+Ax

huhk(x) = f

hk (x), fhk (x) = f

h(tk, x),

tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uh0(x) = ϕh(x), x ∈ Ωh,

uhN(x) =J∑

j=1

αj

(uh[λj/τ](x) +

(uh[λj/τ]+1(x) − u

h[λj/τ]

(x))(λj

τ−[λj

τ

]))+ ψh(x), x ∈ Ωh.

(2.6)

To formulate our result on well-posedness, we will give definition of Cα01([0, 1]τ ,H) and

C([0, 1]τ ,H). Let F([0, 1]τ ,H) be the linear space of mesh functions ϕτ = {ϕk}N−11 with values

in the Hilbert spaceH. We denote C([0, 1]τ ,H) normed space with the norm

∥∥ϕτ∥∥C([0,1]τ ,H) = max

1≤k≤N−1

∥∥ϕk∥∥H, (2.7)


and Cα01([0, 1]τ ,H) normed space with the norm

∥∥ϕτ∥∥Cα

01([0, 1]τ ,H) =∥∥ϕτ

∥∥C([0, 1]τ ,H) + sup

1≤k≤k+r≤N−1

((N − k)τ)α((k + r)τ)α

(rτ)α∥∥ϕk+r − ϕk

∥∥H. (2.8)

Theorem 2.1. Let τ and |h| be sufficiently small positive numbers. Then, the solutions of differenceschemes (2.5) and (2.6) satisfy the following stability and almost coercive stability estimates

∥∥∥∥{uhk

}N−1

1

∥∥∥∥C([0, 1]τ , L2h)

≤M1

[∥∥∥ϕh∥∥∥L2h

+∥∥∥ψh

∥∥∥L2h

+∥∥∥∥{fhk

}N−1

1

∥∥∥∥C([0, 1]τ , L2h)

],

∥∥∥∥∥∥

{uhk+1 − 2uhk + u

hk−1

τ2

}N−1

1

∥∥∥∥∥∥C([0, 1]τ ,L2h)

+∥∥∥∥{uhk

}N−1

1

∥∥∥∥C([0, 1]τ , W

22h)

≤M2

[∥∥∥ϕh∥∥∥W2

2h

+∥∥∥ψh

∥∥∥W2

2h

+ ln1

τ + |h|∥∥∥∥{fhk

}N−1

1

∥∥∥∥C([0, 1]τ , L2h)

].

(2.9)

Here,M1 andM2 do not depend on τ, h, ψh(x), ϕh(x), and fhk (x), 1 ≤ k ≤N − 1.

Theorem 2.2. Let τ and |h| be sufficiently small positive numbers. Then, the solution of differenceschemes (2.5) and (2.6) satisfies the following coercive stability estimate:

∥∥∥∥∥∥

{uhk+1 − 2uh

k+ uh

k−1τ2

}N−1

1

∥∥∥∥∥∥Cα

01([0, 1]τ , L2h)

+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα

01([0, 1]τ , W22h)

≤M3

[∥∥∥ϕh∥∥∥W2

2h

+∥∥∥ψh

∥∥∥W2

2h

+1

α(1 − α)∥∥∥∥{fhk

}N−1

1

∥∥∥∥Cα

01([0, 1]τ , L2h)

].

(2.10)

M3 is independent of τ, h, ψh(x), ϕh(x), and fhk (x), 1 ≤ k ≤N − 1.


Proofs of Theorems 2.1 and 2.2 are based on the symmetry properties of operator Axh

defined by formula (2.3) and on the following formulas:

uhk(x) =(I − R2N

)−1

×{(

Rk − R2N−k)ϕh(x)+

(RN−k − RN+k

)uhN(x)−

(RN−k − RN+k

)

×(I + τB)(2I + τB)−1B−1N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

i=1

(R|k−i| − Rk+i

)fhi (x)τ,

uhN(x) = D−1(

J∑

k=1

αk(I − R2N

)−1

×{(

R[λk/τ]−R2N−[λk/τ])ϕh(x)−

(RN−[λk/τ]−RN+[λk/τ]

)(I+τB)(2I+τB)−1B−1

×N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

}+ (I + τB)(2I + τB)−1B−1

×⎛

⎝[λk/τ]∑

i=1

R[λk/τ]−ifhi (x)τ +N−1∑

i=[λk/τ]+1

Ri−[λk/τ]fhi (x)τ

−N−1∑

i=1

R[λk/τ]+ifhi (x)τ

)+ ψh(x)

),

(2.11)

for difference scheme (2.5), and

uhN(x) = D−1(

J∑

k=1

αk(I − R2N

)−1

×{(

R[λk/τ]−R2N−[λk/τ])ϕh(x)−

(RN−[λk/τ]−RN+[λk/τ]

)(I+τB)(2I+τB)−1

×B−1N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

}+(I + τB)(2I + τB)−1B−1

×⎛

⎝[λk/τ]∑

i=1

R[λk/τ]−ifhi (x)τ+N−1∑

i=[λk/τ]+1

Ri−[λk/τ]fhi (x)τ−N−1∑

i=1

R[λk/τ]+ifhi (x)τ

⎞

⎠

+(λkτ

−[λkτ

])(I − R2N

)−1


×{τB(R

[λk/τ]+1 − R2N−[λk/τ])ϕh(x) −

(RN−[λk/τ]−1 − RN+[λk/τ]

)(2I + τB)−1

×N−1∑

i=1

(RN−i − RN+i

)fhi (x)τ

2

}

+ (2I + τB)−1⎛

⎝[λk/τ]∑

i=1

R[λk/τ]−ifhi (x)τ2 +

N−1∑

i=[λk/τ]+1

Ri−[λk/τ]−1fhi (x)τ2

−N−1∑

i=1

R[λk/τ]+ifhi (x)τ2

)+ψh(x)

),

(2.12)

for difference scheme (2.6). Here,

R = (I + τB)−1,

B =τA

2+

√τ2A2

4+A, A = Ax

h,

D = I − R2N −J∑

k=1

αk(R

N−[λk/τ] − R N+[λk/τ])for (2.5),

D = I − R2N −J∑

k=1

αk

(R

N−[λk/τ] − R N+[λk/τ] − 1τ

(λk −

[λkτ

]τ

)

×B(RN−[λk/τ]−RN+[λk/τ]+1

))for (2.6),

(2.13)

and on the following theorem on the coercivity inequality for the solution of the ellipticdifference problem in L2h.

Theorem 2.3 (see [22]). For the solution of the elliptic difference problem

Axhu

h(x) = ωh(x), x ∈ Ωh,

uh(x)|x∈S1h= 0, Dhu

h(x)|x∈S2h= 0, S1

h ∪ S2h = Sh,

(2.14)

the following coercivity inequality holds:

m∑

r=1

∥∥∥∥(uh)

xrxr , jr

∥∥∥∥L2h

≤M4

∥∥∥ωh∥∥∥L2h, (2.15)

whereM4 does not depend on h and ωh(x).


Note that we have not been able to obtain sharp estimate for the constants figuring inthe stability estimates. Hence, in the following section, we study difference schemes (2.5) and(2.6) by numerical experiments.


For the numerical result, we consider the nonlocal boundary value problem

−∂2u(t, x)∂t2

− ∂2u(t, x)∂x2

+ u = 2 exp(−t)(x − 1

2x2 +

t

2− 1),

0 < t < 1, 0 < x < 1,

u(0, x) = x2 − 2x,

u(1, x) = u(12, x

)+

(x2

2− x)

exp(−1) −(

3x2

4− 3x

2

)exp

(−12

), 0 ≤ x ≤ 1,

u(t, 0) = ux(t, 1) = 0, 0 ≤ t ≤ 1,

(3.1)

for the elliptic equation. The exact solution of (3.1) is

u(t, x) =

(tx − tx2

2+ x2 − 2x

)exp(−t). (3.2)

For the approximate solution of the nonlocal boundary Bitsadze-Samarskii problem (3.1), weconsider the set [0, 1]τ × [0, 1]h of a family of grid points depending on the small parametersτ and h

[0, 1]τ × [0, 1]h = {(tk, xn) : tk = kτ, 1 ≤ k ≤N − 1,Nτ = 1

xn = nh, 1 ≤ n ≤M − 1,Mh = 1}.(3.3)

Firstly, applying difference scheme (2.5), we present the first order of accuracy differencescheme for the approximate solution of problem (3.1) is


−uk+1n − 2ukn + u

k−1n


kn−1

h2+ ukn = f(tk, xn),

1 ≤ k ≤N − 1, 1 ≤ n ≤M − 1,

u0n = ϕ(xn), 0 ≤ n ≤M,

uNn = u[N/2]n +

(x2n

2− xn

)exp(−1)

−(

3x2n

4− 3xn

2

)exp

(−12

), 0 ≤ n ≤M,

uk0 =ukM − ukM−1

h= 0, 0 ≤ k ≤N,

f(tk, xn) = 2 exp(−tk)(xn −

x2n

2+tk2− 1

),

ϕ(xn) = x2n − 2xn.

(3.4)

Then, we have an (N + 1) × (M + 1) system of linear equations and we will write them in thematrix form

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = 0, UM −UM−1 = 0,(3.5)

where

A =

⎡⎢⎢⎢⎢⎢⎣

0 0 0 · 0 · 0 0 00 a 0 · 0 · 0 0 0· · · · · · · · ·0 0 0 · 0 · 0 a 00 0 0 · 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎣

1 0 0 · 0 · 0 0 0c b c · 0 · 0 0 0· · · · · · · · ·0 0 0 · 0 · c b c0 0 0 · −1 · 0 0 1

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

(3.6)

and C = A, D is an (N + 1) × (N + 1) identity matrix and


Us =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

u0s

u1s

·uN−1s

uNs

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, (3.7)

where s = n − 1, n, n + 1,

ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

·ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

. (3.8)

Here,

a = − 1h2, b =

2τ2

+2h2

+ 1, c = − 1τ2,

ϕkn =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(x2n − 2xn

), k = 0,

f(tk, xn), 1 ≤ k ≤N − 1,(x2n

2− xn

)exp(−1) −

(3x2

n

4− 3xn

2

)exp

(−12

), k =N.

(3.9)

So, we have a second-order difference equation with respect to nmatrix coefficients. To solvethis difference equation, we have applied a procedure of modified Gauss elimination methodfor difference equation with respect to nmatrix coefficients. Hence, we seek a solution of thematrix equation in the following form:

Uj = αj+1Uj+1 + βj+1, j =M − 1, . . . , 1,

UM = (I − αM)−1βM,

αj+1 = −(B + Cαj)−1

A,

βj+1 =(B + Cαj

)−1(Dϕj − Cβj

), j = 1, . . . ,M − 1,

(3.10)

where αj (j = 1, . . . ,M) are (N+1)×(N+1) square matrix and βj (j = 1, . . . ,M) are (N+1)×1column matrix and α1 is the (N + 1) × (N + 1) zero matrix and βj is the (N + 1) × 1 zero


matrix. Secondly, applying difference scheme (2.6), we present the following second order ofaccuracy difference scheme for the approximate solutions of problem (3.1):

−uk+1n − 2ukn + u

k−1n


kn−1

h2+ ukn = f(tk, xn),

1 ≤ k ≤N − 1, 1 ≤ n ≤M − 1,

u0n = ϕ(xn), 0 ≤ n ≤M,

uNn = u[N/2]n +(u[N/2]+1n − u[N/2]n

)(N2

−[N

2

])+

(x2n

2− xn

)exp(−1)

−(

3x2n

4− 3xn

2

)exp

(−12

), 0 ≤ n ≤M,

uk0 = 0, ukM−2 − 4ukM−1 + 3ukM = 0, 0 ≤ k ≤N,

f(tk, xn) = 2 exp(−tk)(xn −

x2n

2+tk2− 1

),

ϕ(xn) = x2n − 2xn.

(3.11)

So, we have again an (N + 1) × (M + 1) system of linear equations and we will write in thematrix form

AUn+1 + BUn + CUn−1 = Rϕn, 1 ≤ n ≤M − 1,

U0 = 0, UM−2 − 4UM−1 + 3UM = 0,(3.12)

where

A =

⎡⎢⎢⎢⎢⎢⎣

0 0 0 · 0 0 · 0 0 00 a 0 · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · 0 a 00 0 0 · 0 0 · 0 0 0

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎣

1 0 0 · 0 0 · 0 0 0c b c · 0 0 · 0 0 0· · · · · · · · · ·0 0 0 · 0 0 · c b c0 0 0 · d e · 0 0 1

⎤⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,


C = A, R = D,

Us =

⎡⎢⎢⎢⎢⎢⎣

u0su1s·

uN−1s

uNs

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

(3.13)

where s = n − 1, n, n + 1 and ϕn =

⎡⎢⎣

ϕ0n

ϕ1n·

ϕN−1n

ϕNn

⎤⎥⎦

(N+1)×1

.

Here,

a = − 1h2, b =

2h2

+2τ2

+ 1, c = − 1τ2, d =

[N

2

]− N

2, e = −1 − d,

ϕkn =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(x2n − 2xn

), k = 0,

f(tk, xn), 1 ≤ k ≤N − 1,(x2n

2− xn

)exp(−1) −

(3x2

n

4− 3xn

2

)exp

(−12

), k =N.

(3.14)

Thus, we have a second-order difference equation with respect to n matrix coefficients.To solve this difference equation, we have applied the same procedure of modified Gausselimination method (3.10) for difference equation with respect to nmatrix coefficients with

UM = (3I + αMαM−1 − 4αM)−1(−βMαM−1 − βM−1 + 4βM

). (3.15)

Now, we will give the results of the numerical analysis. The errors computed by

ENM = max1≤k≤N−1

(M−1∑

n=1

∣∣∣u(tk, xn) − ukn∣∣∣2h

)1/2

(3.16)

of the numerical solutions for different values of M and N, where u(tk, xn) represents theexact solution and ukn represents the numerical solution at (tk, xn). Table 1 gives the erroranalysis between the exact solution and solutions derived by difference schemes for N =M = 20, 40, and 60, respectively.

4. Conclusion

In this work, the first and second orders of accuracy difference schemes for the approximatesolution of the Bitsadze-Samarskii nonlocal boundary value problem for elliptic equationsare presented. Theorems on the stability estimates, almost coercive stability estimates, andcoercive stability estimates for the solution of difference schemes for elliptic equations are



Difference schemes N =M = 20 N =M = 40 N =M = 60Difference scheme (2.5) 0.0049 0.0025 0.0012Difference scheme (2.6) 3.7155e − 005 9.4107e − 006 2.3679e − 006

proved. The theoretical statements for the solution of these difference schemes are supportedby the results of numerical examples. The second order of accuracy difference scheme is moreaccurate comparing with the first order of accuracy difference scheme. As a future work, highorders of accuracy difference schemes for the approximate solutions of this problem couldbe established. Theorems on the stability estimates, almost coercive stability estimates, andcoercive stability estimates for the solution of difference schemes for elliptic equations couldbe proved.

References

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Advances and Applications, Birkhauser, Basel, Switzerland, 1997.[4] A. V. Bitsadze and A. A. Samarskiı, “Some elementary generalizations of linear elliptic boundary

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dell’Universita di Trieste, vol. 28, no. suppl., pp. 337–382, 1996.[6] A. Ashyralyev, “Well-posed solvability of the boundary value problem for difference equations of

elliptic type,” Nonlinear Analysis. Theory, Methods & Applications, vol. 24, no. 2, pp. 251–256, 1995.[7] A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148

of Operator Theory: Advances and Applications, Birkhauser, Basel, Switzerland, 2004.[8] S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic

partial differential equations satisfying general boundary conditions—II,” Communications on Pureand Applied Mathematics, vol. 17, pp. 35–92, 1964.

[9] G. Berikelashvili, “On a nonlocal boundary-value problem for two-dimensional elliptic equation,”Computational Methods in Applied Mathematics, vol. 3, no. 1, pp. 35–44, 2003.

[10] D. G. Gordeziani, “On a method of resolution of Bitsadze-Samarskii boundary value problem,”Abstracts of Reports of Institute of Applied Mathematics, Tbilisi State University, vol. 2, pp. 38–40, 1970.

[11] D. V. Kapanadze, “On a nonlocal Bitsadze-Samarskiı boundary value problem,” Differentsial’nyeUravneniya, vol. 23, no. 3, pp. 543–545, 1987.

[12] V. A. Il’in and E. I. Moiseev, “A two-dimensional nonlocal boundary value problem for the Poissonoperator in the differential and the difference interpretation,” Matematicheskoe Modelirovanie, vol. 2,no. 8, pp. 139–156, 1990.


[14] A. P. Soldatov, “A problem of Bitsadze-Samarskiı type for second-order elliptic systems on the plane,”Rossiıskaya Akademiya Nauk. Doklady Akademii Nauk, vol. 410, no. 5, pp. 607–611, 2006.

[15] R. P. Agarwal, D. O’Regan, and V. B. Shakhmurov, “B-separable boundary value problems in Banach-valued function spaces,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 48–63, 2009.

[16] V. Shakhmurov, “Linear and nonlinear abstract equations with parameters,” Nonlinear Analysis.Theory, Methods & Applications, vol. 73, no. 8, pp. 2383–2397, 2010.

[17] A. A. Dosiev and B. S. Ashirov, “On a numerical solution of a multipoint problem for a second-orderordinary differential equation with singular coefficients,” in Approximate Methods for Solving OperatorEquations, pp. 41–46, Bakinskii Gosudarstvennogo Universiteta, Baku, Azerbaijan, 1991.

[18] A. Ashyralyev, “Nonlocal boundary-value problems for abstract elliptic equations: well-posedness inBochner spaces,” AIP Conference Proceedings, vol. 1309, pp. 66–85, 2010.


[19] M. P. Sapagovas, “A difference method of increased order of accuracy for the Poisson equation withnonlocal conditions,” Differentsial’nye Uravneniya, vol. 44, no. 7, pp. 988–998, 2008.

[20] A. Ashyralyev, C. Cuevas, and S. Piskarev, “On well-posedness of difference schemes for abstractelliptic problems in Lp([0, 1, E]) spaces,” Numerical Functional Analysis and Optimization, vol. 29, no.1-2, pp. 43–65, 2008.

[21] A. Ashyralyev, “Well-posedness of the difference schemes for elliptic equations in Cβ,γτ (E) spaces,”

Applied Mathematics Letters, vol. 22, no. 3, pp. 390–395, 2009.[22] P. E. Sobolevskii,Difference Methods for the Approximate Solution of Differential Equations, Voronezh State

University Press, Voronezh, Russia, 1975.


Research ArticleExistence and Nonexistence of Positive Solutionsfor Quasilinear Elliptic Problem

K. Saoudi

Institut Superieur d’Informatique et de Multimedia de Gabe (ISIMG),Campus Universitaire Cite Erriadh, Zirig-Gabes 6075, Tunisia

Correspondence should be addressed to K. Saoudi, [email protected]

Received 18 April 2012; Accepted 21 June 2012


Copyright q 2012 K. Saoudi. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Using variational arguments we prove some existence and nonexistence results for positivesolutions of a class of elliptic boundary-value problems involving the p-Laplacian.

1. Introduction

In a recent paper, Radulescu and Repovs [1] studied the existence and nonexistence ofpositive solutions of the nonlinear elliptic problem

−Δu = λk(x)uq ± h(x)up in Ω,

u|∂Ω = 0, u > 0 in Ω,(1.1)

where Ω is a smooth bounded domain in Rn, λ > 0 is a parameter, 0 < q < 1 < p, and h, k in

L∞(Ω) such that

ess infx∈Ω

k(x) > 0, ess infx∈Ω

h(x) > 0. (1.2)

They showed using sub-supersolutions arguments and monotonicity methods that theproblem (1.1)+ has a minimal solution, provided that λ > 0 is small enough. The next resultis concerned with problem (1.1)− and asserts that there is some λ∗ > 0 such that (1.1)− has anontrivial solution if λ > λ∗ and no solution exists provided that λ < λ∗.


In the present paper we consider that the corresponding quasilinear problem

−Δpu = λk(x)uq ± h(x)ur in Ω,

u|∂Ω = 0, u > 0 in Ω,(1.3)

where Δpu = div(|∇u|p−2∇u), denotes the p-Laplacian operator, 1 < p < ∞, λ > 0, 0 ≤ q <p − 1 < r < p∗ − 1, with p∗ =Np/(N − p) if p < N, and p∗ = +∞ otherwise, and h, k in L∞(Ω)such that

ess infx∈Ω

k(x) > 0, ess infx∈Ω

h(x) > 0. (1.4)

We are concerned with the existence of weak solutions of problems (1.3)+ and (1.3)−, that is,for functions u ∈W1,p

0 (Ω) satisfying ess infKu > 0 over every compact set K ⊂ Ω and

∫

Ω|∇u|p−2∇u · ∇φdx = λ

∫

Ωk(x)uqφdx ±

∫

Ωh(x)urφdx (1.5)

for all φ ∈ C∞c (Ω). As usual, C∞

c (Ω) denotes the space of all C∞ functions φ : Ω → R withcompact support. Using variational methods, we will prove the following theorems.

Theorem 1.1. Assume 0 ≤ q < p − 1 < r < p∗ − 1. Then there exists a positive number Λ such thatthe following properties hold:

(1) for all λ ∈ (0,Λ) problem (1.3)+ has a minimal solution uλ;

(2) Problem (1.3)+ has a solution if λ = Λ;

(3) Problem (1.3)+ does not have any solution if λ > Λ.

Theorem 1.2. Assume 0 ≤ q < p − 1 < r < p∗ − 1. Then there exists a positive number Λ such thatthe following properties hold:

(1) If λ > Λ, then problem (1.3)− has at least one solution;

(2) If λ < Λ, then problem (1.3)− does not have any solution.

2. Proof of Theorem 1.1

At first, we give the definition of weak supersolution and subsolution of (1.3)+. By definitionu ∈W1,p

0 (Ω) is a weak subsolution to (1.3)+ if u > 0 in Ω and

∫

Ω|∇u|p−2∇u · ∇φdx ≤ λ

∫

Ωk(x)uqφdx ±

∫

Ωh(x)urφdx (2.1)

for all φ ∈ C∞c (Ω). Similarly u ∈ W1,p

0 (Ω) is a weak supersolution to (1.3)+ if in the above thereverse inequalities hold.


Let us define

Λ def= sup{λ > 0 : (1.3)+has a weak solution} (2.2)

and the energy functional Eλ :W1,p0 (Ω) → R defined by

Eλ(u) =1p

∫

Ω|∇u|pdx

− λ

q + 1

∫

Ωk(x)uq+1dx − 1

r + 1

∫

Ωh(x)uq+1dx

(2.3)

in the Sobolev spaceW1,p0 (Ω).

The proof of the theorem is organized in several steps.

Step 1 (existence of minimal solution for 0 < λ < Λ). To show the existence of a solution to(1.3)+, we construct a subsolution uλ, and a supersolution uλ, such that uλ ≤ uλ.

We introduce the following Dirichlet problem:

−Δpu = λk(x)uq in Ω,

u|∂Ω = 0, u > 0 in Ω.(2.4)

From [2] we know there exists a unique solution, say u, satisfying the problem (2.4). Defineuλ = εu. Then −Δp(uλ) = λk(x)ε

p−1uq and uλ is a subsolution of the problem (1.3)+ if

λk(x)εp−1uq ≤ λk(x)εquq + h(x)εrur . (2.5)

Indeed, for ε small enough we get

λk(x)εp−1uq ≤ λk(x)εquq ≤ λk(x)εquq + h(x)εrur . (2.6)

(Since q < p − 1 and for ε ∈ (0, 1)). Then εu is a subsolution of the problem (1.3)+.On the other hand, let v the solution to the following problem be:

−Δpv = λ + 1 in Ω,

v|∂Ω = 0, v > 0 in Ω.(2.7)

Then 0 < v < K in Ω. By simplicity of writing we call

F(u) = λk(x)uq + h(x)ur. (2.8)

Define uλ(x) = Tv(x)where T is a constant that will be chosen in such a way that

−Δpuλ ≥ F(TM) ≥ F(uλ), (2.9)


whereM = max{1, ‖v‖∞}. Now −Δpuλ = Tp−1(λ + 1) and

F(uλ) ≡ λk(x)Tqvq + Trvr ≤ λc1TqMq + c2TrMr, (2.10)

where c1 = ‖k‖L∞ et c2 = ‖h‖L∞ . Then, it is sufficient to find T such that

(λ + 1) ≥ λc1Tq+1−pMq + c2Tr+1−pMr. (2.11)

We call

ϕ(T) = λATq+1−p + BTr+1−p, (2.12)

with A = c1Mq,B = c2Mr . Then

limT→ 0+

ϕ(T) = limT→∞

ϕ(T) = ∞, (2.13)

because q + 1 − p < 0 < r + 1 − p; then ϕ attains a minimum in [0,∞). Elementarycomputations shows that this function attains its minimum for T0 = Cλ1/(r−q) where C =

[AB−1(r − p + 1)(p − q − 1)−1]1/(r−q)

. For the validity of (2.11) it suffices that

ϕ(T0) ≤ λ + 1, (2.14)

that is,

Dλ(r+1−p)/(r−q) < λ + 1, (2.15)

where D is a constant, depends on p, q, and M. Then there exists λ0 such that for 0 < λ <λ0, u(x) = T0v is a supersolution of problem (1.3)+. It remains to show that εu ≤ T0v. In turn,fix the supersolution, that is, T , for ε small enough, we get

−Δpuλ = λk(x)εp−1uq ≤ λεp−1 ≤ −Δp(uλ). (2.16)

Consequently, we may apply the weak comparison principle (see Proposition 2.3 in [3]) inorder to conclude that uλ ≤ uλ. Thus, By the classical iteration method (1.3)+ has a solutionbetween the subsolution and supersolution.

Let us now prove that uλ is a minimal solution of (1.3)+. We use here the weakcomparison principle (see Proposition 2.3 in Cuesta and Takac [3]) and the followingmonotone iterative scheme:

−Δpun = λk(x)uqn−1 + h(x)urn−1 in Ω;

un|∂Ω = 0,(2.17)


where u0 = uλ, the unique solution to (2.4). Note that u0 is a weak subsolution to (1.3)+. andu0 ≤ U where U is any weak solution to (1.3)+. Then, from the weak comparison principle,we get easily that u0 ≤ u1 and {un}∞n=1 is a nondecreasing sequence. Furthermore, un ≤ U

and {un}∞n=1 is uniformly bounded inW1,p0 (Ω). Hence, it is easy to prove that {un} converges

weakly in W1,p0 (Ω) and pointwise to uλ, a weak solution to (1.3)+. Let us show that uλ is

the minimal solution to (1.3)+ for any 0 < λ < Λ. Let vλ a weak solution to (1.3)+ for any0 < λ < Λ. Then, u0 = uλ ≤ vλ. From the weak comparison principle, un ≤ vλ for any n ≥ 0.Letting n → ∞, we get uλ ≤ vλ. This completes the proof of the Step 1.

Step 2 (there exists Λ > 0 such that (1.3)+ has no positive solution for λ > Λ). From thedefinition of Λ, problem (1.3)+ does not have any solution if λ > Λ. In what follows we claimthat Λ < ∞. We argue by contradiction: suppose there exists a sequence λn → ∞ such that(1.3)+ admits a solution un. Denote

m := min{ess inf

x∈Ωk(x), ess inf

x∈Ωh(x)

}> 0. (2.18)

There exists λ∗ > 0 such that

m(λtq + tr) ≥ (λ1 + ε)tp−1 ∀t > 0, ε ∈ (0, 1), λ > λ∗, (2.19)

where λ1 is the first Dirichlet eigenvalue of −Δp is positive and is given by

λ1 = minu/= 0

∫Ω |∇u|p∫Ω |u|p (2.20)

(see Lindqvist [4]). Choose λn > λ∗. Clearly un is a supersolution of the problem

−Δpu = (λ1 + ε)up−1 in Ω,

u > 0, u|∂Ω = 0(2.21)

for all ε ∈ (0, 1). We now use the result in [2] to choose μ < λ1 + ε small enough so thatμφ1(x) < un(x) and μφ1 is a subsolution to problem (2.8). By amonotone interation procedurewe obtain a solution to (2.8) for any ε ∈ (0, 1), contradicting the fact that λ1 is an isolated pointin the spectrum of −Δp inW

1,p0 (Ω) (see Anane [5]). This proves the claim and completes the

proof of the Step 2.

Step 3 (there exists at least one positive-weak solution for λ = Λ to (1.3)+). Let {λk}k∈Nbe

such that λk ↑ Λ as k → ∞. Then, from Step 1, there exists uk = uλk ≥ uλk to a weak positivesolution to (1.3)+ for λ = λk. Therefore, for any φ ∈ C∞

c (Ω), we have

∫

Ω|∇uk|p−2∇uk∇φdx = λkk(x)

∫

Ω(uk)qφdx + h(x)

∫

Ωurkφdx. (2.22)


Since uk ∈ W1,p0 (Ω) and uk ≥ uλk it is easy to see that (2.22) holds also for φ ∈ W

1,p0 (Ω).

Moreover, from above

Eλk(uk) ≤ Eλk(uλk

)<

1p

∫

Ω

∣∣∣∇uλk∣∣∣pdx − λkk(x)

q + 1

∫

Ωuλk

q+1dx < 0, (2.23)

it follows that

supk

‖uk‖p <∞. (2.24)

Hence, there exists uΛ ≥ uλk such that uk ⇀ uΛ in W1,p0 (Ω) as k → ∞ and then by

Sobolev imbedding and using the fact that k, h ∈ L∞(Ω):

uk ⇀ u in Lq(Ω) and point wise a.e. as k −→ ∞. (2.25)

From (2.22), (2.24), and (2.25), we get for any φ ∈W1,p0 (Ω)

∫

Ω|∇uΛ|p−2∇uΛ∇φdx = λ

∫

Ωk(x)uqΛφdx +

∫

Ωh(x)urΛφdx (2.26)

which completes the proof of the Step 3 and gives the proof of Theorem 1.1.


At first, we introduce some notation which will be used throughout the proof. The norm inW

1,p0 (Ω)will be denoted by

‖u‖pdef=

(∫

Ω|∇u|pdx

)1/p

. (3.1)

The norm in Lq+1(Ω)will be denoted by

‖u‖q+1def=

(∫

Ω|u|q+1dx

)1/q+1

. (3.2)

The norm in Lr+1(Ω)will be denoted by

‖u‖r+1def=

(∫

Ω|u|r+1dx

)1/r+1

. (3.3)


Let us define the energy functional Jλ :W1,p0 (Ω) → R defined by

Jλ(u) =1p

∫

Ω|∇u|pdx

− λ

q + 1

∫

Ωk(x)uq+1dx +

1r + 1

∫

Ωh(x)ur+1dx

(3.4)

in the Sobolev spaceW1,p0 (Ω).

The proof of the theorem is organized in several steps.

Step 1 (coercivity of Jλ:). For any u ∈W1,p0 (Ω) and all λ > 0

Jλ(u) ≥ 1p‖u‖p − C1‖u‖q+1q+1 + C2‖u‖r+1r+1, (3.5)

where C1 = λ‖k‖L∞/(q + 1) and C2 = (r + 1)−1ess infx∈Ωh(x) are positive constants. We call

φ(T) = ATq+1−p − BTr+1−p. (3.6)

Then

limT→ 0+

φ(T) = limT→∞

φ(T) = ∞, (3.7)

because q + 1 − p < 0 < r + 1 − p; then ϕ attains a minimum m < 0 in [0,∞).By elementary computations shows that this function attains its minimum for T =[A(q + 1 − p)/(Br + 1 − p)]1/(r−q).

Returning to (3.5), we deduce that

Jλ(u) ≥ 1p‖u‖p +m. (3.8)

Hence, from (3.8), we get that

Jλ(u) −→ +∞ as ‖u‖ −→ ∞. (3.9)

Let n �→ un be a minimizing sequence of Jλ inW1,p0 (Ω), which is bounded inW1,p

0 (Ω)by Step 1. Without loss of generality, we may assume that (un)n is nonnegative, convergesweakly to some u in W

1,p0 (Ω), and converges also pointwise. Moreover, by the weak lower

semicontinuity of the norm ‖ · ‖ and the boundedness of (un)n inW1,p0 (Ω)we get

Jλ(u) ≤ limn→∞

inf Jλ(un). (3.10)

Hence u is a global minimizer of Jλ inW1,p0 (Ω), which completes the proof of the Step 1.


Step 2 (the weak limit u is a nonnegative weak solution of (1.3)− if λ > 0 is sufficiently large).Firstly, observe that Jλ(0) = 0. Thus, to prove that the nonnegative solution is nontrivial, itsuffices to prove that there exists λ∗ > 0 such that

infu∈W1,p

0 (Ω)Jλ(u) < 0 ∀λ > λ∗. (3.11)

For this, we consider the constrained minimization problem

λ∗ def= inf{1p

∫

Ω|∇w|pdx +

1r + 1

∫

Ωh(x)|w|r+1dx : w ∈W1,p

0 (Ω) and1

q + 1

×∫

Ωk(x)|w|q+1dx = 1

}.

(3.12)

Let n �→ vn be a minimizing sequence of (3.12) in W1,p0 (Ω), which is bounded in

W1,p0 (Ω), so that we can assume, without loss of generality, that it converges weakly to some

v ∈W1,p0 (Ω), with

1q + 1

∫

Ωk(x)|v|q+1dx = 1, λ∗ =

1p

∫

Ω|∇v|pdx +

1r + 1

∫

Ωh(x)|v|r+1dx. (3.13)

Thus, Jλ(v) = λ∗ − λ < 0 for any λ > λ∗.Now put

Λ def= inf{λ > 0 : (1.3)− admits a non trivial weak solution}. (3.14)

From above λ∗ ≥ Λ and that problem (1.3)− has a solution for all λ > λ∗. The proof of theStep 2 is now completed.

Step 3 (problem (1.3)− has a weak solution for any λ > Λ). By the definition of Λ, thereexists μ ∈ (Λ, λ) such that Jμ has a nontrivial critical point uμ ∈ W

1,p0 (Ω). Since μ < λ,uμ is

a subsolution of the problem (1.3)−. In order to find a super-solution of the problem (1.3)−which dominates uμ, we consider the constrained minimization problem

inf{Jλ(w);w ∈W1,p

0 (Ω) and w ≥ uμ.}. (3.15)

Arguments similar to those used in Step 2 show that the above minimization problem has asolution uλ ≥ uμ which is also a weak solution of problem (1.3)−, provided λ > Λ.

Using similar arguments as in [6]. Thus, from Theorem 2.2 in Pucci and Servadei[7], based on the Moser iteration, it is clear that u ∈ L∞

loc. Next, again by bootstrapregularity [Corollary on p. 830] due to DiBenedetto, [8] shows that the weak solutionu ∈ C1,α(Ω) where α ∈ (0, 1). Finally, the nonnegative follows immediately by the strongmaximum principle since u is a C1 nonnegative weak solution of the differential inequality∇(|∇u|p−2∇u)−h(x)ur ≤ 0 inΩ, with p−1 < r, see, for instance, Section 4.8 of Pucci and Serrin[9]. Thus, u > 0 in Ω. The proof of the Step 3 is now completed.


Step 4 (nonexistence for λ > 0 is small). The same monotonicity arguments as in Step 3 showthat (1.3)− does not have any solution if λ < Λ, which completes the proof of the Theorem 1.2.

References

[1] V. Radulescu and D. Repovs, “Combined effects in nonlinear problems arising in the study of aniso-tropic continuous media,” Nonlinear Analysis, vol. 75, no. 3, pp. 1524–1530, 2012.

[2] C. A. Santos, “Non-existence and existence of entire solutions for a quasi-linear problem with singularand super-linear terms,” Nonlinear Analysis, vol. 72, no. 9-10, pp. 3813–3819, 2010.

[3] M. Cuesta and P. Takac, “A strong comparison principle for positive solutions of degenerate ellipticequations,” Differential and Integral Equations, vol. 13, no. 4–6, pp. 721–746, 2000.

[4] P. Lindqvist, “On the equation div(|∇u|p−2∇u) + λ|u|p−2u = 0,” Proceedings of the American MathematicalSociety, vol. 109, no. 1, pp. 157–164, 1990.

[5] A. Anane, “Simplicite et isolation de la premiere valeur propre du p-laplacien avec poids,” ComptesRendus des Seances de l’Academie des Sciences I, vol. 305, no. 16, pp. 725–728, 1987.

[6] R. Filippucci, P. Pucci, and V. Radulescu, “Existence and non-existence results for quasilinear ellipticexterior problems with nonlinear boundary conditions,” Communications in Partial Differential Equa-tions, vol. 33, no. 4–6, pp. 706–717, 2008.

[7] P. Pucci and R. Servadei, “Regularity of weak solutions of homogeneous or inhomogeneous quasilinearelliptic equations,” Indiana University Mathematics Journal, vol. 57, no. 7, pp. 3329–3363, 2008.

[8] E. DiBenedetto, “C1+α local regularity of weak solutions of degenerate elliptic equations,” NonlinearAnalysis, vol. 7, no. 8, pp. 827–850, 1983.

[9] P. Pucci and J. Serrin, “Maximum principles for elliptic partial differential equations,” in Handbookof Differential Equations: Stationary Partial Differential Equations, M. Chipot, Ed., vol. 4, pp. 355–483,Elsevier, 2007.


Research ArticleOn Generalized Localization ofFourier Inversion Associated with an EllipticOperator for Distributions

Ravshan Ashurov,1 Almaz Butaev,2, 3 and Biswajeet Pradhan2, 3

1 Institute of Mathematics, National University of Uzbekistan, Hodjaeva Street 29,Tashkent 100125, Uzbekistan

2 Institute of Advanced Technology (ITMA), University Putra Malaysia, 43400 Serdang,Selangor, Malaysia

3 Department of Civil Engineering, Faculty of Engineering, Geospatial, Information ScienceCenter (GIS RC), University Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia

Correspondence should be addressed to Biswajeet Pradhan, [email protected]



Copyright q 2012 Ravshan Ashurov et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study the behavior of Fourier integrals summed by the symbols of elliptic operators andpointwise convergence of Fourier inversion. We consider generalized localization principle whichin classical Lp spaces was investigated by Sjolin (1983), Carbery and Soria (1988, 1997) and Alimov(1993). Proceeding these studies, in this paper, we establish sharp conditions for generalizedlocalization in the class of finitely supported distributions.

1. Introduction

In this paper, we study the behavior of spherical Fourier integrals and pointwise convergenceand summability of Fourier inversion.

LetA(D) =

∑

|α|=mcαD

α(1.1)

be a homogeneous elliptic differential operator of orderm. Let us consider its symbol definedas polynomial:

A(x) =∑

|α|=mcαx

α, (1.2)


and assume that the Gaussian curvature of surface S = {x ∈ Rn : A(x) = 1} is always strictlypositive.

We recall that for f ∈ L2(Rn) its Fourier transform is defined as

f(ξ) =∫f(y)e−iyξdy (1.3)

and partial Fourier integral associated with elliptic operator (1.1):

Eλf(x) = (2π)−n∫

A(ξ)<λf(ξ)dξ (1.4)

(note that throughout the paper we consider only Lebesque measure on Rn and∫=∫Rn).

For some functions, Fourier integrals do not converge pointwisely and various summationtechniques are applied to recover convergence property. In this paper, we consider themethod of the Riesz means. The Riesz means of order s are defined as

Esλf(x) = (2π)−n∫

A(ξ)≤λ

(1 − |ξ|2

λ2

)s

f(ξ)eiξxdξ. (1.5)

As an example, one can consider Laplacian A(D) =∑n

i=1(∂2/∂x2

i ), and note that thelevel surfaces of its symbol are Euclidean spheres. Thus, Fourier inversion associated withLaplace operator has the form:

Eλf(x) = (2π)−n∫

|ξ|2<λf(ξ)eiξxdξ (1.6)

and known as spherical partial Fourier integrals. The question of Eλf(x) convergence to f(x)almost everywhere is not solved in Rn, n ≥ 2 even for classical L2 functions and presentsone of the most challenging open problems of classical harmonic analysis, and even specialcases of this problem are of particular interest. One of such special cases is the problemof generalized localization, which for the first time was formulated by V. Ii’in in [1]. Forconvenience, we give its definition for the Riesz means Esλ.

Definition 1.1. We say that, for the Riesz means of order s, the generalized localizationprinciple in function class F is satisfied, if for any function f ∈ F, the equality

limλ→∞

Esλf(x) = 0 (1.7)

is true for a.e. x ∈ Rn \ supp f .

This localization principle generalizes the classical Riemann localization principle andfor Lp functions was intensively investigated by Sjolin [2], Carbery and Soria [3, 4], Bastis[5–7], and Ashurov et al. [8]. It was established that Rn localization holds true in Lp, wherep ∈ [2, 2n/(n − 1)] and fails otherwise.


Over the last several years, a number of Fourier inversion studies considereddistributions and investigated the behavior of their Fourier integrals (see, e.g., [9–12]).In particular, Alimov in [13] considered the classical Riemann localization principle forcompactly supported distributions and established criteria for its validity (see also [14, 15]).

In this paper, we study generalized localization principle for compactly supporteddistributions and present conditions for its fulfillment.

2. Notation and Definitions

We define Schwartz space S(Rn) as the function class of all infinitely differentiable functionsthat are rapidly decreasing at infinity along with all partial derivatives. It is well known thatS(Rn), being equipped with a family of seminorms

dα,β(φ)= sup

x∈Rn

∣∣∣xαDβφ(x)∣∣∣, (2.1)

is a Frechet space (here α, β are multi-indices andD is a partial derivative). As usual, we alsoconsider class of tempered distributions S′ defined as dual to S.

Let E be the space of infinitely differentiable functions with topology τE such that φn →0 in τE if and only if for each multiindex α and compact K

supx∈K

Dαφn(x) −→ 0. (2.2)

As usual we denote its conjugate space by E′.It is known (see, e.g., [16]) that each f ∈ E′ has finite support and equivalent to the

class of finitely supported tempered distributions. Thus, it follows from the Paley-Wiennertheorem that, for each f ∈ E′, its Fourier transform f ∈ C∞. Since f is locally integrable, it isnatural to define Fourier integral of f ∈ E′ and its Riesz means by (1.4) and (1.5), respectively.

We also note that for f ∈ L2 the Riesz mean Esλf can be considered as an integral

operator:

Esλf(x) = (2π)−n∫f(y)θsλ(x − y)dy, (2.3)

with kernel θsλ(y) = msλ(y) where

msλ

(y)=

(1 − A

(y)

λ

)s

+

, (2.4)

where (1 −A(y)/λ)s+ = (1 −A(y)/λ)s · χA(y)<λ(y).


Representation (2.3) has its natural analogue for f ∈ E′. Let ψn be a sequence ofSchwartz functions such that ψn(y) = 0 as |y| > λ and ψn(y) → ms

λ(y) in L1 norm. Then:

Esλf(x) = limn→∞

(2π)−n∫f(ξ)ψn(ξ)eixξdξ

= (2π)−n limn→∞

⟨f(ξ), ψn(ξ)eixξ

⟩

= (2π)−n limn→∞

⟨f(y), ψn(x − y)⟩.

(2.5)

Note that inequality ‖g‖∞ ≤ ‖g‖1 implies that ψn → msλ in E and since f is continuous on E

Esλf(x) = (2π)−n⟨f(·), θsλ(x − ·)⟩. (2.6)

We will need Sobolev’s classes which can be defined for l ∈ R in the following way.

Definition 2.1. We say that tempered distribution f belongs to Sobolev classHl if f is a regulardistribution such that

∥∥f∥∥2Hl =

∫ ∣∣∣f(ξ)∣∣∣2(1 + |ξ|2

)ldξ <∞. (2.7)

One can see that, in particular, H0 = L2. We also remark that for every f ∈ E′ there isl ∈ R such that f ∈ Hl (for proof see, e.g., [16]).

In other respects, we make the following conventions:

(i) symbol Jν is used to denote Bessel function of the first kind and order ν ≥ 0,

(ii) χE is preserved for an indicator function of E ⊂ Rn,

(iii) unless otherwise indicated, all functions are assumed to be defined on Rn and bydefinition Lp(Ω) ≡ {f ∈ Lp(Rn) : supp f ⊂ Ω ⊂ Rn}.

3. Main Result

As has been mentioned above, every f ∈ E′ belongs to some Sobolev classes Hl, in thispaper, we use this fact to establish criterion of generalized localization for finitely supporteddistributions. The following theorems present major results of current study.

Theorem 3.1. Let f ∈ E′ ∩H−l, l ≥ 0. Then, for integer s ≥ l, equality

limλ→∞

Esλf(x) = 0 (3.1)

holds true a.e. on Rn \ supp f .

Our approach is based on the methods by Carbery and Soria [3] and in order to proveTheorem 3.1, wewill follow his idea first proving some auxiliary facts in the following section.


4. Dual Sets

Let a(x) = [A(x)]1/m and K = {x ∈ Rn : a(x) ≤ 1}. Then, K is a symmetric body that isconvex compact symmetric set. We recall that set K∗ = {y : |x · y| ≤ 1, ∀x ∈ K} is called polarset with respect to K.

As it is done in [17], we will introduce the norm ‖ · ‖a generated by a(x) as

‖x‖a = a(x) (4.1)

and dual norm ‖ · ‖∗a as

∥∥y∥∥∗a = sup

‖x‖a≤1

∣∣x · y∣∣ = sup‖x‖a=1

∣∣x · y∣∣. (4.2)

Next, let S and S∗ be the boundaries of K and K∗, respectively.It is not difficult to show that S∗ = {∇a(x), x ∈ S}. Indeed on the one hand a(λx) =

λa(x) and, therefore, for x ∈ S

x · ∇a(x) = da

dr(x) = a(x) = 1,

(−x) · ∇a(x) = −dadr

(x) = a(x) = −1,(4.3)

which means that ‖∇a(x)‖∗a ≥ 1. On the other hand, for any y ∈ S, one can consider F(y) =y · ∇a(x) and examine its local extremums on the surface S. Since S is compact, F(y) reachesits extremum values and it is known that, at extremum points, ∇F(y) must be parallel to thenormal to S at point y, which is parallel to∇a(y). Since∇F(y) = ∇a(x), we can conclude that∇a(x)‖∇a(y) at the extremum points. Since S is strictly convex, it is possible only for y = ±x,that implies ‖∇a(x)‖∗a ≤ 1.

It is convenient for given x ∈ Rn to use the notation θ(x) to denote the point on S suchthat the outer normal to S at θ(x) is parallel to x. Similarly, we denote η(x) the point on S∗

such that the outer normal to S∗ at η(x) is parallel to x. One can remark that we have justseen that for y ∈ S∗

y · θ(y) = 1. (4.4)

5. Technical Lemmas for Theorem 3.1

We will need the asymptotic representation of θsλ(y), which can be derived by stationary-

phase method (see, e.g., [18]):

θsλ(y)= λ(n−1)/2

∣∣y∣∣−(n+1)/2 ·

[Rs

+(y, λ)eiλy·θ(y) + Rs

−(y, λ)e−iλy·θ(y)

], (5.1)


where functions Rs±(y, λ) ∈ C∞({y : |y| > ε} × [1,∞)) and

DαyD

β

λRs±(y, λ)= O(λ−s−β

), (5.2)

uniformly on |y| > δ and λ > δ.Now, let us consider positive numbers ε and R, ε < R and function φ(x) = φ(‖x‖∗a) ∈

C∞0 vanishing on {x : (‖x‖∗a < ε) ∨ (‖x‖∗a > R)}. Then, for s ≥ 0, we set by definition

Θsλ(x) = φ(x)θ

sλ(x), (5.3)

where θsλas in (2.3).

We will need some estimates for the Fourier transform of Θsλ. With this aim, we will

need the following lemmas.

Lemma 5.1. Let t ≥ δ > 0 and |ξ| < 1. Then, for any α > 0

∣∣∣Θst (ξ)∣∣∣ ≤ O(t−α). (5.4)

Proof. This estimate easily follows from the definition of Θst . Indeed,

Θst (ξ) =

∫θst (x)φ(x)e

−iξxdx =∫ (1 − A

(y)

t

)s

+

(x)φ(x)e−iξxdx

=∫ (

1 − A(x)t

)s

+φ(x + ξ)dx =

∫

A(x)<tφ(x + ξ)dx

+s∑

k=1

Ck

tk

∫

A(x)<tφk(·, ξ)(x)dx,

(5.5)

where φk(y, ξ) = Bk(Dy)[φ(y)e−iyξ] and B(D) is formally conjugate to operator A(D). Sinceφ(0) = φk(0, ξ) = 0,

Θst (ξ) =

∫

A(x)>tφ(x + ξ)dx +

s∑

k=1

Ck

tk

∫

A(x)>tφk(·, ξ)(x)dx. (5.6)

Further, we notice that since φk(y, ξ) ∈ C∞0 then for any α > 0 there is Cα such that functions

φk(x, ξ) = Cα/(1 +A(x))α, uniformly for k = 1, . . . , s and |ξ| < 1. For the same reason, for anyα > 0, one has φ(x) ≤ O((1 + x)−α). Now substituting these estimates into (5.6), we completethe proof.

Lemma 5.2. Let t ≥ δ > 0 and |ξ| ≥ 1. Then, for any α > 0,

∣∣∣Θst (ξ)∣∣∣ =

O(1)t−s

(1 + |‖ξ‖a − t|)α. (5.7)


Proof. By definition,

Θst (ξ) =

∫

ε<‖y‖∗a<Rφ(y)θst(y)e−iξ·ydy. (5.8)

Let us pass to a new coordinate system y → (r = ‖y‖∗a, η = η(y)). Then,

Θst (ξ) =

∫R

ε

φ(r)rn−1∫

η∈S∗θst(rη)e−irξ·ηdσ

(η)dr, (5.9)

where dσ(η) is a Lebesgue surface measure of S∗.Using (5.1), we have

Θst (ξ) = t(n−1)/2

∫R

ε

φ(r)r(n−3)/2∫

η∈S∗

∣∣η∣∣−(n+1)/2eitr[η·θ(η)]Rs

+(rη, t)e−irξ·ηdσ

(η)dr

+ t(n−1)/2∫R

ε

φ(r)r(n−3)/2∫

η∈S∗

∣∣η∣∣−(n+1)/2e−itr[η·θ(η)]Rs

−(rη, t)e−irξ·ηdσ

(η)dr.

(5.10)

We will focus on the first term since the second one can be handled alike

Ist (ξ) = t(n−1)/2

∫R

ε

φ(r)r(n−3)/2∫

η∈S∗

∣∣η∣∣−(n+1)/2eitr[η·θ(η)]Rs


(η)dr (5.11)

and note that due to (4.4) η · θ(η) = 1, and thus

Ist (ξ) = t(n−1)/2

∫R

ε

φ(r)r(n−3)/2eitr∫

η∈S∗

∣∣η∣∣−(n+1)/2Rs


(η)dr. (5.12)

One can use the expression ξ = ‖ξ‖aθ(ξ) and employ stationary phase method to obtain

∫

η∈S∗

∣∣η∣∣−(n+1)/2Rs


(η)= ‖ξ‖−(n−1)/2a eir‖ξ‖a[θ(ξ)·η(θ(ξ))]Ps+(rξ, t)

+ ‖ξ‖−(n−1)/2a e−ir‖ξ‖a[θ(ξ)·η(θ(ξ))]Ps−(rξ, t),

(5.13)

where Ps± are smooth functions such that Dαt D

βzP

s±(z, t) = O(t−s−α). Using this expression, we

have

Ist (ξ) =(

t

‖ξ‖a

)(n−1)/2 ∫R

ε

φ(r)r(n−3)/2eir(t−‖ξ‖a)Ps+(rξ, t)dr

+(

t

‖ξ‖a

)(n−1)/2 ∫R

ε

φ(r)r(n−3)/2e−ir(t−‖ξ‖a)Ps−(rξ, t)dr.

(5.14)


Further integrating by parts the integrals, one can see that for anyN > 0 both integralsare controlled by (CNt

−s)/(1 + |t − ‖ξ‖a|)N . As a result, we have

∣∣Ist (ξ)∣∣ ≤(

t

‖ξ‖a

)(n−1)/2 CNt−s

(1 + |t − ‖ξ‖a|)N≤ DNt

−s

(1 + |t − ‖ξ‖a|)N−((n−1)/2) , (5.15)

uniformly for |ξ| > 1 and t > δ. Finally, substituting into (5.10), we obtain (5.7).

Now, combining Lemmas 5.1 and 5.2, we can claim that, in fact, for t > δ and anyξ ∈ Rn,

∣∣∣Θst (ξ)∣∣∣ ≤ O(1)t−s

(1 + |‖ξ‖a − t|)α. (5.16)

Lemma 5.3. Let Θsλ(x) be defined by (5.3). Then, for any δ > 0 there is Cδ > 0 such that

∫∞

δ

∣∣∣Θst (ξ)∣∣∣2dt ≤ Cδ

(1 + |ξ|)2s. (5.17)

Proof. As it follows from (5.16),

∫∞

δ

∣∣∣Θst (ξ)∣∣∣2dt ≤ O(1)

∫∞

δ

t−2sdt

(1 + |‖ξ‖a − t|)2α, (5.18)

where α > 0 can be chosen arbitrary large. Changing the variables u = ξ − t, one has

∫∞

δ

|t|−2sdt(1 + |‖ξ‖a − t|)2α

=∫

δ<|‖ξ‖a−u|<‖ξ‖a/2

|‖ξ‖a − u|−2sdt(1 + |u|)2α

+∫

max(δ,‖ξ‖a/2)<|‖ξ‖a−u|

|‖ξ‖a − u|−2sdt(1 + |u|)2α

.

(5.19)

It is not difficult to see that for the values u in the first integral |u| > |‖ξ‖a− | u−‖ξ‖a| > ‖ξ‖a/2,and thus choosing α > max(2s, 1)

∫

δ<|‖ξ‖a−u|<‖ξ‖a/2

|‖ξ‖a − u|−2sdt(1 + |u|)2α

≤ O(1)(1 + ‖ξ‖a)α

≤ O(1)

(1 + ‖ξ‖a)2s. (5.20)

Moreover, it is clear that for such α

∫

δ<|‖ξ‖a−u|<‖ξ‖a/2

|‖ξ‖a − u|−2sdt(1 + |u|)2α

≤ O(1)

(1 + ‖ξ‖a)2s. (5.21)


Therefore,

∫∞

δ

∣∣∣Θst (ξ)∣∣∣2dt ≤ O(1)

(1 + ‖ξ‖a)2s. (5.22)

Since all norms in Rn are equivalent, the lemma is proved.

Lemma 5.4. Let Θsλbe defined by (5.3). Then, for any δ > 0, there is Cδ such that

∫∞

δ

∣∣∣∣d

dtΘst (ξ)∣∣∣∣2

dt ≤ Cδ

(1 + |ξ|)2s. (5.23)

Proof. For any t > 1, using the Fubini theorem, one has

∫ t

1

∫d

duΘsu

(y)e−iξydy du =

∫e−iξy

∫ t

1

d

duΘsu

(y)dudy = Θs

t (ξ) − Θs1(ξ), (5.24)

which implies (d/dt)Θst (ξ) = (d/dt)Θs

t (ξ).If s > 0

d

dtΘst (x) = φ(x)

d

dtθst (x)

= φ(x)d

dt

∫ t

0

(1 − u2

t2

)s

dθu(x)

=2sφ(x)

t

∫ t

0

(1 − u2

t2

)s−1u2

t2dθu(x)

=2st

(Θs−1t (x) −Θs

t (x)).

(5.25)

Thus, using inequality (a + b)2 ≤ 2a2 + 2b2, one has

∫∞

δ

∣∣∣∣d

dtΘst (ξ)∣∣∣∣2

dt ≤ C∫∞

δ

t−2∣∣∣Θs−1

t (ξ)∣∣∣2dt + C

∫∞

δ

t−2∣∣∣Θs

t (ξ)∣∣∣2dt. (5.26)

Now, one can use estimate (5.16) to each integral on the right side and complete the proof.


If s = 0, then for any ξ ∈ Rn,

∣∣∣∣d

dtΘt(ξ)

∣∣∣∣ =

∣∣∣∣∣d

dt

∫

A(y)≤tφ(ξ + y

)dy

∣∣∣∣∣

=

∣∣∣∣∣

∫

A(y)=tφ(ξ + y

)n(y)dσ(y)∣∣∣∣∣

≤ O(1)(1 + |‖ξ‖a − t|)α

, ∀α > 0.

(5.27)

Using this estimate and the reasoning presented in the previous lemma, we obtain therequired estimate.


Let f ∈ H−l ∩ E′ be such that supp f ⊂ Ω. For 0 < ε < 1/2, we set

Eε ={x : 2ε < dist(x,Ω) < (2ε)−1

}(6.1)

and consider an arbitrary radial function φε ∈ C∞0 such that

φε(x) =

⎧⎨

⎩1,

3ε2

≤ |x| ≤ 1ε+ diam Ω;

0, |x| ≤ ε.(6.2)

It is clear that to prove the theorem it is sufficient to show that for any ε > 0,limλ→∞Esλf(x) = 0, a.e. x ∈ Eε

In this case, as x ∈ Eε due to (2.6)

Esλf(x) =∫f(ξ)

[χΩ(·)θsλ(x − ·)] (−ξ)dξ

=∫f(ξ)

[φε(x − ·)θsλ(x − ·)] (−ξ)dξ

=∫f(ξ)

[φεθ

sλ

](ξ)eiξxdξ,

(6.3)

or using notation (5.3)

Esλf(x) =∫f(ξ)Θs

λ(ξ)eiξxdξ =

(f(ξ)Θs

λ(ξ))(−x). (6.4)


Further, we consider maximal operator:

Es∗f(x) = supλ>1

∣∣Esλf(x)∣∣. (6.5)

We recall that to prove a.e. convergence on Eε one can use the standard technique of Banachprinciple (see, e.g., [19]) according to which it is sufficient to estimate maximal operator onEε ⊂ Rn \ supp f as

∥∥Es∗f(x)∥∥L2(Eε)

≤ C∥∥f∥∥H−l . (6.6)

Let γ(t) : R → R+ be a C∞ function such that

γ(t) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

0, t ≤ 13;

1, t ≥ 23.

(6.7)

If we set Esλf(x) = γ(λ)Es

λf(x), then by (6.4),

Esλf(x) = γ(λ) < f(·), Θsλ(x − ·) >= γ(λ)

(f(ξ)Θs

λ(ξ))(−x). (6.8)

According to Sobolev’s embedding theorem (see, e.g., [20]) for any f ∈ H1(R1),

∥∥f∥∥L∞

≤ C∥∥f∥∥H1 . (6.9)

Using this fact, we have

Es∗f(x) ≤∥∥∥Esλf(x)

∥∥∥L∞(R)

≤∥∥∥Esλf(x)

∥∥∥H1(R)

. (6.10)

And, therefore, in order to obtain (6.6), it is sufficient to show that there are constants C1, C2

such that the following estimates are true:

∫ ∥∥∥Esλf(x)∥∥∥2

L2(R)dx ≤ C1

∥∥f∥∥H−l ,

∫ ∥∥∥∥d

dλEsλf(x)

∥∥∥∥2

L2(R)dx ≤ C2

∥∥f∥∥H−l .

(6.11)

First, we note that estimate (5.16) and f ∈ H−l imply that fΘsλ∈ L2 which in turn with

(6.8) implies the fact Esλf ∈ L2.


Further, using the Plancherel theorem, we have

∫ ∫ ∣∣∣Esλf(x)∣∣∣2dλdx ≤

∫γ2(λ)

∫ ∣∣∣f(ξ)∣∣∣2∣∣∣Θs

λ(ξ)∣∣∣2dξ dλ

≤∫∞

1/3γ2(t)

∫ (1 + |ξ|2

)l∣∣∣Θst (ξ)∣∣∣2(1 + |ξ|2

)−l∣∣∣f(ξ)∣∣∣2dξ dt

≤ supξ∈Rn

(1 + |ξ|2

)l ∫∞

1/3

∣∣∣Θst (ξ)∣∣∣2dt × ∥∥f∥∥2H−l ≤ C

∥∥f∥∥2H−l

(6.12)

(the last inequality follows from Lemma 5.3).For the same reason, (6.11) can be proved using Lemmas 5.3 and 5.4:

∫ ∫∞

1/3

∣∣∣∣d

dt

[γ(t)Θs

t

] ∗ f(x)∣∣∣∣2

dt dx ≤∥∥∥γ

′2(λ)∥∥∥∞

∫ ∫∞

1/3

∣∣Θst ∗ f(x)

∣∣2dt dx

+∫ ∫∞

1/3

∣∣∣∣d

dλΘsλ ∗ f(x)

∣∣∣∣2

dλdx ≤ C∥∥f∥∥2H−l .

(6.13)

Acknowledgments

The authors are thankful to the University PutraMalaysia for the support under RUGS (Grantno. 05-03-11-1450RU). A. Butaev would also like to expresses his gratitude to for the supportunder IGRF scheme.

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Research ArticleA Note on the Stability of the Integral-DifferentialEquation of the Parabolic Type in a Banach Space

Maksat Ashyraliyev

Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas,34353 Istanbul, Turkey

Correspondence should be addressed to Maksat Ashyraliyev, [email protected]



Copyright q 2012 Maksat Ashyraliyev. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The integral-differential equation of the parabolic type in a Banach space is considered. The uniquesolvability of this equation is established. The stability estimates for the solution of this equationare obtained. The difference scheme approximately solving this equation is presented. The stabilityestimates for the solution of this difference scheme are obtained.

1. Introduction

We consider the integral-differential equation

du(t)dt

+ sgn(t)Au(t) =∫ t

−tB(s)u(s)ds + f(t), −1 ≤ t ≤ 1 (1.1)

in an arbitrary Banach space E with unbounded linear operators A and B(t) in E with densedomain D(A) ⊂ D(B(t)) and

∥∥∥B(t)A−1∥∥∥E→E

≤M, −1 ≤ t ≤ 1. (1.2)

A function u(t) is called a solution of (1.1) if the following conditions are satisfied:

(i) u(t) is continuously differentiable on [−1, 1]. The derivatives at the endpoints areunderstood as the appropriate unilateral derivatives.


(ii) The element u(t) belongs to D(A) for all t ∈ [−1, 1], and the functions Au(t) andB(t)u(t) are continuous on [−1, 1].

(iii) u(t) satisfies (1.1).

A solution of (1.1) defined in this manner will from now on be referred to as a solutionof (1.1) in the space C(E) = C([−1, 1], E) of all continuous functions ϕ(t) defined on [−1, 1]with values in E equipped with the norm

∥∥ϕ∥∥C(E) = max

−1≤t≤1

∥∥ϕ(t)∥∥E. (1.3)

We consider (1.1) under the assumption that the operator −A generates an analyticsemigroup exp{−tA}(t ≥ 0), that is, the following estimates hold:

∥∥∥e−tA∥∥∥E→E

≤M,∥∥∥tAe−tA

∥∥∥E→E

≤M, 0 ≤ t ≤ 1. (1.4)

Integral inequalities play a significant role in the theory of differential and integral-differential equations. They are useful to investigate some properties of the solutions ofequations, such as existence, uniqueness and stability, see for instance [1–11].

Mathematical modelling of real-life phenomena is widely used in various appliedfields of science. This is based on the mathematical description of real-life processes andthe subsequent solving of the appropriate mathematical problems on the computer. Themathematical models of many real-life problems lead to already known or new differentialand integral-differential equations. In most of the cases it is difficult to find the exact solutionsof the differential and integral-differential equations. For this reason discrete methods playa significant role, especially with the appearance of highly efficient computers. A well-known and widely applied method of approximate solutions for differential and integral-differential equations is the method of difference schemes. Modern computers allow us toimplement highly accurate difference schemes. Hence, the task is to construct and investigatehighly accurate difference schemes for various types of differential and integral-differentialequations. The investigation of stability and convergence of these difference schemes is basedon the discrete analogues of integral inequalities.

Gronwall in 1919 showed the following result [12].

Lemma 1.1. If M = const > 0, δ = const > 0, and continuous function x(t) ≥ 0 satisfies theinequalities

x(t) ≤ δ +M∫ t

0x(s)ds, 0 ≤ t ≤ T, (1.5)

then

x(t) ≤ δ exp[Mt], 0 ≤ t ≤ T. (1.6)

A number of different generalizations of Gronwall’s integral inequality with one andtwo dependent limits have been obtained, see for instance [13, 14].


In numerical analysis literature, see for instance [15, 16], one can find the followingdiscrete analogue of Lemma 1.1.

Lemma 1.2. If xj , j = 0, . . . ,N is a sequence of real numbers with

|xi| ≤ δ + hMi−1∑

j=0

∣∣xj∣∣, i = 1, . . . ,N, (1.7)

whereM = const > 0 and δ = const > 0, then

|xi| ≤ (hM|x0| + δ) exp[Mih], i = 1, . . . ,N. (1.8)

In the current paper, we will derive the discrete analogue of generalization of theGronwall’s integral inequality. It is used to obtain the generalization of Gronwall’s integralinequality with two dependent limits. We will consider the applications of these inequalitiesto the integral-differential equation (1.1) of the parabolic type with two dependent limits in aBanach space E. The unique solvability of this equation is established. The stability estimatesfor the solution of this equation are obtained. The difference scheme approximately solvingthis equation is presented. The stability estimates for the solution of this difference schemeare obtained.

2. Gronwall’s Type Integral Inequality with Two Dependent Limits andIts Discrete Analogue

First of all, let us obtain the theorems on the Gronwall’s type integral inequalities with twodependent limits and their discrete analogues. We will use these results in the remaining partof the paper.

Theorem 2.1. Assume that vi ≥ 0, ai ≥ 0, δi ≥ 0, i = −N, . . . ,N + 2M are the sequences of realnumbers and the inequalities

vi ≤ δi + h⎛

⎝|i−M|+M−1∑

j=−|i−M|+M+1

ajvj − aMvM⎞

⎠, i = −N, . . . ,N + 2M (2.1)

hold. Then for vi the inequalities

vM−1 ≤ δM−1, vM+1 ≤ δM+1, vM ≤ δM + h(aM−1δM−1 + aM+1δM+1), (2.2)

vi ≤ δi + h|i−M|+M−1∑

j=M+1

(ajδj + a2M−jδ2M−j

)B|i−M|+M−1,j , i = −N, . . . ,M − 2,M + 2, . . . ,N + 2M

(2.3)


are satisfied, where

Bk,j =

⎧⎪⎨

⎪⎩

k∏n=j+1

[1 + h(an + a2M−n)], if j =M + 1, . . . , k − 1,

1, if j = k.(2.4)

Proof. By putting i = M − 1,M + 1,M directly in (2.1), we obtain the inequalities (2.2),correspondingly. Let us prove (2.3). We denote

yi = h

⎛

⎝|i−M|+M−1∑

j=−|i−M|+M+1

ajvj − aMvM⎞

⎠, i = −N, . . . ,N + 2M. (2.5)

Then (2.1) gets the form

vi ≤ δi + yi, i = −N, . . . ,N + 2M. (2.6)

Moreover, we have

y2M−i = h

⎛

⎝|M−i|+M−1∑

j=−|M−i|+M+1

ajvj − aMvM⎞

⎠ = yi, i = −N, . . . ,N + 2M. (2.7)

Then, using (2.5)–(2.7) for i =M + 1, . . . ,N + 2M − 1, we obtain

yi+1 − yi = h⎛

⎝i∑

j=2M−iajvj − aMvM

⎞

⎠ − h⎛

⎝i−1∑

j=2M−i+1ajvj − aMvM

⎞

⎠

= h(aivi + a2M−iv2M−i)

≤ hai(yi + δi

)+ ha2M−i

(y2M−i + δ2M−i

)

= h(ai + a2M−i)yi + h(aiδi + a2M−iδ2M−i).

(2.8)

So,

yi+1 ≤ [1 + h(ai + a2M−i)]yi + h(aiδi + a2M−iδ2M−i), i =M + 1, . . . ,N + 2M − 1. (2.9)

Then by induction we can prove that

yi ≤i−M−1∏

n=1

[1 + h(aM+n + aM−n)]yM+1 +i−1∑

j=M+1

h(ajδj + a2M−jδ2M−j

)Bi−1,j (2.10)

hold for i = M + 2, . . . ,N + 2M. Since yM+1 = 0, using (2.6), we obtain (2.3) for i = M +2, . . . ,N + 2M.


Let us prove (2.3) for i = −N, . . . ,M− 2. Using (2.5)–(2.7) for i = −N + 1, . . . ,M− 1, wehave

yi−1 − yi = h⎛

⎝2M−i∑

j=i

ajvj − aMvM⎞

⎠ − h⎛

⎝2M−i−1∑

j=i+1

ajvj − aMvM⎞

⎠

= h(aivi + a2M−iv2M−i)

≤ hai(yi + δi

)+ ha2M−i

(y2M−i + δ2M−i

)

= h(ai + a2M−i)yi + h(aiδi + a2M−iδ2M−i).

(2.11)

So,

yi−1 ≤ [1 + h(ai + a2M−i)]yi + h(aiδi + a2M−iδ2M−i), i = −N + 1, . . . ,M − 1. (2.12)

Then by induction we can prove that

yi ≤M−i−1∏

n=1

[1 + h(aM+n + aM−n)]yM−1 +2M−i−1∑

j=M+1

h(ajδj + a2M−jδ2M−j

)B2M−i−1,j (2.13)

hold for i = −N, . . . ,M − 2. Since yM−1 = 0, using (2.6), we obtain (2.3) for i = −N, . . . ,M − 2.The proof of Theorem 2.1 is complete.

By putting M = 0, δi ≡ const, ai ≡ const, i = −N, . . . ,N, and using the inequality1 + x < exp[x] for x > 0 in the Theorem 2.1, we get the following result.

Theorem 2.2. Assume that vi ≥ 0, i = −N, . . . ,N is the sequence of real numbers and theinequalities

vi ≤ δ + Lh

⎛

⎝|i|−1∑

j=−|i|+1vj − v0

⎞

⎠, i = −N, . . . ,N (2.14)

hold. Then for vi the inequalities

v0 ≤ δ exp[2Lh], vi ≤ δ exp[2Lh(|i| − 1)], i = −N, . . . ,−1, 1, . . . ,N (2.15)

are satisfied.

By puttingNh = 1, 2Mh = T and passing to limit h → 0 in the Theorem 2.1, we obtainthe following generalization of Gronwall’s integral inequality with two dependent limits.


Theorem 2.3. Assume that v(t) ≥ 0, δ(t) ≥ 0 are the continuous functions on [−1, 1 + T] anda(t) ≥ 0 is an integrable function on [−1, 1 + T] and the inequalities

v(t) ≤ δ(t) + sgn(t − T

2

)∫ t

T−ta(s)v(s)ds, −1 ≤ t ≤ 1 + T (2.16)

hold. Then for v(t) the inequalities

v(t)

≤ δ(t) +∫ t

T/2(a(s)δ(s) + a(T − s)δ(T − s)) exp

[∫ t

s

(a(τ) + a(T − τ))dτ]ds,

T

2≤ t ≤ 1 + T,

v(t)

≤ δ(t) +∫T−t

T/2(a(s)δ(s) + a(T − s)δ(T − s)) exp

[∫T−t

s

(a(τ) + a(T − τ))dτ]ds, −1 ≤ t < T

2(2.17)

are satisfied.

Finally, by putting δ(t) ≡ const, a(t) ≡ const, −1 ≤ t ≤ 1, and T = 0 in the Theorem 2.3,we get the following result.

Theorem 2.4. Assume that v(t) ≥ 0 is a continuous function on [−1, 1] and the inequalities

v(t) ≤ C + L sgn(t)∫ t

−tv(s)ds, −1 ≤ t ≤ 1 (2.18)

hold, where C = const ≥ 0 and L = const ≥ 0. Then for v(t) the inequalities

v(t) ≤ C exp(2L|t|), −1 ≤ t ≤ 1 (2.19)

are satisfied.

3. The Integral-Differential Equation of the Parabolic Type

Now, we consider the application of the generalizations of Gronwall’s integral inequalitywith two dependent limits and their discrete analogues to the integral-differential equation(1.1) of the parabolic type with two dependent limits in a Banach space E.

First of all, let us give one theorem that will be needed below.


Theorem 3.1. Suppose that F(t) ∈ C([−1, 1], E), K(t, s) ∈ C([−1, 1], E). Then there is a uniquesolution of the integral equation

z(t) = F(t) + sgn(t)∫ t

−tK(t, s)z(s)ds, −1 ≤ t ≤ 1. (3.1)

Proof. The proof of this theorem is based on a fixed-point theorem. It is easy to see that theoperator

Bz(t) = F(t) + sgn(t)∫ t

−tK(t, s)z(s)ds, −1 ≤ t ≤ 1 (3.2)

maps C([−1, 1], E) into C([−1, 1], E). By using a special value of λ in the norm

‖v‖C∗([−1,1],E) = max−1≤t≤1

e−λ|t|‖v(t)‖E, (3.3)

we can prove that A is the contracting operator on C∗([−1, 1], E). Indeed, we have

e−λ|t|‖Bz(t) − Bu(t)‖E ≤∫ |t|

−|t|‖K(t, s)‖E→Ee

−λ(|t|−|s|)e−λ|s|‖z(s) − u(s)‖Eds

≤ max−1≤s,t≤1

‖K(t, s)‖E→E

∫ |t|

−|t|e−λ(|t|−|s|)‖z − u‖C∗([−1,1],E)ds

= 2 max−1≤s,t≤1

‖K(t, s)‖E→E

∫ |t|

0e−λ(|t|−s)ds‖z − u‖C∗([−1,1],E)

= 2 max−1≤s,t≤1

‖K(t, s)‖E→E‖z − u‖C∗([−1,1],E)1 − e−λ|t|

λ

≤ ‖z − u‖C∗([−1,1],E)2(1 − e−λ)

λmax

−1≤s,t≤1‖K(t, s)‖E→E

(3.4)

for any t ∈ [−1, 1]. So,

‖Bz − Bu‖C∗([−1,1],E) ≤ ‖z − u‖C∗([−1,1],E)αλ, (3.5)

where αλ = (2(1− e−λ)/λ)max−1≤s,t≤1‖K(t, s)‖E→E and αλ → 0 when λ → ∞. Finally, we notethat the norms

‖v‖C∗([−1,1],E) = max−1≤t≤1

e−λ|t|‖v(t)‖E,

‖v‖C([−1,1],E) = max−1≤t≤1

‖v(t)‖E(3.6)

are equivalent in C([−1, 1], E). The proof of Theorem 3.1 is complete.


Theorem 3.2. Suppose that assumptions (1.2) and (1.4) for the operators A and B(t) hold. Assumethat f(t) is continuously differentiable on [−1, 1] function. Then there is a unique solution of (1.1)and stability inequality

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥E

+ max−1≤t≤1

‖Au(t)‖E ≤M∗[∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

](3.7)

holds, whereM∗ does not depend on f(t) and t.

Proof. The proof of the existence and uniqueness of the solution of (1.1) is based on thefollowing formula:

u(t) = sgn(t)A−1f(t) − sgn(t)e−|t|AA−1f(0) − sgn(t)∫ t

0e−(|t|−|s|)AA−1f ′(s)ds

+ sgn(t)∫ t

−t

[I − e−(|t|−|s|)A

]A−1B(s)u(s)ds, −1 ≤ t ≤ 1

(3.8)

and the Theorem 3.1.First, we note that the solution of (1.1) satisfies u(0) = 0. Indeed, assume that u(t) is

the solution of (1.1)with B ≡ 0. Then

u′(t) +Au(t) = f(t), 0 < t ≤ 1,

u′(t) −Au(t) = f(t), −1 ≤ t < 0,(3.9)

and from the continuity of f, u′(t), and Au at t = 0 we get

u′(0) +Au(0) = f(0),

u′(0) −Au(0) = f(0).(3.10)

This leads to 2Au(0) = 0, and it follows that u(0) = 0.Let us now prove (3.8). First, we consider the case when 0 ≤ t ≤ 1. It is well known

that the Cauchy problem

du(t)dt

+Au(t) = F(t), 0 ≤ t ≤ 1,

u(0) = 0(3.11)

for differential equations in an arbitrary Banach space E with positive operator A has theunique solution

u(t) =∫ t

0e−(t−s)AF(s)ds, 0 ≤ t ≤ 1 (3.12)


for smooth F(t). By putting

F(t) =∫ t

−tB(s)u(s)ds + f(t), (3.13)

we have

u(t) =∫ t

0e−(t−s)Af(s)ds +

∫ t

0e−(t−s)A

∫s

−sB(τ)u(τ)dτ ds, 0 ≤ t ≤ 1. (3.14)

Since

∫ t

0

∫s

−se−(t−s)AB(τ)u(τ)dτ ds =

∫ t

0

∫ t

τ

e−(t−s)AB(τ)u(τ)dsdτ

+∫0

−t

∫ t

−τe−(t−s)AB(τ)u(τ)dsdτ

=∫ t

0

(I − e−(t−τ)A

)A−1B(τ)u(τ)dτ

+∫0

−t

(I − e−(t+τ)A

)A−1B(τ)u(τ)dτ

=∫ t

−t

(I − e−(t−|s|)A

)A−1B(s)u(s)ds,

∫ t

0e−(t−s)Af(s)ds = A−1f(t) − e−tAA−1f(0) −

∫ t

0e−(t−s)AA−1f ′(s)ds,

(3.15)

we obtain (3.8) for 0 ≤ t ≤ 1.Now, let −1 ≤ t ≤ 0. Then we consider the problem

du(t)dt

−Au(t) = F(t), −1 ≤ t ≤ 0,

u(0) = 0(3.16)

for differential equations in an arbitrary Banach space E with positive operator A, which hasthe unique solution

u(t) = −∫0

t

e(t−s)AF(s)ds, −1 ≤ t ≤ 0. (3.17)

By putting

F(t) =∫ t

−tB(s)u(s)ds + f(t), (3.18)


we have

u(t) = −∫0

t

e(t−s)Af(s)ds +∫0

t

e(t−s)A∫−s

s

B(τ)u(τ)dτ ds, −1 ≤ t ≤ 0. (3.19)

Since

∫0

t

∫−s

s

e(t−s)AB(τ)u(τ)dτ ds =∫−t

0

∫−τ

t

e(t−s)AB(τ)u(τ)dsdτ

+∫0

t

∫ τ

t

e(t−s)AB(τ)u(τ)dsdτ

=∫−t

0

(I − e(t+τ)A

)A−1B(τ)u(τ)dτ

+∫0

t

(I − e(t−τ)A

)A−1B(τ)u(τ)dτ

=∫−t

t

(I − e−(−t−|s|)A

)A−1B(s)u(s)ds,

∫0

t

e(t−s)Af(s)ds = A−1f(t) − etAA−1f(0) +∫0

t

e−(−t+s)AA−1f ′(s)ds,

(3.20)

we obtain (3.8) for −1 ≤ t ≤ 0.From (3.8) it follows that

Au(t) = sgn(t)f(t) − sgn(t)e−|t|Af(0) − sgn(t)∫ t

0e−(|t|−|s|)Af ′(s)ds

+ sgn(t)∫ t

−t

[I − e−(|t|−|s|)A

]B(s)u(s)ds, −1 ≤ t ≤ 1.

(3.21)

Applying the triangle inequality and assumptions (1.2) and (1.4), we get

‖Au(t)‖E ≤ ∥∥f(t)∥∥E +∥∥∥e−|t|A

∥∥∥E→E

∥∥f(0)∥∥E +∫ |t|

−|t|

∥∥∥e−(|t|−|s|)A∥∥∥E→E

∥∥f ′(s)∥∥Eds

+∫ |t|

−|t|

[1 +∥∥∥e−(|t|−|s|)A

∥∥∥E→E

]∥∥∥B(s)A−1∥∥∥E→E

‖Au(s)‖Eds

≤ (M + 1)

[∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

]

+ sgn(t)M(M + 1)∫ t

−t‖Au(s)‖Eds, −1 ≤ t ≤ 1.

(3.22)


Then, using the Theorem 2.4, we have

‖Au(t)‖E ≤ (M + 1)

[∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

]e2M(M+1)|t|

≤ (M + 1)e2M(M+1)

[∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

], −1 ≤ t ≤ 1.

(3.23)

So,

max−1≤t≤1

‖Au(t)‖E ≤ (M + 1)e2M(M+1)

[∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

]. (3.24)

By applying the triangle inequality in (1.1) and assumptions (1.2) and (1.4), we obtain

∥∥∥∥du(t)dt

∥∥∥∥E

≤ ‖Au(t)‖E +∫ |t|

−|t|

∥∥∥B(s)A−1∥∥∥E→E

‖Au(s)‖Eds +∥∥f(t)

∥∥E

≤ (2M + 1)max−1≤t≤1

‖Au(t)‖E +∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds, −1 ≤ t ≤ 1.

(3.25)

So,

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥E

≤ (2M + 1)max−1≤t≤1

‖Au(t)‖E +∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds. (3.26)

Then using (3.24), we have

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥E

+ max−1≤t≤1

‖Au(t)‖E ≤ 2(M + 1)max−1≤t≤1

‖Au(t)‖E +∥∥f(0)

∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

≤(2(M + 1)2e2M(M+1) + 1

)[∥∥f(0)∥∥E +∫1

−1

∥∥f ′(s)∥∥Eds

].

(3.27)

So, stability inequality (3.7) holds withM∗ = 2(M+1)2e2M(M+1) +1. The proof of Theorem 3.2is complete.

Note that it does not hold, generally speaking

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥E

+ max−1≤t≤1

‖Au(t)‖E ≤M∗max−1≤t≤1

∥∥f(t)∥∥E (3.28)

in an arbitrary Banach space E for the general strong positive operatorA, see [17, Section 1.5,Chapter 1]. Nevertheless, we can establish the following theorem.


Theorem 3.3. Suppose that assumptions (1.4) for the operator A hold and

∥∥∥B(t)A−1∥∥∥Eα →Eα

≤M, −1 ≤ t ≤ 1. (3.29)

Assume that f(t) is a continuous on [−1, 1] function. Then there is a unique solution of (1.1) andstability inequality

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥Eα

+ max−1≤t≤1

‖Au(t)‖Eα ≤M∗(α)max−1≤t≤1

∥∥f(t)∥∥Eα

(3.30)

holds, where M∗(α) does not depend on f(t) and t. Here the fractional spaces Eα = Eα(E,A) (0 <α < 1), consisting of all v ∈ E for which the following norms are finite:

‖v‖Eα = sup0<z

z1−α∥∥A exp{−zA}v∥∥E. (3.31)

Proof. First, we rewrite (3.8) as

u(t) =∫ t

0e−(|t|−|s|)Af(s)ds + sgn(t)

∫ t

−t

[I − e−(|t|−|s|)A

]A−1B(s)u(s)ds, −1 ≤ t ≤ 1. (3.32)

The proof of the existence and uniqueness of the solution of (1.1) is based on the formula(3.32) and an analogue of the Theorem 3.1. Let us prove (3.30). From (3.32) it follows that

Au(t) =∫ t

0Ae−(|t|−|s|)Af(s)ds + sgn(t)

∫ t

−t

[I − e−(|t|−|s|)A

]B(s)u(s)ds, −1 ≤ t ≤ 1. (3.33)

Applying the triangle inequality, the definition of the norm of the space Eα and assumptions(1.4) and (3.29), we obtain

‖Au(t)‖Eα ≤∥∥∥∥∥

∫ |t|

−|t|Ae−(|t|−|s|)Af(s)ds

∥∥∥∥∥Eα

+∫ |t|

−|t|

[1 +∥∥∥e−(|t|−|s|)A

∥∥∥E→E

]∥∥∥B(s)A−1∥∥∥Eα →Eα

‖Au(s)‖Eαds

≤∥∥∥∥∥

∫ |t|


∥∥∥∥∥Eα

+ sgn(t)M(M + 1)∫ t

−t‖Au(s)‖Eαds.

(3.34)

By [17, Chapter 1, Theorem 4.1], we obtain

∥∥∥∥∥

∫ |t|


∥∥∥∥∥Eα

≤ M

α(1 − α) max−1≤t≤1

∥∥f(t)∥∥Eα. (3.35)


So,

‖Au(t)‖Eα ≤M

α(1 − α) max−1≤t≤1

∥∥f(t)∥∥Eα

+ sgn(t)M(M + 1)∫ t

−t‖Au(s)‖Eαds (3.36)

for −1 ≤ t ≤ 1. Then, using the Theorem 2.4, we have

‖Au(t)‖Eα ≤M

α(1 − α) max−1≤t≤1

∥∥f(t)∥∥Eαe2M(M+1)|t|, −1 ≤ t ≤ 1. (3.37)

So,

max−1≤t≤1

‖Au(t)‖Eα ≤M

α(1 − α) max−1≤t≤1

∥∥f(t)∥∥Eαe2M(M+1). (3.38)

Then, using the triangle inequality in (1.1) yields

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥Eα

≤ M(M + 1)α(1 − α) max

−1≤t≤1

∥∥f(t)∥∥Eαe2M(M+1). (3.39)

Combining last two inequalities, we obtain (3.30)withM∗(α) = (M(M+2)/α(1−α))e2M(M+1).The proof of Theorem 3.3 is complete.

Now, we consider the Rothe difference scheme for approximate solutions of (1.1).

uk − uk−1τ

+Auk =k−1∑

i=−k+1Biuiτ + ϕk, k = 1, . . . ,N,

uk − uk−1τ

−Auk−1 = −−k∑

i=k

Biuiτ + ϕk, k = −N + 1, . . . , 0,

Bk = B(tk), tk = kτ, k = −N, . . . ,N,

u0 = 0.

(3.40)

Theorem 3.4. Suppose that the requirements of the Theorem 3.2 are satisfied. Then for the solution ofdifference scheme (3.40), the following stability inequalities

maxk=−N+1,...,N

∥∥∥∥uk − uk−1

τ

∥∥∥∥E

+ maxk=−N,...,N

‖Auk‖E ≤M∗[∥∥ϕ0∥∥E +

N∑

k=−N+1

∥∥ϕk − ϕk−1∥∥E

](3.41)

hold, whereM∗ does not depend on ϕk, k = −N, . . . ,N.

Proof. By induction we can prove that the initial value problem

uk − uk−1τ

+Auk = ψk, k = 1, . . . ,N, u0 = 0 (3.42)


for difference equations in an arbitrary Banach space Ewith positive operatorA has a uniquesolution

uk =k∑

i=1

Rk−i+1ψiτ, k = 1, . . . ,N, (3.43)

where R = (I + τA)−1. By putting ψk =∑k−1

i=−k+1 Biuiτ + ϕk, we obtain

Auk = Ak∑

i=1

Rk−i+1ϕiτ +Ak∑

i=1

Rk−i+1i−1∑

j=−i+1Bjujτ

2, k = 1, . . . ,N. (3.44)

Using

τk∑

i=±j+1Rk−i+1

= τ(R + R2 + · · · + Rk∓j

)= τR(I − R)−1

(I − Rk∓j

)= A−1

(I − Rk∓j

), k = 1, . . . ,N,

(3.45)

we have

Ak∑

i=1

Rk−i+1i−1∑

j=−i+1Bjujτ

2 = A0∑

j=−k+1τ

k∑

i=−j+1Rk−i+1Bjujτ +A

k−1∑

j=1

τk∑

i=j+1

Rk−i+1Bjujτ

= A0∑

j=−k+1A−1(I − Rk+j

)Bjujτ +A

k−1∑

j=1

A−1(I − Rk−j

)Bjujτ

=k−1∑

i=−k+1

[I − Rk−|i|

]Biuiτ.

(3.46)

Furthermore,

Ak∑

i=1

Rk−i+1ϕiτ =k∑

i=1

(I − R)Rk−iϕi

=k∑

i=1

Rk−iϕi −k∑

i=1

Rk−i+1ϕi

=k+1∑

i=2

Rk−i+1ϕi−1 −k∑

i=1

Rk−i+1ϕi


=k∑

i=1

Rk−i+1ϕi−1 − Rkϕ0 + ϕk −k∑

i=1

Rk−i+1ϕi

= ϕk − Rkϕ0 −k∑

i=1

Rk−i+1(ϕi − ϕi−1).

(3.47)

Putting (3.46)-(3.47) in (3.44), we get

Auk = ϕk − Rkϕ0 −k∑

i=1

Rk−i+1(ϕi − ϕi−1)+

k−1∑

i=−k+1i /= 0

[I − Rk−|i|

]Biuiτ (3.48)

for k = 1, . . . ,N.Since

Rk = (I + τA)−k =1

(k − 1)!

∫∞

0tk−1e−te−τtAdt, k = 1, . . . ,N, (3.49)

applying estimates (1.4) gives

∥∥∥Rk∥∥∥E→E

≤ M

(k − 1)!

∫∞

0tk−1e−tdt =M, k = 1, . . . ,N. (3.50)

Then, applying the triangle inequality and the estimate (1.2) in (3.48), we obtain

‖Auk‖E ≤ ∥∥ϕk∥∥E +∥∥∥Rk∥∥∥E→E

∥∥ϕ0∥∥E +

k∑

i=1

∥∥∥Rk−i+1∥∥∥E→E

∥∥ϕi − ϕi−1∥∥E

+k−1∑

i=−k+1i /= 0

(1 +∥∥∥Rk−|i|

∥∥∥E→E

)∥∥∥BiA−1∥∥∥E→E

‖Aui‖Eτ

≤∥∥∥∥∥

k∑

i=1

(ϕi − ϕi−1

)+ ϕ0

∥∥∥∥∥E

+M∥∥ϕ0∥∥E +M

k∑

i=1

∥∥ϕi − ϕi−1∥∥E +M(M + 1)

k−1∑

i=−k+1i /= 0

‖Aui‖Eτ

≤ (M + 1)N∑

i=−N+1

∥∥ϕi − ϕi−1∥∥E + (M + 1)

∥∥ϕ0∥∥E

+M(M + 1)k−1∑

i=−k+1i /= 0

‖Aui‖Eτ, k = 1, . . . ,N.

(3.51)


So, for k = 1, . . . ,N,

‖Auk‖E ≤ M(∥∥ϕ0∥∥E +

N∑

i=−N+1

∥∥ϕi − ϕi−1∥∥E

)+M(M + 1)

k−1∑

i=−k+1i /= 0

‖Aui‖Eτ (3.52)

holds, where M = max{M(1 + τ(M + 1)),M + 1}.In a similar way, we can prove that the initial value problem

uk − uk−1τ

−Auk−1 = −−k∑

i=k

Biuiτ + ϕk, k = −N + 1, . . . , 0, u0 = 0 (3.53)

for difference equations in an arbitrary Banach space Ewith positive operatorA has a uniquesolution and inequalities

‖Auk‖E ≤ M(∥∥ϕ0∥∥E +

N∑

i=−N+1

∥∥ϕi − ϕi−1∥∥E

)+M(M + 1)

−k−1∑

i=k+1i /= 0

‖Aui‖Eτ (3.54)

hold for k = −N . . . , 0.Now, the proof of this theorem is based on the Theorem 2.2 and the inequalities (3.52)

and (3.54). The proof of Theorem 3.4 is complete.

Note that it does not hold, generally speaking

maxk=−N+1,...,N

∥∥∥∥uk − uk−1

τ

∥∥∥∥E

+ maxk=−N,...,N

‖Auk‖E ≤M∗ maxk=−N,...,N

∥∥ϕk∥∥E (3.55)

in the arbitrary Banach space E for the general strong positive operator A, see [17, 18].This approach and theory of difference schemes of [17] permit us to obtain the

following two theorems on stability estimates for the solution of difference scheme (3.40).

Theorem 3.5. Suppose that the requirements of the Theorem 3.2 are satisfied. Then for the solution ofdifference scheme (3.40) the following stability inequalities

maxk=−N+1,...,N

∥∥∥∥uk − uk−1

τ

∥∥∥∥E

+ maxk=−N,...,N

‖Auk‖E

≤M∗ min{ln

1τ, 1 + |ln ‖A‖E→E|

}max

k=−N,...,N

∥∥ϕk∥∥E

(3.56)

hold, whereM∗ does not depend on ϕk, k = −N, . . . ,N.


Theorem 3.6. Suppose that the requirements of the Theorem 3.3 are satisfied. Then for the solution ofdifference scheme (3.40), the following stability inequalities

maxk=−N+1,...,N

∥∥∥∥uk − uk−1

τ

∥∥∥∥E′α

+ maxk=−N,...,N

‖Auk‖E′α≤M∗(α) max

k=−N,...,N

∥∥ϕk∥∥E′α

(3.57)

hold, where M∗(α) does not depend on ϕk, k = −N, . . . ,N. Here the fractional spaces E′α =

E′α(E,A) (0 < α < 1), consisting of all v ∈ E for which the following norms are finite:

‖v‖E′α= sup

0<zzα∥∥∥A(z +A)−1v

∥∥∥E. (3.58)

Stability estimates could be also proved for the more general Pade difference schemesof the high order of accuracy, see [17, 19].

4. Conclusion

In this paper, the integral-differential equation of the parabolic type with two dependentlimits in a Banach space is studied. The unique solvability of this equation is established. Thestability estimates for the solution of this equation are obtained. The Rothe difference schemeapproximately solving this equation is presented. The stability estimates for the solution ofthis difference scheme are obtained.

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Research ArticleFinite Difference Method for the ReverseParabolic Problem

Charyyar Ashyralyyev,1, 2 Ayfer Dural,3 and Yasar Sozen4

1 Department of Computer Technology of the Turkmen Agricultural University, Gerogly Street,74400 Ashgabat, Turkmenistan

2 Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey3 Gaziosmanpasa Lisesi, 34245 Istanbul, Turkey4 Department of Mathematics, Fatih University, 34500 Istanbul, Turkey

Correspondence should be addressed to Yasar Sozen, [email protected]

Received 17 April 2012; Accepted 12 June 2012


Copyright q 2012 Charyyar Ashyralyyev et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A finite difference method for the approximate solution of the reverse multidimensional parabolicdifferential equation with a multipoint boundary condition and Dirichlet condition is applied.Stability, almost coercive stability, and coercive stability estimates for the solution of the firstand second orders of accuracy difference schemes are obtained. The theoretical statements aresupported by the numerical example.

1. Introduction

In the study of boundary value problems for partial differential equations, the role played bythe well-posedness (coercivity inequalities) is well known (see, e.g., [1–3]). Well-posednessof nonlocal boundary value problems for partial differential equations of parabolic type hasbeen studied extensively by many researchers (see, e.g., [4–15] and the references therein).

In the paper [4], Ashyralyev studied the positivity of second-order differential anddifference operators with nonlocal condition and the structure of interpolation spacesgenerated by these operators in a Banach space. Applying this result, he obtained the coerciveinequalities for the solutions of the nonlocal boundary value problem for differential anddifference equations.

In [5], Ashyralyev et al. considered the nonlocal boundary value problem

v′(t) +Av(t) = f(t), 0 < t < 1, v(0) = v(λ) + μ, 0 < λ ≤ 1, (1.1)


in a Banach space with strongly positive operator A. They established the well-posednessof problem (1.1) in Holder spaces. Moreover, they obtained the exact Schauder’s estimates inHolder norms of solutions of the boundary values problem for 2m-th order multidimensionalparabolic equations.

Ashyralyev established in [6] the well-posedness of the nonlocal boundary-valueproblem (1.1) in Bochner spaces. He considered the first and second order of accuracydifference schemes for the approximate solutions of problem (1.1). He also established thecoercive inequalities for the solutions of these difference schemes. Moreover, in applications,he obtained the almost coercive stability and coercive stability estimates for the solutions ofdifference schemes for the approximate solutions of the nonlocal boundary-value problemfor parabolic equation.

Clement and Guerre-Delabriere studied in [8]maximal regularity (in the Lp-sense) forabstract Cauchy problems of order one and boundary value problems of order two. As iswell-known regularity of the first problems implies regularity of the second ones; they alsoproved that the converse to hold if the underlying Banach space has the UMD property. Astronger notion of regularity, which is introduced by Sobolevskii, plays an important role inthe proofs.

In [9], Gulin et al. considered the linear heat equation:

ut = uxx + uyy, 0 < x < 1, 0 < y < 1, (1.2)

with Dirichlet condition u(x, 0, t) = u(x, 1, t) = 0, 0 ≤ x ≤ 1 and nonlocal boundary conditionsu(0, y, t) = 0, ux(0, y, t) = ux(1, y, t), 0 ≤ y ≤ 1. They constructed an explicit difference schemewith second order of approximation with respect to the space variables and first order ofapproximation with respect to t. Moreover, using previous results of Ionkin and Morozovafor the one-dimensional heat equation with nonlocal boundary conditions, they proved thestability of this scheme with respect to theD1-norm ‖y‖D1 = (D1y, y)

1/2, which is induced bythe symmetric and positive-definite matrix D1.

Liu et al. studied in [10] a finite difference method for multidimensional nonlinearcoupled system of parabolic and hyperbolic equations. By using a variational method, theyobtained an a priori estimate. They also proved that the finite difference scheme is uniquelysolvable and unconditionally stable. To support the theory, they gave numerical example oftwo-dimensional problem.

In [11, 12], Martin-Vaquero and Vigo-Aguiar provided algorithms improving the CPUtime and accuracy of Crandall’s formula. They studied the convergence of the algorithms andcompared the efficiency of the methods with well-known numerical examples.

In [13], Sapagovas applied finite difference approximations to a nonhomogeneousheat equation in one space dimension, subject to nonlocal boundary conditions. He presenteda stable difference approximation for the equation and a piecewise constant discretization ofthe integrals appearing in the boundary conditions. He discussed the stability of the completeproblem with respect to two parameters included in the integral terms. He constructed astability region in the plane of the parameters and gave practical examples with specificchoices of the integral conditions. Sapagovas investigated in [14] the stability of implicitdifference schemes for the equation of a thermoelastic rod, which is a parabolic equationsubject to integral conditions for the boundaries.

In [15], Shakhmurov dealt with a nonlocal boundary value problem for a degenerateequation in a Banach space E with unbounded operators in E. He proved the maximal Lp


regularity and Fredholmness of the problem. He also applied the results to nonlocal boundaryvalue problems for degenerate elliptic and quasielliptic differential equations and their finiteor infinite systems on cylindrical domains.

It is well known that reverse problems arise in various applications, for example,boundary layer problems in fluid dynamics [16, 17], plasma physics, and astrophysics inthe study of propagation of an electron beam through the solar corona [18]. For furtherapplications of such problems, we refer the reader to [19–22] and the references therein.

In the paper [23], Ashyralyev et al. considered themultipoint nonlocal boundary valueproblem for reverse parabolic equations

du(t)dt

−Au(t) = f(t), (0 ≤ t ≤ 1),

u(1) =p∑

k=1

αku(θk) + ϕ,

0 ≤ θ1 < θ2 < · · · < θp < 1

(1.3)

in a Hilbert spaceH with self-adjoint positive definite operator A.u(t) is called a solution of problem (1.3) if the following conditions hold:

(1) u(t) is continuously differentiable on the segment [0, 1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.

(2) The element u(t) belongs to D(A) for all t ∈ [0, 1] and the function Au(t) iscontinuous on the segment [0, 1].

(3) u(t) satisfies the equation and the nonlocal boundary conditions (1.3).

A solution of problem (1.3) defined in this manner will be from now referred to as asolution of problem (1.3) in the space C(H) = C([0, 1],H) of all continuous functions ϕ(t)defined on [0, 1] with values inH equipped with the norm

∥∥ϕ∥∥C(H) = max

0≤t≤1

∥∥ϕ(t)∥∥H. (1.4)

Problem (1.3) is well posed in C(H), if for the solutions of (1.3), we have the followingcoercivity inequality:

∥∥u′∥∥C(H) + ‖Au(t)‖C(H) ≤M

(∥∥f∥∥C(H) +

∥∥Aϕ∥∥H

). (1.5)

Here, 1 ≤M is independent of f(t) ∈ C(H), ϕ ∈ D(A).Throughout the paper,M indicates positive constants which can be different from time

to time and we are not interested to make precise. We writeM(α, β, . . .) to stress the fact thatthe constant depends only on α, β, . . .

Under the assumption:

p∑

k=1

|αk| ≤ 1. (1.6)


Ashyralyev et al. established in [23] the well-posedness of these problems in the spaceof smooth functions. In applications, they obtained coercivity estimates for the solution ofparabolic differential equations.

Moreover, in [24], Ashyralyev et al. considered the first order of accuracy Rothedifference scheme:

τ−1(uk − uk−1) −Auk−1 = ϕk, ϕk = f(tk),

tk = kτ, 1 ≤ k ≤N, Nτ = 1,

uN =p∑

m=1

αmu�m + ϕ,

�m =[θmτ

], 1 ≤ m ≤ p,

(1.7)

for approximately solving problem (1.3). They established some stability estimates andalmost coercivity of the solution for the difference scheme.

In the present paper, multipoint nonlocal boundary value problem for themultidimen-sional parabolic equation with Dirichlet condition,

ut(t, x) +n∑

r=1

(ar(x)uxr )xr = f(t, x),

x = (x1, . . . , xn) ∈ Ω, 0 < t < 1,

u(1, x) =p∑

i=1

αiu(θi, x) + ϕ(x), x ∈ Ω,

0 ≤ θ1 < θ2 < · · · < θp < 1,

u(t, x) = 0, x ∈ S, 0 ≤ t ≤ 1

(1.8)

under the condition (1.6) is considered. Here, ar(x), (x ∈ Ω), ϕ(x) (x ∈ Ω), and f(t, x) (t ∈(0, 1), x ∈ Ω) are given smooth functions and ar(x) ≥ a > 0, and Ω = (0, �) × · · · × (0, �) is theopen cube in the n-dimensional Euclidean space with boundary S, Ω = Ω ∪ S. In the HilbertspaceH = L2(Ω), we introduce the self-adjoint positive definite operator A defined by

Au(x) = −n∑

r=1

(ar(x)uxr (x))xr , (1.9)

with domain

D(A) ={u, uxr , uxrxr ∈ L2

(Ω), 1 ≤ r ≤ n : u(x)|x∈S = 0

}. (1.10)

Then, problem (1.8) can be written in the abstract form as the nonlocal boundary valueproblem for reverse parabolic equation (1.3).


The first and second orders of accuracy in t and the second order of accuracy in spacevariables for the approximate solution of problem (1.8) are presented. Applying the methodof papers [23, 24], the stability, almost coercive stability, and coercive stability estimatesfor the solution of these difference schemes are obtained. The modified Gauss eliminationmethod for solving these difference schemes in the case of one-dimensional parabolic partialdifferential equations is used.

2. Difference Schemes: Stability Estimates

We will discretize problem (1.8) in two steps. In the first step, we define the grid spaces

Ωh = {x = xm = (h1m1, . . . , hnmn);m = (m1, . . . , mn), mr = 0, . . . ,Nr, hrNr = 1, r = 1, . . . , n},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.(2.1)

We denote that |h| =√h21 + · · · + h21. Let L2h = L2(Ωh) denote the Banach space of grid

functions:

ϕh(x) ={ϕ(h1m1, . . . , hnmn)

}, (2.2)

defined on Ωh, equipped with the norm

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

. (2.3)

To the differential operator A generated by problem (1.8), we assign the second-orderapproximation difference operator Ax

h= Cx

h+ Bx

hacting in the space of grid functions uh(x),

satisfying the condition uh(x) = 0 for all x ∈ Sh. Assume that Cxhis self-adjoint, positive-

definite operator in L2h and (Cxh)−1Bx

his bounded operator in L2h.

By using Axh, we arrive at the multipoint nonlocal boundary value problem:

duh(t, x)dt

−Axhu

h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,

uh(1, x) =p∑

m=1

αmuh(θm, x) + ϕh(x), x ∈ Ωh,

(2.4)

for a finite system of ordinary differential equations with a fixed |h|. Note that |h| =√h21 + · · · + h2n → 0. Therefore, we will try to obtain stability, coercivity stability, and almost

coercivity estimates with constants independent of |h|.


In the second step, problem (2.4) is replaced by the first order of accuracy differencescheme

uhk(x) − uhk−1(x)τ

−Axhu

hk−1(x) = f

hk (x),

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N, x ∈ Ωh,

uhN(x) =p∑

m=1

αmuh�m(x) + ϕh(x), x ∈ Ωh,

�m =[θmτ

], m = 1, . . . , p,

(2.5)

and the second order of accuracy difference scheme


−AxhB

xhu

hk−1(x) = f

hk (x),

Bxh = I +τAx

h

2, fhk (x) = B

xhf

h(tk−(τ/2), x),

tk = kτ, 1 ≤ k ≤N, Nτ = 1, x ∈ Ωh,

uhN(x) =p∑

m=1

αm{(I + dmAx

h

)uh�m(x) + dmB

xhψ�m+1

}+ ϕh(x), x ∈ Ωh,

dm = θm −[θmτ

]τ, �m =

[θmτ

], m = 1, . . . , p,

(2.6)

where [·] denotes the greatest integer function.To formulate our results, let L2h = L2(Ωh) and W2

2h = W22 (Ωh) be spaces of the grid

functions ϕh(x) = {ϕ(h1m1, . . . , hnmn)} defined on Ωh, equipped with the norms

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣∣(ϕh)

xr

∣∣∣∣2

h1 · · ·hn⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣∣(ϕh(x)

)

xrxr ,mr

∣∣∣∣2

h1 · · ·hn⎞

⎠1/2

.

(2.7)

Furthermore, let [0, 1]τ = {tk = kτ, 1 ≤ k ≤ N, Nτ = 1} be the uniform grid space withstep size τ > 0, whereN is a fixed positive integer. We denote Fτ(H) = F([0, 1]τ ,H) for thelinear space of grid functions ϕτ = {ϕk}N1 with values in the Hilbert spaceH.


For α ∈ [0, 1], let Cα(H) = Cα([0, 1]τ ,H) and Cα1(H) = Cα1([0, 1]τ ,H) be, respectively,the Holder space and the weighted Holder space with the norms

∥∥ϕτ∥∥Cα(H) =

∥∥ϕτ∥∥Cτ (H) + max

1≤k<k+r≤N

∥∥ϕk+r − ϕk∥∥H

(rτ)α,

∥∥ϕτ∥∥Cα1 (H) =

∥∥ϕτ∥∥Cτ (H) + max

1≤k<k+r≤N

((N − k)τ)α∥∥ϕk+r − ϕk∥∥H

(rτ)α.

(2.8)

Here, Cτ(H) = C([0, 1]τ ,H) is the Banach space of bounded grid functions with norm:

∥∥ϕτ∥∥Cτ (H) = max

1≤k≤N

∥∥ϕk∥∥H. (2.9)

Theorem 2.1. Let τ and |h| be sufficiently small positive numbers. Then, for the solutions of differenceschemes (2.5) and (2.6), the following stability estimate holds:

∥∥∥∥{uhk

}N0

∥∥∥∥Cτ (L2h)

≤M(δ, θp

)[∥∥∥ϕh

∥∥∥L2h

+∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (L2h)

], (2.10)

whereM(δ, θp) is independent of τ , h, ϕh(x), and fhk (x) and k = 1, . . . ,N.

Proof. The proof of Theorem 2.1 is based on the formulas for the solution of difference scheme(2.5)

uhk = RN−kuhN −N∑

j=k+1

Rj−kfhj τ, 0 ≤ k ≤N − 1, (2.11)

uhN = Tτ

⎛

⎝−p∑

k=1

N∑

j=�k+1

αkRj−�kfhj τ + ϕh

⎞

⎠, (2.12)

and for the solution of difference scheme (2.6)

uhk = DN−kuhN −N∑

j=k+1

Dj−kfhj τ, 0 ≤ k ≤N − 1, (2.13)

uhN = T ′τ

⎧⎨

⎩−p∑

m=1

N∑

j=�m+1

αm(I + dmAx

h

)Dj−�mfhj τ +

p∑

m=1

αmdm(Bxh)−1

fh�m+1 + ϕh

⎫⎬

⎭. (2.14)


Here,

R =(I + τAx

h

)−1, D =

⎛

⎝I + τAxh +

(τAx

h

)2

2

⎞

⎠−1

,

Tτ =

(I −

p∑

k=1

αkRN−[θk/τ]

)−1, T ′

τ =

(I −

p∑

k=1

αk(I + dkAx

h

)DN−[θk/τ]

)−1.

(2.15)

By the spectral representation of self-adjoint positive definite operator and the triangleinequality, we have

‖Tτ‖H→H ≤ supδ≤μ

1∣∣∣1 −∑p

k=1 αk(1 + τμ

)−N+[θp/τ]∣∣∣≤M(

δ, θp). (2.16)

Similarly, we have

∥∥T ′τ

∥∥H→H ≤M(

δ, θp). (2.17)

Estimates (2.16) and (2.17) conclude the proof of Theorem 2.1.

Theorem 2.2. Let τ and |h| be sufficiently small positive numbers. Then, for the solutions of differenceproblem (2.5) and (2.6), the following almost coercivity inequality

∥∥∥∥{τ−1(uhk − ukk−1

)}N1

∥∥∥∥Cτ (L2h)

≤M(δ, θp

)[∥∥∥ϕh

∥∥∥W2

2h

+∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (L2h)

ln1

τ + |h|

](2.18)

is valid, whereM(δ, θp) does not depend on τ, h, ϕh(x),fhk (x), k = 1, . . . ,N.

Proof. Using formulas (2.11)–(2.14), estimates (2.16) and (2.17), the triangle inequality,assumption (1.6), we obtain

∥∥∥∥{τ−1(uhk − ukk−1

)}N1

∥∥∥∥Cτ (L2h)

≤M(δ, θp

)[∥∥∥ϕh

∥∥∥W2

2h

+min{ln

1τ, 1 +

∣∣∣∣ln∥∥∥Ah

x

∥∥∥L2h →L2h

∣∣∣∣}∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (L2h)

].

(2.19)

Since

∣∣∣∣ln∥∥∥Ah

x

∥∥∥L2h →L2h

∣∣∣∣ ≤M ln1|h| , (2.20)


we have that

min{ln

1τ, 1 +

∣∣∣∣ln∥∥∥Ah

x

∥∥∥L2h →L2h

∣∣∣∣}

≤M1 ln1

τ + |h| . (2.21)

From that, inequality (2.19), and the following theorem on the coercivity inequality for thesolution of the elliptic difference problem in L2h it follows inequality (2.18). Theorem 2.2 isproved.

Theorem 2.3 (see [25, 26]). For the solution of the elliptic difference problem:

Axhu

h(x) = ωh(x), x ∈ Ωh,

uh(x) = 0, x ∈ Sh,(2.22)


n∑

r=1

∥∥∥∥(uhk

)

xrxr ,jr

∥∥∥∥L2h

≤M∥∥∥ωh

∥∥∥L2h, (2.23)

whereM does not depend on h and ωh.

Theorem 2.4. Let τ and |h| be sufficiently small positive numbers. Then, the solutions of differenceproblem (2.5) and (2.6) satisfy the following coercivity stability estimate:

∥∥∥∥{τ−1(uhk − uhk−1

)}N1

∥∥∥∥Cα1 (L2h)

≤M(δ, θp, α

)[∥∥∥ϕh

∥∥∥W2

2h

+∥∥∥∥{fhk

}N1

∥∥∥∥Cα1 (L2h)

], (2.24)

whereM(δ, θp, α) is independent of τ , h, fhk (x), and ϕh(x), k = 1, . . . ,N.

Theorem 2.5. Let Axhϕ

h(x) = fhN(x) − ∑p

k=1 αmfh�m(x). Then, for solutions of problem (2.5) and

(2.6), the following coercive stability estimate holds:

∥∥∥∥{τ−1(uhk − uhk−1

)}N1

∥∥∥∥Cα(L2h)

+∥∥∥∥{uhk

}N1

∥∥∥∥Cα(W2

2h)≤M(

δ, θp, α)∥∥∥∥{fhk

}N1

∥∥∥∥Cα(L2h)

, (2.25)

whereM(δ, θp, α) does not depend on τ, h, fhk(x), and ϕh(x), k = 1, . . . ,N.


The proofs of Theorems 2.4–2.5 are based on the formulas:

Axhu

hk−1 = R

N−k+1Axhu

hN −

N∑

j=k+1

τAxhR

j−k+1(fhj − fhk

)+(RN−k+1 − I

)fhk ,

AxhB

xhu

hk−1 = D

N−k+1AxhB

xhu

hN

−N∑

j=k+1

τAxh

(I +

τAxh

2

)Dj−k+1

(fhj − fhk

)+(DN−k+1 − I

)fhk ,

(2.26)

the self-adjoint positive definiteness of the operatorAxhin L2h, estimates (2.16) and (2.17), the

triangle inequality, and assumption (1.6).


For the numerical result, we consider the nonlocal boundary value problem:

∂u(t, x)∂t

+ (2 + cos(x))∂2u(t, x)∂x2

− sin(x)∂u(t, x)∂x

= f(t, x), 0 < x < π, 0 < t < 1,

f(t, x) =(2t − 6t2 + 4t3

)sin(x)

− t2(1 − t)2(2 + cos(x)) sin(x) − t2(1 − t)2 cos(x) sin(x),

u(0, x) = u(1, x), 0 ≤ x ≤ π,

u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1,(3.1)

for the reverse parabolic equation. It is easy to see that u(t, x) = t2(1 − t)2 sin(x) is the exactsolution of (3.1).

For the approximate solution of nonlocal boundary value problem (3.1), consider theset [0, 1]τ × [0, π]h of a family of grid points depending on the small parameters τ and h

[0, 1]τ × [0, π]h

= {(t, xn) : tk = kτ, k = 1, . . . , N − 1,Nτ = 1, xn = nh, n = 1, . . . ,M − 1,Mh = π}.(3.2)


Applying (2.5), we get the first order of accuracy in t and the second order of accuracyin x

ukn − uk−1n

τ+ (2 + cos(xn))

uk−1n+1 − 2uk−1n + uk−1n−1h2

− sin(xn)uk−1n+1 − uk−1n−1

2h= f(tk, xn), k = 1, . . . ,N, n = 1, . . . ,M − 1,

uk0 = ukM = 0, k = 0, . . . ,N,

u0n = uNn , n = 0, . . . ,M,

(3.3)

for the approximate solutions of the nonlocal boundary value problem (3.1).Note that for difference scheme (3.3), we have that

{−(2 + cos(xn))

uk−1n+1 − 2uk−1n + uk−1n−1h2

− sin(xn)uk−1n+1 − uk−1n−1

2h

}M−1

1

= Cxhu

hk−1(x) + B

xhu

hk−1(x),

(3.4)

where

Cxhu

h(x) ={− (2 + cos(xn+1))((un+1 − un)/h) − (2 + cos(xn))((un − un−1)/h)

h

}M−1

1,

Bxhuh(x) =

{cos(xn+1) − cos(xn)

h

un − un−1h

− sin(xn)un+1 − un

2h

}M−1

1.

(3.5)

It is easy to see that Cxh = (Cx

h)∗ and Cx

h ≥ δIh, and

∥∥∥(Cxh

)−1Bxh

∥∥∥L2h →L2h

≤M, (3.6)

where Ih is the identity operator.So, Theorems 2.1, 2.2, 2.4, and 2.5 are compatible for the solution of (3.3).We can write (3.3) as in the matrix form

Anun+1 + Bnun + Cnun−1 = Iϕn, n = 1, . . . ,M − 1,

u0 = �0, uM = �0.(3.7)


Here, ϕn is an (N + 1) × 1 column matrix, An, Bn, Cn are (N + 1) × (N + 1) square matrices,An = anR, Cn = cnR,

R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0 0 01 0 0 0 · · · 0 0 0 00 1 0 0 · · · 0 0 0 00 0 1 0 · · · 0 0 0 0...

......

... · · · ......

......

0 0 0 0 · · · 0 0 0 00 0 0 0 · · · 1 0 0 00 0 0 0 · · · 0 1 0 00 0 0 0 · · · 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Bn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 · · · 0 0 0 −1bn d 0 0 · · · 0 0 0 00 bn d 0 · · · 0 0 0 00 0 bn d · · · 0 0 0 0...

......

... · · · ......

......

0 0 0 0 · · · d 0 0 00 0 0 0 · · · bn d 0 00 0 0 0 · · · 0 bn d 00 0 0 0 · · · 0 0 bn d

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

an =2 + cos(xn)

h2− sin(xn)

2h, bn = − 1

τ− 2(2 + cos(xn))

h2,

cn =2 + cos(xn)

h2+sin(xn)

2h, d =

1τ.

ϕn =

⎡⎢⎣

ϕ0n...ϕNn

⎤⎥⎦,

ϕ0n = 0, n = 1, . . . ,M − 1,

ϕkn = f(tk, xn), n = 1, . . . ,N, n = 1, . . . ,M − 1,

(3.8)

here and in the future I is the (N + 1) × (N + 1) identity matrix,

us =

⎡⎢⎣

u0s...uNs

⎤⎥⎦

(N+1)×1

, s = n − 1, n, n + 1. (3.9)

Samarskii and Nikolaev studied this type of system in [27] for difference equations.We seek the solution of (3.7) by the formula

un = αn+1un+1 + βn+1, n =M − 1, . . . , 1, (3.10)


where uM = �0, αn (n = 1, . . . ,M − 1) are (N + 1) × (N + 1) square matrices and βn (n =1, . . . ,M − 1) are (N + 1)× 1 column matrices. For the solution of difference equation (3.7)weneed to use the following formulas for αn, βn:

αn = −(Bn + Cnαn−1)−1An,

βn = (Bn + Cnαn−1)−1(Iϕn − Cnβn−1

), n = 2, . . . ,M − 1,

(3.11)

where α1 is the (N + 1) × (N + 1) zero matrix and β1 is the (N + 1) × 1 zero column vector.Second, we consider again the nonlocal boundary value problem (3.1). Applying (2.6)

and formulas:

u(xn+1) − u(xn−1)2h

− u′(xn) = O(h2),

u(xn+1) − 2u(xn) + u(xn−1)h2

− u′′(xn) = O(h2),

u(xn+2) − 4u(xn+1) + 6u(xn) − 4u(xn−1) + u(xn−2)h4

− u(4)(xn) = O(h2),

2u(0) − 5u(h) + 4u(2h) − u(3h)h2

− u′′(0) = O(h2),

2u(1) − 5u(1 − h) + 4u(1 − 2h) − u(1 − 3h)h2

− u′′(1) = O(h2),

(3.12)

we get the second order of accuracy in t and x

ukn − uk−1n

τ− (sin(xn) + τ sin(xn) + τ sin(xn) cos(xn))

uk−1n+1 − uk−1n−12h

+((2 + cos(xn)) +

τ

2

(6 cos(xn) + 5cos2(xn) − 2

))uk−1n+1 − 2uk−1n + uk−1n−1h2

+ τ(4 sin(xn) + 2 sin(xn) cos(xn))uk−1n+2 − 2uk−1n+1 + 2uk−1n−1 − uk−1n−2

2h3

−τ2(2 + cos(xn))2

(uk−1n+2 − 4uk−1n+1 + 6uk−1n − 4uk−1n−1 + u

k−1n−2)

h4= ϕkn,

ϕkn = f(tk − τ

2, xn)− τ

2

(1h2

(2 + cos(xn))

×(f(tk − τ

2, xn+1

)− 2f

(tk − τ

2, xn)+ f(tk − τ

2, xn−1

))

− 12h

sin(xn)(f(tk − τ

2, xn+1

)− f(tk − τ

2, xn−1

))),


k = 1, . . . ,N, n = 2, . . . ,M − 2,

uk0 = ukM = 0, uk1 =45uk2 −

15uk3 , k = 0, . . . ,N,

ukM−1 =45ukM−2 −

15ukM−3, k = 0, . . . ,N,

u0n − uNn = 0, n = 0, . . . ,M,

(3.13)

for the approximate solutions of the nonlocal boundary value problem (3.1).We can rewrite this system in the following matrix form:

Anun+2 + Bnun+1 + Cnun +Dnun−1 + Enun−2 = Iϕn, n = 2, . . . ,M − 2,

u0 = �0, uM = �0, u1 =45u2 − 1

5u3, uM−1 =

45uM−2 − 1

5uM−3,

(3.14)

where ϕn is an (N + 1) × 1 column matrix, An, Bn, Cn, Dn, En are (N + 1) × (N + 1) squarematrices

ϕn =

⎡⎢⎣

ϕ0n...ϕNn

⎤⎥⎦, (3.15)

An = vnR, Bn = ynR,Dn = znR, En = wnR,

Cn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 · · · 0 −1rn p 0 · · · 0 00 rn p · · · 0 0...

...... · · · ...

...0 0 0 · · · p 00 0 0 · · · rn p

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (3.16)

where p = 1/τ ,

vn =τ

2h3(4 sin(xn) + 2 sin(xn) cos(xn)) − τ

2h4(2 + cos(xn))2,

yn = − 12h

(sin(xn) + τ sin(xn) + τ sin(xn) cos(xn))

+1h2

((2 + cos(xn)) +

τ

2

(6 cos(xn) + 5 cos2(xn) − 2

))

− τ

h3(4 sin(xn) + 2 sin(xn) cos(xn)) +

2τh4

(2 + cos(xn))2,


rn = − 1τ− 2h2

((2 + cos(xn)) +

τ

2

(6 cos(xn) + 5cos2(xn) − 2

))

− 3τh4

(2 + cos(xn))2,

zn =12h

(sin(xn) + τ sin(xn) + τ sin(xn) cos(xn))

+1h2

((2 + cos(xn)) +

τ

2

(6 cos(xn) + 5cos2(xn) − 2

))

+τ

h3(4 sin(xn) + 2 sin(xn) cos(xn)) +

2τh4

(2 + cos(xn))2,

wn = − τ

2h4(2 + cos(xn))2 − τ

2h3(4 sin(xn) + 2 sin(xn) cos(xn)).

(3.17)

For the solution of the last matrix equation, we use the modified variant of Gausselimination method. We seek a solution of the matrix equation of the matrix equation in thefollowing form:

un = αn+1un+1 + βn+1un+2 + γn+1, n =M − 2, . . . , 0,

uM = �0, DM =(βM−2 + 5I

) − (4I − αM−2)αM−1,

uM−1 = D−1M

[(4I − αM−2)γM−1 − γM−2

],

(3.18)

where γ1 = γ2 = �0, α1 = β1 are (N + 1) × (N + 1) zero matrices, α2 = −4β2 = (4/5)I, and

βn+1 = −F−1n An,

αn+1 = −F−1n

(Bn +Dnβn + Enαn−1βn

),

γn+1 = −F−1n

(Iϕn −Dnγn − Enαn−1γn − Enγn−1

),

Fn =(Cn +Dnαn + Enβn−1 + Enαn−1αn

), n = 2, . . . ,M − 2.

(3.19)

Now, let us give the results of the numerical analysis. In order to get the solution,we used MATLAB programs. The numerical solutions are recorded for different values ofN = M and ukn represents the numerical solutions of these difference schemes at (tk, xn). Fortheir comparison, the errors are computed by

ENM = max−N≤k≤N,1≤n≤M−1

∣∣∣u(tk, xn) − ukn∣∣∣. (3.20)

Table 1 gives the error analysis between the exact solution and solutions derived by differenceschemes. Table 1 is constructed for N = M = 20, 40, and 60, respectively. Hence, the secondorder of accuracy difference scheme is more accurate compared with the first order of accu-racy difference scheme.



Difference schemes N =M = 20 N =M = 40 N =M = 60Difference scheme (3.3) 5.693 × 10−3 2.865 × 10−3 1.914 × 10−3

Difference scheme (3.13) 2.390 × 10−4 6.408 × 10−5 2.901 × 10−5

Table 1 is the error analysis between the exact solution and solutions derived by difference schemes.

Acknowledgment

The authors would like to thank Prof. Dr. Allaberen Ashyralyev (Fatih University, Turkey)on his very helpful comments and suggestions in improving the quality of this work.

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Research ArticleOn the Regularized Solutions of Optimal ControlProblem in a Hyperbolic System

Yesim Sarac and Murat Subası

Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey

Correspondence should be addressed to Yesim Sarac, [email protected]



Copyright q 2012 Y. Sarac and M. Subası. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We use the initial condition on the state variable of a hyperbolic problem as control function andformulate a control problem whose solution implies the minimization at the final time of thedistance measured in a suitable norm between the solution of the problem and given targets.We prove the existence and the uniqueness of the optimal solution and establish the optimalitycondition. An iterative algorithm is constructed to compute the required optimal control as limit ofa suitable subsequence of controls. An iterative procedure is implemented and used to numericallysolve some test problems.

1. Introduction and Statement of the Problem

Optimal control problems for hyperbolic equations have been investigated by Lions in hisfamous book [1]. Lions examined the problems in detail when the control function is at theright hand side and in the boundary condition of the hyperbolic problem. Furthermore, whenthe control is in the boundaries [2–4], in the coefficient [5, 6], and at the right hand side ofthe equation [7, 8], there have been some control problem studies for different types of costfunctionals. As for the control of initial conditions, Lions mentioned the control of the initialvelocity of the system in detail but stated briefly the control of initial status of the systemsolving the system in L2.

In this study, we consider the following problem of minimizing the cost functional:

Jα(ϕ)=∫ l

0

[ut(x, T ;ϕ

) − y1(x)]2dx +

∫ l

0

[ux(x, T ;ϕ

) − y2(x)]2dx + α

∫ l

0ϕ2xdx, (1.1)


under the following condition:

utt − a2uxx = F(x, t), (x, t) ∈ Ω := (0, l) × (0, T]

u(x, 0) = ϕ(x), ut(x, 0) = ψ(x), x ∈ (0, l)

u(0, t) = 0, u(l, t) = 0, t ∈ (0, T].

(1.2)

Since the problem is usually ill posed for α = 0, we use the parameter α > 0 as theregularization parameter which is the strong convexity constant, and this guaranties theuniqueness and stability of the regularized solution. The functional Jα(ϕ) is called costfunctional and the term ‖ϕ‖2

H10is called penalization term; its role is, on one hand, to avoid

using “too large” controls in the minimization of Jα(ϕ) and, on the other hand, to assurecoercivity for Jα(ϕ).

Lions in [1] mentioned the observation of u(x, T ;ϕ) in L2(0, l) and ut(x, T ;ϕ) inH−1(0, l) for the control ϕ ∈ L2(0, l). Except this study, there is no investigation in the literatureabout the control of initial status of the hyperbolic system up to now. In this study, weinvestigate different targets. With the choice of the functional in (1.1), we use ut(x, T ;ϕ)and ux(x, T ;ϕ), which correspond to final velocity and force, respectively, for the controlϕ ∈ H1

0(0, l). Since the Frechet differential of the cost functional cannot be obtained withthe usage of usual norm in H1

0 , we get the differentiability with the only use of H10 -Poincare

norm.The spaceH1

0(0, l) is a Hilbert subspace ofH1(0, l) and theH10 -Poincare inner product

and theH10 -Poincare norm are defined, respectively, as

(u, v)H10= (∇u,∇v)L2

, ‖u‖H10= ‖∇u‖L2

. (1.3)

Let

Φad = closed, convex subset of H10(0, l). (1.4)

We search for

Infϕ∈Φad

Jα(ϕ). (1.5)

We organize this paper as follows. In Section 2, we establish the existence and the uniquenessof the optimal solution. In Section 3, we derive the necessary optimality condition. InSection 4, we construct an algorithm for the numerical approximation of the optimal solutionaccording to steepest descent algorithm. In Section 5, we give symbolic representation foroptimal solution by using this algorithm on some examples.

2. Existence and Uniqueness of the Optimal Solution

First we state the generalized solution of the hyperbolic problem (1.2).


Definition 2.1. The generalized solution of (1.2) will be defined as the function u ∈ H10(Ω),

u(x, 0) = ϕ(x)which satisfies the following integral identity:

∫T

0

∫ l

0

(−utvt + a2uxvx

)dx dt =

∫T

0

∫ l

0Fv dx dt +

∫ l

0ψ(x)v(x, 0)dx, (2.1)

for all v ∈ H10(Ω)with v(x, T) = 0. To have this solution the following is needed:

F ∈ L2(Ω), ϕ ∈ H1(0, l), ψ ∈ L2(0, l). (2.2)

Theorem 2.2. Suppose that (2.2) holds, then (1.2) has a unique generalized solution and the follow-ing estimate is valid for the solution:

‖u‖2H1

0 (Ω) ≤ c1(∥∥ϕ∥∥2H1

0 (0,l)+∥∥ψ∥∥2L2(0,l)

+ ‖F‖2L2(Ω)

). (2.3)

Proof of this theorem can easily be obtained by Galerkin method used in [9].

The strategy to prove existence and uniqueness of this optimal control is to usethe relationship between minimization of quadratic functionals and variational problemscorresponding to symmetric bilinear forms. The key point is to write Jα(ϕ) in the followingway:

Jα(ϕ)= π(ϕ, ϕ) − 2Lϕ + b. (2.4)

Here

π(ϕ, ϕ)=∫ l

0

[ut(x, T ;ϕ

) − ut(x, T ; 0)]2dx

+∫ l

0

[ux(x, T ;ϕ

) − ux(x, T ; 0)]2dx + α

∫ l

0ϕ2xdx,

(2.5)

is bilinear (since the mapping ϕ → u[ϕ] − u[0] is linear) and symmetric.Also, the difference function δu = u(x, t;ϕ) − u(x, t; 0) is the solution of the following

problem:

δutt − a2δuxx = 0,

δu(x, 0) = ϕ, δut(x, 0) = 0,

δu(0, t) = 0, δu(l, t) = 0,

(2.6)

and for the solution of this problem the following estimates are valid:

‖δut(x, T)‖2L2(0,l) ≤ c2∥∥ϕ∥∥2H1

0 (0,l), ‖δux(x, T)‖2L2(0,l) ≤ c3

∥∥ϕ∥∥2H1

0 (0,l). (2.7)


Hence, we write the following:

∣∣π(ϕ, ϕ)∣∣ =∫ l

0(δut)

2dx +∫ l

0(δux)

2dx + α∫ l

0ϕ2xdx

= ‖δut‖2L2(0,l) + ‖δux‖2L2(0,l) + α∥∥ϕ∥∥2H1

0 (0,l)

≥ α∥∥ϕ∥∥2H10 (0,l)

,

(2.8)

and this implies the coercivity of π(ϕ, ϕ). Since

π(ϕ, η)=∫ l

0

[ut(x, T ;ϕ

) − ut(x, T ; 0)][ut(x, T ;η

) − ut(x, T ; 0)]dx

+∫ l

0

[ux(x, T ;ϕ

) − ux(x, T ; 0)][ux(x, T ;η

) − ux(x, T ; 0)]dx + α

∫ l

0ϕxηxdx

(2.9)

applying Cauchy-Schwartz inequality, we get

∣∣π(ϕ, η)∣∣ ≤ ∥∥ut

(x, T ;ϕ

) − ut(x, T ; 0)∥∥L2(0,l)

∥∥ut(x, T ;η

) − ut(x, T ; 0)∥∥L2(0,l)

+∥∥ux(x, T ;ϕ

) − ux(x, T ; 0)∥∥L2(0,l)

∥∥ux(x, T ;η

) − ux(x, T ; 0)∥∥L2(0,l)

+ α∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l),

∣∣π(ϕ, η)∣∣ ≤ ∥∥δut

(x, T ;ϕ

)∥∥L2(0,l)

∥∥δut(x, T ;η

)∥∥L2(0,l)

+∥∥δux

(x, T ;ϕ

)∥∥L2(0,l)

∥∥δux(x, T ;η

)∥∥L2(0,l)

+ α∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l)

(2.10)

for δu(x, T ;ϕ) = u(x, T ;ϕ) − u(x, T ; 0) and δu(x, T ;η) = u(x, T ;η) − u(x, T ; 0).So, we obtain

∣∣π(ϕ, η)∣∣ ≤ c4

∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l)+ c5∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l)+ α∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l)(2.11)

using (2.7) and write

∣∣π(ϕ, η)∣∣ ≤ c6

∥∥ϕ∥∥H1

0 (0,l)

∥∥η∥∥H1

0 (0,l), (2.12)

for c6 = max{c4, c5, α}. Then π(ϕ, η) is continuous.The functional Lϕ in (2.4) is defined as

Lϕ =∫ l

0

[ut(x, T ;ϕ

) − ut(x, T ; 0)][y1(x) − ut(x, T ; 0)

]dx

+∫ l

0

[ux(x, T ;ϕ

) − ux(x, T ; 0)][y2(x) − ux(x, T ; 0)

]dx.

(2.13)


We can easily write that

Lϕ ≤ ‖δut(x, T)‖L2(0,l)

∥∥y1(x) − ut(x, T ; 0)∥∥L2(0,l)

+ ‖δux(x, T)‖L2(0,l)

∥∥y2(x) − ux(x, T ; 0)∥∥L2(0,l)

, Lϕ

≤ c7∥∥ϕ∥∥H1

0 (0,l)

(2.14)

using (2.7). Hence we see that the functional Lϕ is continuous.The number b ∈ R in (2.4) is defined as

b =∫ l

0

[y1(x) − ut(x, T ; 0)

]2dx +

∫ l

0

[y2(x) − ux(x, T ; 0)

]2dx. (2.15)

Therefore we have established the conditions of the following existence and uniquenesstheorem for the problem.

Theorem 2.3. Let π(ϕ, ϕ) be a continuous symmetric bilinear coercive form and Lϕ a continuouslinear form onH1

0 . Then there exists a unique element ϕ∗ ∈ Φ ad such that

Jα(ϕ∗) = Inf

ϕ∈Φ ad

Jα(ϕ). (2.16)

Proof of this theorem can easily be obtained by showing the weak lower semicontinuity of Jα(ϕ) as in[1].

3. Lagrange Multipliers and Optimality Condition

To derive the optimality condition, let us introduce the Lagrangian L(u, ϕ, zt), given by

L(u, ϕ, zt

)=∫ l

0

[ut(x, T ;ϕ

) − y1(x)]2dx +

∫ l

0

[ux(x, T ;ϕ

) − y2(x)]2dx

+ α∫ l

0

(ϕx)2dx +

∫T

0

∫ l

0

[utt − a2uxx − F(x, t)

]ztdx dt.

(3.1)

Notice that L is linear in zt, therefore

L′zt

(u, ϕ, zt

)= 0 (3.2)

corresponds to the state equation (1.2). Moreover,

L′u

(u, ϕ, zt

)= 0 (3.3)


generates the following adjoint problem:

ztt − a2zxx = 0,

zt(x, T) = −2[ut(x, T ;ϕ

) − y1(x)],

zx(x, T) = − 2a2[ux(x, T ;ϕ

) − y2(x)],

z(0, t) = 0, z(l, t) = 0,

(3.4)

while

L′ϕ

(u, ϕ, zt

)= 0 (3.5)

constitutes the following Euler equation:

⟨J ′α(ϕ), δϕ⟩H1

0 (0,l)=∫ l

0

(−a2zx(x, 0) + 2αϕx

)(δϕ)xdx = 0, ∀δϕ ∈ Φad. (3.6)

So, we can state the following theorem in view of [10].

Theorem 3.1. The control ϕ∗ and the state u∗ = u(ϕ∗) are optimal if there exists a multiplier z∗t ∈Φ ad such that z∗ and ϕ∗ satisfy the following optimality conditions:

⟨−a2z∗(x, 0) + 2αϕ∗, ϕ − ϕ∗

⟩

H10 (0,l)

≥ 0, (3.7)

for ∀ϕ ∈ Φ ad.

4. An Iterative Algorithm and Its Convergence

Now, we can apply standard steepest descent iteration. Gradient of Jα at any ϕ is given by

∇Jα(ϕ)= −a2z(x, 0) + 2αϕ. (4.1)

It turns out that −∇Jα(ϕ) plays the role of the steepest descent direction for Jα. This suggestsan iterative procedure to compute a sequence of controls {ϕk} convergent to the optimal one.

Select an initial control ϕ0. If ϕk is known (k ≥ 0) then ϕk+1 is computed according tothe following scheme.

(1) Solve the state problem (1.2) in the sense (2.1) and get corresponding uk.

(2) Knowing uk solve the adjoint problem (3.4).

(3) Using zk get the gradient (∇Jα)k.(4) Set

ϕk+1 = ϕk − βk∇Jα(ϕk), (4.2)


and select the relaxation parameter βk in order to assure that

Jα(ϕk+1) − Jα

(ϕk)= βk

[−∥∥J ′α(ϕk)∥∥2 +

o(βk)

βk

]< 0, (4.3)

for sufficiently small βk > 0.

Concerning the choice of the relaxation parameter, there are several possibilities andthese can be found in any optimization books.

One of the following can be taken as a stopping criterion to the iteration process:

∥∥ϕk+1 − ϕk∥∥ < ε1,

∣∣Jα(ϕk+1) − Jα

(ϕk)∣∣ < ε2,

∥∥J ′α(ϕk)∥∥ < ε3. (4.4)

Lemma 4.1. The cost functional (1.1) is strongly convex with the strong convexity constant α.From the following strongly convex functional definition:

Jα(βϕ1 +

(1 − β)ϕ2

) ≤ βJα(ϕ1)+(1 − β)Jα

(ϕ2) − χβ(1 − β)∥∥ϕ1 − ϕ2

∥∥2H1

0 (0,l), (4.5)

we can see that the cost functional (1.1) is strongly convex with the constant χ = α.

So, we can give the following theorem which states the convergence of the minimizerto optimal solution.

Theorem 4.2. Let ϕ∗ be optimum solution of the problem (1.1)–(1.5). Then the minimizer given in(4.2) satisfies the following inequality:

∥∥ϕk − ϕ∗∥∥2 ≤ 2α

(Jα(ϕk) − Jα

(ϕ∗)), k = 0, 1, 2, . . . . (4.6)

Proof. If we take β = 1/2 in the definition of the strongly convex functional, we write

Jα

(12ϕk +

12ϕ∗)

≤ 12Jα(ϕk)+12Jα(ϕ∗) − α1

4∥∥ϕk − ϕ∗∥∥2

L2(0,l). (4.7)

Since

Jα(ϕ∗) ≤ Jα

(12ϕk +

12ϕ∗), (4.8)

we find

Jα(ϕ∗) ≤ 1

2Jα(ϕk)+12Jα(ϕ∗) − α1

4∥∥ϕk − ϕ∗∥∥2

L2(0,l),

∥∥ϕk − ϕ∗∥∥2 ≤ 2α

(Jα(ϕk) − Jα

(ϕ∗)).

(4.9)


5. Numerical Examples

Example 5.1. Let us consider the following problem of minimizing the cost functional:

Jα(ϕ)=∫1

0

⎡⎢⎣ut(x, 1;ϕ

) −

⎛⎜⎝− sin(1)

⎧⎪⎨

⎪⎩

14x2 0 ≤ x ≤ 1

2−x3 +

54x2 − 1

4x

12≤ x ≤ 1

⎞⎟⎠

⎤⎥⎦

2

dx

+∫1

0

⎡⎢⎣ux(x, 1;ϕ

) −

⎛⎜⎝cos(1)

⎧⎪⎨

⎪⎩

12x 0 ≤ x ≤ 1

2−3x2 +

52x − 1

412≤ x ≤ 1

⎞⎟⎠

⎤⎥⎦

2

dx

+ α∫1

0ϕ2xdx

(5.1)

under the following condition:

utt − uxx = cos(t)

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

−14(x2 + 2

)0 ≤ x < 1

2, t ∈ (0, 1]

14(4x3 − 5x2 + 25x − 10

) 12< x ≤ 1, t ∈ (0, 1],

u(x, 0) = ϕ(x), ut(x, 0) = 0, x ∈ (0, 1),

u(0, t) = 0, u(1, t) = 0, t ∈ (0, 1].

(5.2)

Rewrite the functional as

Jα(ϕ)= J1α(ϕ)+ αJ2α

(ϕ), (5.3)

where

J1α(ϕ)=∫1

0

⎡⎢⎣ut(x, 1;ϕ

) −

⎛⎜⎝− sin (1)

⎧⎪⎨

⎪⎩

14x2 0 ≤ x ≤ 1

2−x3 +

54x2 − 1

4x

12≤ x ≤ 1

⎞⎟⎠

⎤⎥⎦

2

dx

+∫1

0

⎡⎢⎣ux(x, 1;ϕ

) −

⎛⎜⎝cos (1)

⎧⎪⎨

⎪⎩

12x 0 ≤ x ≤ 1

2−3x2 +

52x − 1

412≤ x ≤ 1

⎞⎟⎠

⎤⎥⎦

2

dx,

J2α(ϕ)=∫1

0ϕ2xdx

(5.4)

Choosing α = 0.1 and starting the initial element ϕ0 = sinπx, then we get the minimizingsequence. Here the relaxation parameter βk = 0.01 assures the inequality J0.1(ϕk+1) < J0.1(ϕk).


In this example if we use the stopping criteria J0.1(ϕk+1) − J0.1(ϕk) > −0.166 × 10−9, we get thefollowing minimizing element after 250 iterations:

ϕ250 = 0.00134242 sin(15.70796327x) + 0.00859668 sin(9.424777962x)

− 0.00037644 sin(25.13274123x) − 0.00148524 sin(18.84955592x)

+ 0.06814824 sin(3.141592654x) − 0.03968284 sin(6.283185308x)

− 0.00032149 sin(31.41592654x) + 0.00024578 sin(28.27433389x)

− 0.00299971 sin(12.56637062x) + 0.00061268 sin(21.99114858x)

(5.5)

and for this optimal control the values of the J10.1(ϕ250) and J20.1(ϕ250) are such as

J10.1(ϕ250)= 0.00005173, J20.1

(ϕ250)= 0.05881965. (5.6)

For different α the values of J1α(ϕ), J2α(ϕ) and optimal controls ϕ∗ are given in Table 1.

Example 5.2. We consider the following problem of minimizing the cost functional:

Jα(ϕ)=∫3

0

⎡⎢⎢⎢⎢⎢⎢⎣ut(x, 2;ϕ

) −

⎛⎜⎜⎜⎜⎜⎜⎝

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

12(5x3 − 13x2 + 9x

)0 ≤ x ≤ 1

12x2 − 2x + 2 1 ≤ x ≤ 2

0 2 ≤ x ≤ 3

⎞⎟⎟⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎦

2

dx

+∫3

0

⎡⎢⎢⎣ux(x, 2;ϕ

) −

⎛⎜⎜⎝3

⎧⎪⎪⎨

⎪⎪⎩

12(15x2 − 26x + 9

)0 ≤ x ≤ 1

x − 2 1 ≤ x ≤ 20 2 ≤ x ≤ 3

⎞⎟⎟⎠

⎤⎥⎥⎦

2

dx + α∫3

0ϕ2xdx

(5.7)

subject to

utt − 4uxx = −4(t + 1)

⎧⎪⎪⎨

⎪⎪⎩

15x − 13 0 ≤ x < 1, t ∈ (0, 2]1 1 ≤ x < 2, t ∈ (0, 2]0 2 ≤ x < 3, t ∈ (0, 2]

u(x, 0) = ϕ(x), ut(x, 0) =

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

12(5x3 − 13x2 + 9x

)0 ≤ x ≤ 1

12x2 − 2x + 2 1 ≤ x ≤ 2

0 2 ≤ x ≤ 3

u(0, t) = 0, u(3, t) = 0, t ∈ (0, 2].

(5.8)


Table 1

α J1α(ϕ∗) J2α(ϕ

∗) ϕ∗

0.5 0.00066463 0.05016802

0.06066266 sin(3.14159265x) − 0.03734277 sin(6.28318530x)+0.00825020 sin(9.42477796x)− 0.00290788 sin(12.5663706x)+0.00130929 sin(15.7079632x)− 0.00145454 sin(18.8495559x)+0.00060179 sin(21.9911485x)− 0.00037057 sin(25.1327412x)+0.00024238 sin(28.2743338x)− 0.00031746 sin(31.4159265x)

0.3 0.00027601 0.05420087


0.1 0.00005173 0.05881966


0.03 0.00002314 0.06059838


0.01 0.00002053 0.06112368


0.001 0.00002020 0.06133626


We can rewrite the cost functional as

Jα(ϕ)= J1α(ϕ)+ αJ2α

(ϕ)

(5.9)

For

J1α(ϕ)=∫3

0

⎡⎢⎢⎢⎢⎢⎢⎣ut(x, 2;ϕ

) −

⎛⎜⎜⎜⎜⎜⎜⎝

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

12(5x3 − 13x2 + 9x

)0 ≤ x ≤ 1

12x2 − 2x + 2 1 ≤ x ≤ 2

0 2 ≤ x ≤ 3

⎞⎟⎟⎟⎟⎟⎟⎠

⎤⎥⎥⎥⎥⎥⎥⎦

2

dx

+∫3

0

⎡⎢⎢⎣ux(x, 2;ϕ

) −

⎛⎜⎜⎝3

⎧⎪⎪⎨

⎪⎪⎩

12(15x2 − 26x + 9

)0 ≤ x ≤ 1

x − 2 1 ≤ x ≤ 20 2 ≤ x ≤ 3

⎞⎟⎟⎠

⎤⎥⎥⎦

2

dx,

J2α(ϕ)=∫3

0ϕ2xdx.

(5.10)


Table 2

α J1α(ϕ∗) J2α(ϕ

∗) ϕ∗

0.9 5.78860382 8.69671794

0.02889657 sin(8.37758041x) + 0.04493685 sin(7.33038285x)+0.24991184 sin(1.04719755x) + 0.52713734 sin(2.09439510x)+0.45790546 sin(3.14159265x) + 0.26256973 sin(4.18879020x)+0.15223865 sin(5.23598775x) + 0.09116834 sin(6.28318530x)+0.02580495 sin(9.42477796x) + 0.01946190 sin(10.4719755x)

0.6 3.70770920 11.9063673


0.2 0.73684310 20.7888427


0.04 0.26519950 25.3541218


0.02 0.25964159 25.7401337


0.002 0.25769444 26.0957452


Taking α = 0.2 and the initial element ϕ0 = 0, we obtain a minimizing sequence. In thisexample β = 0.015 and stopping criteria 0.5 × 10−6 are chosen.

Optimal control function after 37 iterations is

ϕ37 = 0.02815886 sin(9.42477796x) + 0.71530518 sin(3.14159265x)

+ 0.60922355 sin(1.04719755x) + 0.99053011 sin(2.09439510x)

+ 0.36120182 sin(4.18879020x) + 0.19210117 sin(5.23598775x)

+ 0.02095115 sin(10.4719755x) + 0.10849356 sin(6.28318530x)

+ 0.05138069 sin(7.33038285x) + 0.03214310 sin(8.37758041x).

(5.11)

J10.2(ϕ37) and J20.2(ϕ37) are 0.7368431097 and 20.78884275, respectively, for this optimal control.For different α, the values of J1α(ϕ), J

2α(ϕ), and optimal controls ϕ∗ are given in Table 2.


6. Conclusions

In a hyperbolic problem, the initial condition u(x, 0) = ϕ(x) can be controlled from the targetsut(x, T ;ϕ) and ux(x, T ;ϕ) using H1

0 -Poincare norm. The Lagrange multiplier is zt while thefunction z(x, t) is the solution of adjoint problem. The symbolic optimal control function iseasily obtained in numerical examples.

References

[1] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Translated from theFrench by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170, Springer,New York, NY, USA, 1971.

[2] M. Negreanu and E. Zuazua, “Uniform boundary controllability of a discrete 1-D wave equation,”Systems & Control Letters, vol. 48, no. 3-4, pp. 261–279, 2003.

[3] L. I. Bloshanskaya and I. N. Smirnov, “Optimal boundary control by an elastic force at one end anda displacement at the other end for an arbitrary sufficiently large time interval in the string vibrationproblem,” Differential Equations, vol. 45, no. 6, pp. 878–888, 2009.

[4] A. Smyshlyaev and M. Krstic, “Boundary control of an anti-stable wave equation with anti-dampingon the uncontrolled boundary,” Systems & Control Letters, vol. 58, no. 8, pp. 617–623, 2009.

[5] X. Feng, S. Lenhart, V. Protopopescu, L. Rachele, and B. Sutton, “Identification problem for the waveequation with Neumann data input and Dirichlet data observations,” Nonlinear Analysis, vol. 52, no.7, pp. 1777–1795, 2003.

[6] A. Munch, P. Pedregal, and F. Periago, “Optimal design of the damping set for the stabilization of thewave equation,” Journal of Differential Equations, vol. 231, no. 1, pp. 331–358, 2006.

[7] J. D. Benamou, “Domain decomposition, optimal control of systems governed by partial differentialequations, and synthesis of feedback laws,” Journal of Optimization Theory and Applications, vol. 102,no. 1, pp. 15–36, 1999.

[8] F. Periago, “Optimal shape and position of the support for the internal exact control of a string,”Systems & Control Letters, vol. 58, no. 2, pp. 136–140, 2009.

[9] O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, vol. 49 of Applied MathematicalSciences, Springer, New York, NY, USA, 1985.

[10] E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization,Springer, New York, NY, USA, 1985.


Research ArticleStability of Difference Schemes forFractional Parabolic PDE withthe Dirichlet-Neumann Conditions

Zafer Cakir

Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey

Correspondence should be addressed to Zafer Cakir, [email protected]



Copyright q 2012 Zafer Cakir. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The stable difference schemes for the fractional parabolic equation with Dirichlet and Neumannboundary conditions are presented. Stability estimates and almost coercive stability estimateswith ln (1/(τ + |h|)) for the solution of these difference schemes are obtained. A procedureof modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.

1. Introduction

Theory and applications, methods of solutions of problems for fractional differential equa-tions have been studied extensively by many researchers [1–18]. In this study, initial-boundary-value problem for the fractional parabolic equation

∂u(t, x)∂t

+D1/2t u(t, x) −

m∑

p=1

(ap(x)uxp

)

xp+ σu(t, x) = f(t, x),

x = (x1, . . . , xm) ∈ Ω, 0 < t < T,

u(t, x)|S1= 0,

∂u(t, x)∂�n

∣∣∣∣S2

= 0, 0 ≤ t ≤ T, S1 ∪ S2 = S = ∂Ω,

u(0, x) = 0, x ∈ Ω,

(1.1)


with Dirichlet and Neumann conditions is considered. Here D1/2t = D1/2

0+ is the standardRiemann-Liouville’s derivative of order 1/2 and Ω is the open cube in the m-dimensionalEuclidean space

Rm :{x ∈ Ω : x = (x1, . . . , xm); 0 < xj < 1, 1 ≤ j ≤ m}, (1.2)

with boundary S,Ω = Ω ∪ S, ap(x) (x ∈ Ω) and f(t, x) (t ∈ (0, T), x ∈ Ω) are given smoothfunctions, ap(x) ≥ a > 0, σ > 0, and �n is the normal vector to Ω.

The first and second orders of accuracy stable difference schemes for the numericalsolution of problem (1.1) are presented. Stability estimates and almost coercive stabilityestimates with ln(1/(τ + |h|)) for the solution of these difference schemes are obtained. Themethod is illustrated by numerical examples.

2. The First and Second Orders of Accuracy Stable Difference Schemesand Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us definethe grid space

Ωh ={x = xp =

(h1p1, . . . , hmpm

), p =

(p1, . . . , pm

), 0 ≤ pj ≤Mj, hjMj = 1, j = 1, . . . , m

},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.(2.1)

We introduce the Hilbert space L2h = L2(Ωh) of the grid function ϕh(x) = {ϕ(h1p1, . . . , hmpm)}defined on Ω, equipped with the norm

∥∥∥ϕh∥∥∥L2(Ωh)

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hm

⎞

⎠1/2

. (2.2)

To the differential operator Ax generated by problem (1.1), we assign the difference operatorAxhby the formula

Axhu

h = −m∑

p=1

(ap(x)uhxp

)

xp,jp+ σuhx, (2.3)

acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ S1h

andDhuh(x) = 0 for all x ∈ S2h. HereDhuh(x) is the approximation of ∂u/∂�n. It is known that


Axh is a self-adjoint positive definite operator in L2(Ωh). With the help of Ax

h we arrive at theinitial-boundary-value problem

dvh(t, x)dt

+D1/2t vh(t, x) +Ax

hvh(t, x) = fh(t, x), 0 < t < T, x ∈ Ωh,

vh(0, x) = 0, x ∈ Ωh,

(2.4)

for a finite system of ordinary fractional differential equations.In the second step, applying the first order of approximation formula

D1/2τ uk =

1√π

k∑

r=1

Γ(k − r + 1/2)(k − r)!

(ur − ur−1τ1/2

), 1 ≤ k ≤N, (2.5)

for

D1/2t u(tk) =

1Γ(1/2)

∫ tk

0(tk − s)−1/2u′(s)ds, (2.6)

(see [19]) and using the first order of accuracy stable difference scheme for parabolicequations, one can present the first order of accuracy difference scheme with respect to t


+D1/2τ uhk +A

xhu

hk(x) = f

hk (x), x ∈ Ωh,

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N, Nτ = T,

uh0(x) = 0, x ∈ Ωh,

(2.7)

for the approximate solution of problem (2.4). Here

Γ(k − r + 1

2

)=∫∞

0tk−r+1/2e−tdt. (2.8)


Moreover, applying the second order of approximation formula

D1/2τ uk =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[−d√2

3

]u0 +

[d√2

3

]u1, k = 1,

[2d

√6

5

]u0 +

[d√6

5

]u1 +

[d√6

5

]u2, k = 2,

dk−1∑

m=2{[(k −m)b1(k −m) + b2(k −m)]um−2

+[(2m − 2k − 1)b1(k −m) − 2b2(k −m)]um−1+[(k −m + 1)b1(k −m) + b2(k −m)]um}+

d

6√2[−uk−2 − 4uk−1 + 5uk], 3 ≤ k ≤N,

(2.9)

for

D1/2t u

(tk − τ

2

)=

1Γ(1/2)

∫ tk−τ/2

0

(tk − τ

2− s)−1/2

u′(s)ds (2.10)

and the Crank-Nicholson difference scheme for parabolic equations, one can present the sec-ond order of accuracy difference scheme with respect to t and to x and


+D1/2τ uhk(x) +

12Axh

(uhk(x) + u

hk−1(x)

)= fhk (x), x ∈ Ωh,

fhk (x) = f(tk − τ

2, x), tk = kτ, 1 ≤ k ≤N, Nτ = T,

uh0(x) = 0, x ∈ Ωh,

(2.11)

for the approximate solution of problem (2.4). Here

d =2√πτ

, b1(r) =

√

r +12−√

r − 12, b2(r) = −1

3

((r +

12

)3/2

−(r − 1

2

)3/2).

(2.12)


difference scheme (2.7) and (2.11) satisfy the following stability estimate:

max1≤k≤N


≤ C1 max1≤k≤N

∥∥∥fhk∥∥∥L2h. (2.13)

Here C1 does not depend on τ, h, and fhk, 1 ≤ k ≤N.


Proof. For the solution of difference scheme (2.7), we have the following formulas:

uhk(x) =k∑

s=1

Rk−s+1Fhs (x)τ, 1 ≤ k ≤N, (2.14)

where

R =(I + τAx

h

)−1,

Fhk (x) = fhk (x) −D1/2

τ uhk(x),

D1/2τ uhk(x) =

1√π

k∑

m=1

Γ(k −m + 1/2)(k −m)!

τ−1/2[−D1/2

τ uhm(x) + fhm(x)

].

(2.15)

The proof of (2.13) for (2.7) is based on (2.14) and estimate

∥∥∥AxhR

k∥∥∥L2h−→L2h

≤ 1kτ

,∥∥∥Rk

∥∥∥Lh2−→L2h

≤ 1, 1 ≤ k ≤N, (2.16)

and the triangle inequality.In the same manner, we can obtain (2.13) for (2.11) using the inequality

∥∥∥AxhB

kC2∥∥∥L2h−→L2h

≤ 1kτ

,∥∥∥Bk

∥∥∥L2h−→L2h

≤ 1, 1 ≤ k ≤N. (2.17)


difference scheme (2.7) satisfy the following almost coercive stability estimate:

max1≤k≤N

∥∥∥∥∥uhk − uhk−1

τ

∥∥∥∥∥L2h

+ max1≤k≤N

m∑

p=1

∥∥∥∥(uhk

)

xpxp,jp

∥∥∥∥L2h

≤ C2 ln1

τ + |h| max1≤k≤N

∥∥∥fhk∥∥∥L2h. (2.18)

Here C2 is independent of τ, h, and fhk , 1 ≤ k ≤N.

Proof. The proof of (2.18) for (2.7) is based on (2.14) and estimate (2.16) and the triangleinequality.

Theorem 2.3. Let τ and |h| =√h21 + · · · + h2m be sufficiently small numbers. Then, the solutions of

difference scheme (2.11) satisfy the following almost coercive stability estimate:

max1≤k≤N

∥∥∥∥∥uhk− uh

k−1τ

∥∥∥∥∥L2h

+ max1≤k≤N

12

m∑

p=1

∥∥∥∥(uhk + u

hk−1)

xpxp,jp

∥∥∥∥L2h

≤ C3 ln1

τ + |h| max1≤k≤N

∥∥∥fhk∥∥∥L2h. (2.19)

Here C3 does not depend on τ, h, and fhk, 1 ≤ k ≤N.


Proof. The proof of (2.19) for (2.11) is based on (2.14) and estimate (2.17) and the triangleinequality.

Remark 2.4. The method of proofs of Theorems 2.1–2.3 enables us to obtain the estimate ofconvergence of difference schemes of the first and second orders of accuracy for approximatesolutions of the initial-boundary-value problem

∂u(∂t, x)∂t

−n∑

p=1

ap(x)uxpxp +n∑

p=1

bp(x)uxp +Dαt u(t, x) = f(t, x;u(t, x), ux1(t, x), . . . , uxn(t, x)),

x = (x1, . . . , xn) ∈ Ω, 0 < t < T,

u(0, x) = 0, x ∈ Ω,

u(t, x)|S1= 0,

∂u(t, x)∂�n

∣∣∣∣S2

= 0

(2.20)

for semilinear fractional parabolic partial differential equations.

Note that one has not been able to obtain a sharp estimate for the constants figuringin the stability estimates of Theorems 2.1, 2.2, and 2.3. Therefore, our interest in thepresent paper is studying the difference schemes (2.7) and (2.11) by numerical experiments.Applying these difference schemes, the numerical methods are proposed in the followingsection for solving the one-dimensional fractional parabolic partial differential equation. Themethod is illustrated by numerical experiments.

3. Numerical Applications

For numerical results we consider two examples.

Example 3.1. We consider the initial-boundary-value problem

∂u(t, x)∂t

+D1/2t u(t, x) − ∂

∂x

((1 + x)

∂u(t, x)∂x

)+ u(t, x) = f(t, x),

f(t, x) =

[3 + t +

16√t

5√π

+ (1 + x)2π2t

]t2sin2πx − (1 + x)2π2t3cos2πx

−2πt3 sinπx cosπx, 0 < t < 1, 0 < x < 1,

u(t, 0) = ux(t, 1) = 0, 0 ≤ t ≤ 1,

u(0, x) = 0, 0 ≤ x ≤ 1,

(3.1)

for the one-dimensional fractional parabolic partial differential equation. The exact solutionof problem (3.1) is

u(t, x) = t3sin2πx. (3.2)


First, applying difference scheme (2.7), we obtain

ukn − uk−1n

τ+

1√π

k∑

r=1

Γ(k − r + 1/2)(k − r)!

urn − ur−1n

τ1/2

− 1h

[(1 + xn+1)

ukn+1 − uknh

− (1 + xn)ukn − ukn−1

h

]+ ukn = ϕkn,

ϕkn = f(tk, xn), tk = kτ, 1 ≤ k ≤N, xn = nh, 1 ≤ n ≤M − 1,

uk0 = 0, ukM−1 = ukM, 0 ≤ k ≤N,

u0n = 0, 0 ≤ n ≤M.

(3.3)

We get the system of equations in the matrix form

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = 0, UM−1 = UM,(3.4)

where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 00 an 0 · · · 0 00 0 an · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · an 00 0 0 · · · 0 an

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.5)

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

b11 0 0 · · · 0 0b21 b22 0 · · · 0 0b31 b32 b33 · · · 0 0· · · · · · · · · · · · · · · · · ·bN1 bN2 bN3 · · · bNN 0bN+1,1 bN+1,2 bN+1,3 · · · bN+1,N bN+1,N+1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.6)

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 00 cn 0 · · · 0 00 0 cn · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · cn 00 0 0 · · · 0 cn

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.7)


D =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 00 1 0 · · · 0 00 0 1 · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · 1 00 0 0 · · · 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.8)

ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

ϕ2n...

ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, Uq =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U0q

U1q

U2q

...UN−1q

UNq

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, q = n ± 1, n, (3.9)

an = −1 + xn+1h2

, cn = −1 + xnh2

,

b11 = 1, b21 = − 1√τ− 1τ, b22 =

1√τ+1τ+2 + xn+1 + xn

h2+ 1,

b31 = −Γ(1 + 1/2)√πτ

, b32 =Γ(1 + 1/2) − Γ(1/2)√

πτ− 1τ, b33 =

1√τ+1τ+2 + xn+1 + xn

h2+ 1,

bij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−Γ(i − 2 + 1/2)√πτ(i − 2)!

, j = 1,

√πτ

[Γ(i − j + 1/2

)(i − j)! − Γ

(i − j − 1 + 1/2

)(i − j − 1

)!

], 2 ≤ j ≤ i − 2,

√πτ

[Γ(1 +

12

)− Γ(12

)]− 1τ, j = i − 1,

1√τ+1τ+2 + xn+1 + xn

h2+ 1, j = i,

0, i < j ≤N + 1,

(3.10)

for i = 4, 5, . . . ,N + 1 and

ϕkn =

[3 + kτ +

16√kτ

5√π

+ (1 + nh)2π2kτ

](kτ)2sin2(πnh)

− (1 + nh)2π2(kτ)3cos2(πnh) − 2π(kτ)3 sin(πnh) cos(πnh).

(3.11)


So, we have the second-order difference equation with respect to nmatrix coefficients.This type system was developed by Samarskii and Nikolaev [20]. To solve this differenceequation we have applied a procedure for difference equation with respect to k matrixcoefficients. Hence, we seek a solution of the matrix equation in the following form:

Uj = αj+1Uj+1 + βj+1, UM = (I − αM)−1βM, j =M − 1, . . . , 2, 1, (3.12)

where αj (j = 1, 2, . . . ,M) are (N + 1) × (N + 1) square matrices and βj (j = 1, 2, . . . ,M) are(N + 1) × 1 column matrices defined by

αj+1 = −(B + Cαj)−1

A,

βj+1 =(B + Cαj

)−1(Dϕj − Cβj

), j = 1, 2, . . . ,M − 1,

(3.13)

where j = 1, 2, . . . ,M − 1, α1 is the (N + 1) × (N + 1) zero matrix and β1 is the (N + 1) × 1 zeromatrix.

Second, applying difference scheme (2.11), we obtain the second order of accuracydifference scheme in t and in x

ukn − uk−1n

τ+D1/2

τ ukn −12

[(1 + xn)

ukn+1 − 2ukn + ukn−1

h2+ukn+1 − ukn−1

2h− ukn + (1 + xn)

·uk−1n+1 − 2uk−1n + uk−1n−1

h2+uk−1n+1 − uk−1n−1

2h− uk−1n

]= ϕkn,

ϕkn = f(tk − τ

2, xn

), tk = kτ, xn = nh, 1 ≤ k ≤N, 1 ≤ n ≤M − 1,

uk0 = 0, 3ukM − 4ukM−1 + ukM−2 = 0, 0 ≤ k ≤N,

u0n = 0, 0 ≤ n ≤M.

(3.14)

Here D1/2τ ukn is defined same as in (2.9). We get the system of equations in the matrix form

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = 0, 3UM − 4UM−1 +UM−2 = 0,(3.15)


where

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 0an an 0 · · · 0 00 an an · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · an 00 0 0 · · · an an

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.16)

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

b11 0 0 · · · 0 0b21 b22 0 · · · 0 0b31 b32 b33 · · · 0 0· · · · · · · · · · · · · · · · · ·bN1 bN2 bN3 · · · bNN 0bN+1,1 bN+1,2 bN+1,3 · · · bN+1,N bN+1,N+1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.17)

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 0cn cn 0 · · · 0 00 cn cn · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · cn 00 0 0 · · · cn cn

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.18)

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 00 1 0 · · · 0 00 0 1 · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · 1 00 0 0 · · · 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, (3.19)

ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

ϕ2n...

ϕN−1n

ϕNn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, Uq =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

U0q

U1q

U2q

...UN−1q

UNq

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, q = n ± 1, n, (3.20)


an = −12

(1 + xnh2

+12h

), cn = −1

2

(1 + xnh2

− 12h

), d =

2√πτ

,

b1(r) =

√

r +12−√

r − 12, b2(r) = −1

3

[(r +

12

)3/2

−(r − 1

2

)3/2],

b11 = 1, b21 = −d√2

3− 1τ+1 + xnh2

+12, b22 =

d√2

3+1τ+1 + xnh2

+12,

b31 =d2

√6

5, b32 =

d√6

5− 1τ+1 + xnh2

+12, b33 =

d√6

5+1τ+1 + xnh2

+12,

b41 = d[1b1(1) + b2(1)], b42 = d[−3b1(1) − 2b2(1)] − d

6√2,

b43 = d[2b1(1) + b2(1)] − 4d

6√2− 1τ+1 + xnh2

+12, b44 = 5

d

6√2+1τ+1 + xnh2

+12,

b51 = d[2b1(2) + b2(2)], b52 = d[−5b1(2) − 2b2(2) + 1b1(1) + b2(1)],

b53 = d[3b1(2) + b2(2) − 3b1(1) − 2b2(1)] − d

6√2,

b54 = d[2b1(1) + b2(1)] − 4d

6√2− 1τ+1 + xnh2

+12, b55 = 5

d

6√2+1τ+1 + xnh2

+12,

bij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

d[(i − 3)b1(i − 3) + b2(i − 3)], j = 1,

d[(5 − 2i)b1(i − 3) − 2b2(i − 3) + (i − 4)b1(i − 4) + b2(i − 4)], j = 2,

d[(i − j + 1

)b1(i − j) + b2

(i − j) + (2j − 2i + 1

)b1(i − j − 1

)

−2b2(i − j − 1

)+(i − j − 2

)b1(i − j − 2

)+ b2

(i − j − 2

)], 3 ≤ j ≤ i − 3,

d[3b1(2) + b2(2) − 3b1(1) − 2b2(1)] − d

6√2, j = i − 2,

d[2b1(1) + b2(1)] − 4d

6√2− 1τ+1 + xnh2

+12, j = i − 1,

5d

6√2+1τ+1 + xnh2

+12, j = i,

0, i < j ≤N + 1,

(3.21)

for i = 6, 7, . . . ,N + 1 and

ϕkn =

[3 + kτ +

16√kτ

5√π

+ (1 + nh)2π2kτ

](kτ)2sin2(πnh)

− (1 + nh)2π2(kτ)3cos2(πnh) − 2π(kτ)3 sin(πnh) cos(πnh).

(3.22)

For the solution of the matrix equation (3.15), we use the same algorithm as in the first orderof accuracy difference scheme, where uM = [3I − 4αM + αM−1αM]−1 ∗ [(4I − αM−1)βM − βM−1].


Example 3.2. We consider the initial-boundary-value problem

∂u(t, x)∂t

+D1/2t u(t, x) − ∂

∂x

((1 + x)

∂u(t, x)∂x

)+ u(t, x) = f(t, x),

f(t, x) =

[3 + t +

16√t

5√π

+ (1 + x)2π2t

]t2sin2πx − (1 + x)2π2t3cos2πx

−2πt3 sinπx cosπx, 0 < t < 1, 0 < x < 1,

ux(t, 0) = 0, u(t, 1) = 0, 0 ≤ t ≤ 1,

u(0, x) = 0, 0 ≤ x ≤ 1,

(3.23)

for the one-dimensional fractional parabolic partial differential equation. The exact solutionof problem (3.23) is

u(t, x) = t3sin2πx. (3.24)

First, applying difference scheme (2.7), we obtain

ukn − uk−1n

τ+

1√π

k∑

r=1

Γ(k − r + 1/2)(k − r)!

urn − ur−1n

τ1/2

− 1h

[(1 + xn+1)

ukn+1 − uknh

− (1 + xn)ukn − ukn−1

h

]+ ukn = ϕkn,

ϕkn = f(tk, xn), tk = kτ, 1 ≤ k ≤N, xn = nh, 1 ≤ n ≤M − 1,

uk0 = uk1 , ukM = 0, 0 ≤ k ≤N,

u0n = 0, 0 ≤ n ≤M.

(3.25)

We get the system of equations in the matrix form

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

U0 = U1, UM = 0,(3.26)

where matrices A, B, C, D, ϕn, Uq (q = n ∓ 1, n) are defined same as in (3.5), (3.6), (3.7),(3.8), (3.9), respectively.

So, we have the second-order difference equation with respect to nmatrix coefficients.To solve this difference equation we have applied a procedure for difference equation with


respect to k matrix coefficients. Hence, we seek a solution of the matrix equation in the fol-lowing form:

Uj = αj+1Uj+1 + βj+1, UM = 0, j =M − 1, . . . , 2, 1, (3.27)

where αj (j = 1, 2, . . . ,M) are (N + 1) × (N + 1) square matrices and βj (j = 1, 2, . . . ,M) are(N + 1) × 1 column matrices defined by

αj+1 = −(B + Cαj)−1

A,

βj+1 =(B + Cαj

)−1(Dϕj − Cβj

), j = 1, 2, . . . ,M − 1,

(3.28)

where j = 1, 2, . . . ,M − 1, α1 is the (N + 1) × (N + 1) identity matrix and β1 is the (N + 1) × 1zero matrix.

Second, applying the formulas

ux(tk, 0) =uk1 − uk0

h− h

2uxx(tk, 0) + o

(h2), 0 ≤ k ≤N,

ux(tk,M) =3ukM − 4ukM−1 + u

kM−2

2h+ o(h2), 0 ≤ k ≤N,

ut(tk, 0) =uk+10 − uk−10

2τ+ o(τ2), 1 ≤ k ≤N − 1,

ut(tN, 0) =3uN0 − 4uN−1

0 + uN−20

2τ+ o(τ2), k =N,

(3.29)

and applying difference scheme (2.11), we obtain the second order of accuracy differencescheme in t and in x

ukn − uk−1n

τ+D1/2

τ ukn −12

[(1 + xn)


h2+ukn+1 − ukn−1

2h− ukn

+(1 + xn)uk−1n+1 − 2uk−1n + uk−1n−1

h2+uk−1n+1 − uk−1n−1

2h− uk−1n

]= ϕkn,

ϕkn = f(tk − τ

2, xn

), tk = kτ, xn = nh, 1 ≤ k ≤N, 1 ≤ n ≤M − 1,

u00 = 0, k = 0,

− h

4τuk−10 +

[1h+h

2D1/2t +

h

2

]uk0 +

h

4τuk+10 =

1huk1 +

h

2ϕk0 , 1 ≤ k ≤N − 1,

h

4τuN−20 − h

τuN−10 +

[1h+3h4τ

+h

2D1/2t +

h

2

]uN0 =

1huN1 +

h

2ϕN0 , k =N,

ukM = 0, 0 ≤ k ≤N,

u0n = 0, 0 ≤ n ≤M.

(3.30)


Here D1/2τ ukn is defined similar to (2.9). We get the system of equations in the matrix form

AUn+1 + BUn + CUn−1 = Dϕn, 1 ≤ n ≤M − 1,

EU0 = FU1 + Rϕ0, UM = 0,(3.31)

where matricesA, B, C, D, ϕn, Uq (q = n∓1, n) are defined same as in (3.16), (3.17), (3.18),(3.19), (3.20), respectively.

For the solution of the matrix equation (3.31), we use the same algorithm as in the firstorder of accuracy difference scheme, where

uM = 0, α1 = E−1F, β1 = E−1Rϕ0,

F =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 0

01h

0 · · · 0 0

0 01h

· · · 0 0

· · · · · · · · · · · · · · · · · ·0 0 0 · · · 1

h0

0 0 0 · · · 01h

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

, R =

⎡⎢⎢⎣

0 0 · · · 00 1 · · · 0· · · · · · · · · · · ·0 0 · · · 1

⎤⎥⎥⎦

(N+1)×(N+1)

,

E =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

e11 0 0 · · · 0 0e21 e22 0 · · · 0 0e31 e32 e33 · · · 0 0· · · · · · · · · · · · · · · · · ·eN1 eN2 eN3 · · · eNN 0eN+1,1 eN+1,2 eN+1,3 · · · eN+1,N eN+1,N+1

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

e11 = 1, e21 = − h

4τ− 4h3√πτ

, e22 =1h+h

2+

4h3√πτ

, e23 =h

4τ,

e31 =2√2h

15√πτ

, e32 =−16√2h15√πτ

− h

4τ, e33 =

1h+h

2+

14√2h

15√πτ

, e34 =h

4τ,

e41 =dh

2

[(1 +

12

)b1(1) + b2(1)

],

e42 =dh

2

[−4b1(1) − 2b2(1) +

12b1

(0) + b2(0)],

e43 =dh

2

[(2 +

12

)b1(1) + b2(1) − 2 − 2

(−13

)]− h

4τ,

e44 =1h+h

2+dh

2

[(1 +

12

)b1(0) + b2(0)

], e45 =

h

4τ,


e51 =dh

2

[(2 +

12

)b1(2) + b2(2)

],

e52 =dh

2

[−23b1(2) − 2b2(2) +

(1 +

12

)b1(1) + b2(1)

],

e53 =dh

2

[(2 + 1 +

12

)b1(2) + b2(2) − 22b1(1) − 2b2(1) +

12b1

(0) + b2(0)],

e54 =dh

2

[(1 + 1 +

12

)b1(1) + b2(1) − 2b1(0) − 2b2(0)

]− h

4τ,

e55 =1h+h

2+dh

2

[(1 +

12

)b1(0) + b2(0)

], e56 =

h

4τ,

eij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

dh

2

[(i − 3 +

12

)b1(i − 3) + b2(i − 3)

], j = 1,

dh

2

[−2(i − 2) b1(i − 3) − 2b2(i − 3) +

(i − 4 +

12

)b1(i − 4) + b2(i − 4)

], j = 2,

dh

2

[(i − j + 1 +

12

)b1(i − j) + b2

(i − j) − 2

(i − j) b1

(i − j − 1

)

−2b2(i − j − 1

)+(i − j − 2 +

12

)b1(i − j − 2

)+ b2

(i − j − 2

)], 3 ≤ j ≤ i − 2,

− h

4τ+dh

2

[(2 +

12

)b1(1) + b2(1) − 2 b1(0) − 2 b2(0)

], j = i − 1,

1h+h

2+dh

2

[(1 +

12

)b1(0) + b2(0)

], j = i,

h

4τ, j = i + 1,

h

4τ+dh

2

[(i −N + 2 +

12

)b1(i −N + 1) + b2(i −N + 1)

−2(i −N + 1) b1(i −N) − 2 b2(i −N)

+(i −N − 1 +

12

)b1(i −N − 1) + b2(i −N − 1)

], j =N − 1,

−hτ+dh

2

[(2 +

12

)b1(1) + b2(1) − 2 b1(0) − 2 b2(0)

], j =N,

1h+h

2+3h4τ

+dh

2

[(1 +

12

)b1(0) + b2(0)

], j =N + 1,

0, j > i + 1,

(3.32)

for i = 6, 7, . . . ,N + 1 and

ϕk0 = −2π2(kτ)3. (3.33)


Table 1: Error analysis of first and second order of accuracy difference schemes for Example 3.1.

Method N =M = 25 N =M = 50 N =M = 100

1st order of accuracy 0.2553 0.1256 0.06222nd order of accuracy 0.0062 9.771 × 10−4 1.963 × 10−4

Table 2: Error analysis of first and second order of accuracy difference schemes for Example 3.2.

Method N =M = 25 N =M = 50 N =M = 100

1st order of accuracy 0.1653 0.0807 0.03992nd order of accuracy 0.0025 5.943 × 10−4 1.453 × 10−4

3.1. Error Analysis

Finally, we give the results of the numerical analysis. The error is computed by the followingformula:

ENM = max1≤k≤N,1≤n≤M−1

∣∣∣u(tk, xn) − ukn∣∣∣, (3.34)

where u(tk, xn) represents the exact solution and ukn represents the numerical solutions ofthese difference schemes at (tk, xn). The numerical solutions are recorded for different valuesofN andM. Tables 1 and 2 are constructed forN =M = 25, 50, and 100, respectively.

The results in Tables 1 and 2 show that, by using the Crank-Nicholson differencescheme, more accurate approximate results can be obtained rather than the first order ofaccuracy difference scheme.

Acknowledgment

The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for hiscomments and suggestions to improve the quality of the paper.

References

[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, San Diego, Calif, USA, 1999.

[2] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and BreachScience Publishers, Yverdon, Switzerland, 1993.

[3] A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,North-Holland Mathematics Studies, 2006.


[5] V. E. Tarasov, “Fractional derivative as fractional power of derivative,” International Journal ofMathematics, vol. 18, no. 3, pp. 281–299, 2007.

[6] R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractionalorder,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 223–276, Springer,Vienna, Austria, 1997.

[7] D. Matignon, “Stability results for fractional differential equations with applications to controlprocessing,” in Computational Engineeringin System Application 2, Lille, France, 1996.


[8] A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42,pp. 369–381, 1910.

[9] F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” inFractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378,pp. 291–348, Springer, Vienna, Austria, 1997.

[10] A. Ashyralyev, F. Dal, and Z. Pinar, “On the numerical solution of fractional hyperbolic partialdifferential equations,” Mathematical Problems in Engineering, vol. 2009, Article ID 730465, 11 pages,2009.


[12] A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and differenceequations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011.

[13] F. Dal, “Application of variational iteration method to fractional hyperbolic partial differentialequations,”Mathematical Problems in Engineering, vol. 2009, Article ID 824385, 10 pages, 2009.

[14] E. Cuesta, C. Lubich, and C. Palencia, “Convolution quadrature time discretization of fractionaldiffusion-wave equations,” Mathematics of Computation, vol. 75, no. 254, pp. 673–696, 2006.

[15] P. E. Sobolevskii, “Some properties of the solutions of differential equations in fractional spaces,”Trudy Naucno-Issledovatel’skogi Instituta Matematiki VGU, vol. 14, pp. 68–74, 1975.

[16] G. Da Prato and P. Grisvard, “Sommes d’operateurs lineaires et equations differentiellesoperationnelles,” Journal de Mathematiques Pures et Appliquees, vol. 54, no. 3, pp. 305–387, 1975.

[17] A. Ashyralyev and Z. Cakir, “On the numerical solution of fractional parabolic partial differentialequations with the Dirichlet condition,” in Proceedings of the 2nd International Symposium on Computingin Science and Engineering (ISCSE ’11), M. Gunes, Ed., pp. 529–530, 2011.





Research ArticleAsymptotic Solutions of SingularPerturbed Problems with an Instable Spectrum ofthe Limiting Operator

Burkhan T. Kalimbetov, Marat A. Temirbekov,and Zhanibek O. Khabibullayev

Department of Mathematics, A. Yasawi International Kazakh-Turkish University,Turkestan 161200, Kazakhstan

Correspondence should be addressed to Marat A. Temirbekov, [email protected]



Copyright q 2012 Burkhan T. Kalimbetov et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The regularizationmethod is applied for the construction of algorithm for an asymptotical solutionfor linear singular perturbed systems with the irreversible limit operator. The main idea of thismethod is based on the analysis of dual singular points of investigated equations and passage inthe space of the larger dimension, what reduces to study of systems of first-order partial differentialequations with incomplete initial data.

1. Introduction

The investigation of singular perturbed systems for ordinary and partial differentialequations occurring in systems with slow and fast variables, chemical kinetics, the mathe-matical theory of boundary layer, control with application of geoinformational technologies,quantum mechanics, and plasma physics (the Samarsky-Ionkin problem) has been studiedby many researchers (see, e.g., [1–19]).

In this work, the algorithm for construction of an asymptotical solution for linearsingular perturbed systems with the irreversible limit operator is given—the regularizationmethod [1]. The main idea of this method is based on the analysis of dual singular points ofinvestigated equations and passage in the space of the larger dimension, what reduces to thestudy of systems of first-order partial differential equations with incomplete (more exactly,point) initial data.


In this paper, we consider linear singular perturbed systems in the form

εy = A(t)y + h(t), y(0, ε) = y0, t ∈ [0, T], (1.1)

where y = {y1, . . . , yn}, A(t) is a matrix of order (n × n), h(t) = {h1, . . . , hn} is a knownfunction, y0 ∈ Cn is a constant vector, and ε > 0 is a small parameter, in the case of violation ofstability of a spectrum {λj(t)} of the limiting operator A(t).

Difference of such type problems from similar problems with a stable spectrum (i.e.,in the case of λi(t)/= 0, λi(t)/=λj(t), i /= j, i, j = 1, n for all t ∈ [0, T]) is that the limiting system0 = A(t)y + h(t) at violation of stability of the spectrum can have either no solutions oruncountable set of them. In the last case, presence of discontinuous on the segment [0, T]solutions y(t) of the limiting system is not excluded. Under conditions, one can prove (see,e.g., [1, 6]) that the exact solution y(t, ε) of problem (1.1) tends (at ε → +0) to a smoothsolution of the limiting system. However, there is a problematic problem about constructionof an asymptotic solution of problem (1.1). When the spectrum is instable, essentially specialsingularities are arising in the solution of system (1.1). These singularities are not selectedby the spectrum {λj(t)} of the limiting operator A(t). As it was shown in [3–7], they wereinduced by instability points tj of the spectrum.

In the present work, the algorithm of regularization method [1] is generalized onsingular perturbed systems of the form (1.1), the limiting operator of which has someinstable points of the spectrum. In order to construct the spectrum, we use the new algorithmrequiring more constructive theory of solvability of iterative problems. These problems arosein application of the algorithm.

We will consider the problem (1.1) at the following conditions. Assume that

(i) A(t) ∈ C∞([0, T], Cn), h(t) ∈ C∞[0, T]; for any t ∈ [0, T], the spectrum {λj(t)} ofthe operator A(t) satisfies the conditions:

(ii) λi(t) = −(t− ti)siki(t), ki(t)/= 0, ti ∈ [0, T], i = 1, m, m < n (here si- are even naturalnumbers),

(iii) λi(t)/= 0, j = m + 1, n,

(iv) λi(t)/=λj(t), i /= j, i, j = 1, n,

(v) Reλj(t) � 0, j = 1, n.

2. Regularization of the Problem

We introduce basic regularized variables by the spectrum of the limiting operator

τj = ε−1∫ t

0λj(s)ds ≡

ϕj(t)ε

, j = 1, n. (2.1)

Instable points ti ∈ [0, T] of the spectrum {λj(t)} induce additional regularized variablesdescribed by the formulas

σiqi = eϕi(t)/ε

∫ t

0e−ϕi(s)/ε

(s − ti)qiqi

ds ≡ ψiqi(t, ε), i = 1, m, qi = 0, si − 1. (2.2)


We consider a vector function y(t, τ, σ, ε) instead of the solution y(t, ε) to be found forproblem (1.1). This vector function is such that

y(t, τ, σ, ε)|τ=ϕ, σ=ψ ≡ y(t, ε). (2.3)

For y(t, τ, σ, ε), it is natural to set the following problem:

Lεy(t, τ, σ, ε) ≡ ε∂y

∂t+

n∑

j=1

λj(t)∂y

∂τj+

m∑

i=1

si−1∑

qi=0

[λi(t)σiqi + ε

(t − ti)qiqi!

]∂y

∂σiqi

−A(t)y = h(t), y(0, 0, 0, ε) = y0.

(2.4)

We determine the solution of problem (2.4) in the form of a series

y(t, τ, σ, ε) =∞∑

k=−1εkyk(t, τ, σ), (2.5)

with coefficients yk(t, τ, σ) ∈ C∞[0, T].If we substitute (2.5) in (2.4) and equate coefficients at identical degrees of ε, we obtain

the systems for coefficients yk(t, τ, σ):

Ly−1(t, τ, σ) ≡n∑

j=1

λj(t)∂y−1∂τj

+m∑

i=1

si−1∑

qi=0

λi(t)σiqi∂y−1∂σiqi

−A(t)y−1 = 0, y−1(0, 0, 0) = 0, (ε−1)

Ly0(t, τ, σ) = −∂y−1∂t

−m∑i=1

si−1∑qi=0

(t − ti)qiqi!

∂y−1∂σiqi

+ h(t), y0(0, 0, 0) = y0,

...

(ε0)

Lyk+1(t, τ, σ) = −∂yk∂t

−m∑i=1

si−1∑qi=0

(t − ti)qiqi!

∂yk∂σiqi

, k � 1, yk+1(0, 0, 0) = 0,

...

(εk+1)

3. Resolvability of Iterative Problems

We solve each of the iterative problems (εk) in the following space of functions:

U =

⎧⎨

⎩y(t, τ, σ) : y =n∑

k=1

n∑

j=1

ykj(t)ck(t)eτj +n∑

k=1

m∑

i=1

si−1∑

qi=0

ykiqi(t)ck(t)σiqi

+n∑

k=1

yk(t)ck(t), ykj(t), ykiqi(t), yk(t) ∈ C∞([0, T], C1

)},

(3.1)


where ck(t) are eigenvectors of the operator A(t) corresponding eigenvalues λk(t), k = 1, n.We representU in the form ofU(1) ⊕U(0) where

U(0) =

⎧⎨

⎩y(0)(t) : y(0) =n∑

j=1

y(0)j (t)cj(t), y

(0)j (t) ∈ C∞

([0, T], C1

)⎫⎬

⎭,

U(1) =U

U(0).

(3.2)

It is easy to note that each of the systems (εk+1) can be written in the form

Ly(t, τ, σ) = h(t, τ, σ), (3.3)

where h(t, τ, σ) are the corresponding right hand side. Using representations of space U, wecan write system (3.3) in the equivalent form

Ly(1)(t, τ, σ) = h(1)(t, τ, σ), (3.4)

−A(t)y(0)(t) = h(0)(t), (3.5)

where y(1)(t, τ, σ), h(1)(t, τ, σ) ∈ U(1), y(0)(t), h(0)(t) ∈ U(0).We have the following result.

Theorem 3.1. Let h(1)(t, τ, σ) ∈ U(1) and satisfy conditions (i)–(iv). Then, system (3.4) is solvablein theU(1) if and only if

⟨h(1)(t, τ, σ), νj(t, τ, σ)

⟩≡ 0 ∀t ∈ [0, T], j = 1, n,

⟨h(1)(t, τ, σ), νiqi(t, τ, σ)

⟩≡ 0, i = 1, m, qi = 0, si − 1,

(3.6)

where νj(t, τ, σ), νiqi(t, τ, σ) are basic elements of the kernel of the operator

L∗ ≡n∑

j=1

λj(t)∂

∂τj+

m∑

i=1

si−1∑

qi=0

λi(t)σiqi∂

∂σiqi−A∗(t). (3.7)

Proof. Let h(1)(t, τ, σ) =∑n

k=1∑n

j=1 hkj(t)cj(t)eτk +

∑nk=1∑m

i=1∑si−1

qi=0hkiqi(t)ck(t)σiqi .

Determine solutions of system (3.4) in the form

y(1)(t, τ, σ) =n∑

k=1

n∑

j=1

ykj(t)ck(t)eτj +n∑

k=1

m∑

i=1

si−1∑

qi=0

ykiqi(t)ck(t)σiqi . (3.8)


Substituting (3.8) in (3.4) and equating separately coefficients at eτj and σiqi , we obtain theequations

[λk(t) − λj(t)

]ykj(t) = hkj(t), k, j = 1, n,

[λi(t) − λk(t)]yiqik(t) = hkiqi(t), i = 1, m, qi = 0, si − 1, k = 1, n.(3.9)

One can see from this that obtained equations are solvable if and only if

hkk(t) ≡ 0, k = 1, n, hiiqi(t) ≡ 0, i = 1, m, qi = 0, si − 1, (3.10)

and these conditions coincide with conditions (3.6). Theorem 3.1 is proved.

Remark 3.2. Equations (1.1) imply that under conditions (3.6), system (3.4) has a solution inU(1) representable in the form

y(1)(t, τ, σ) =n∑

k=1

n∑

j=1,j /= k

hkj(t)[λk(t) − λj(t)

]cj(t)eτj +n∑

k=1

αk(t)ck(t)eτk

+m∑

i=1

si−1∑

qi=0

γiqi(t)ci(t)σiqi +n∑

k=1

m∑

i=1,i /= k

si−1∑

qi=0

hkiqi(t)[λi(t) − λk(t)]ck(t)σiqi ,

(3.11)

where αk(t), γiqi(t) ∈ C∞([0, T], C1) are arbitrary functions.Consider now system (3.5). As det A(t) ≡ 0 in points t = ti, i = 1, m, this system

does not always have a solution in U(0). Introduce the space V (0) ⊂ U(0) consisting of vectorfunctions

z(0)(t) =n∑

j=1

zj(t)cj(t), zj(t) ∈ C∞([0, T], C1

), j = 1, n, (3.12)

having the properties

[Dli(z(0)(t), di(t)

)]

t=ti=(Dlizi

)(ti) = 0, ∀li = 0, si − 1, i = 1, m, (3.13)

where di(t) are eigenvectors of the operator A∗(t) with regard to eigenvalues λi(t), i = 1, m.Let h(0)(t) =

∑nj=1 hj(t)cj(t) ∈ V (0), that is,

(Dlihi

)(ti) = 0 ∀li = 0, si − 1, i = 1, m. (3.14)

Determine a solution of system (3.5) in the

y(0)(t) =n∑

j=1

yj(t)cj(t). (3.15)


Substituting this function in (3.5), we obtain

−n∑

j=1

yj(t)λj(t)cj(t) =n∑

j=1

hj(t)cj(t). (3.16)

Since {cj(t)} is a basis in Cn, we get

−λi(t)yi(t) = hi(t), i = 1, m, (3.17)

−λj(t)yj(t) = hj(t), j = m + 1, n. (3.18)

It is easy to see that (3.18) has the unique solution

yj(t) =−hj(t)λj(t)

, j = m + 1, n. (3.19)

By virtue of conditions (3.14), the function hi(t) can be represented in the form

hi(t) = (t − ti)si hi(t), i = 1, m, (3.20)

where hi(t) ∈ C∞([0, T], C1 is the certain scalar function, −(t − ti)siki(t)yi(t) = (t − ti)si hi(t),and we see that

yi(t) =

⎧⎪⎨

⎪⎩

−hi(t)ki(t)

, t /= ti,

γi, t = ti,(3.21)

where γi are arbitrary constants, i = 1, m. However, the solution of system (3.5) should belongto the space U(0), and it means that yi(t) ∈ C∞([0, T], C1). Therefore, constants in (3.21) γi =(hi(t)/ki(t))|t=ti and functions are determined uniquely in the form

yi(t) =−hi(t)ki(t)

, ∀t ∈ [0, T], i = 1, m. (3.22)

Thus, under conditions (3.14), system (3.5) has the solution y(0)(t) inU(0) of

y(0)(t) = −m∑

i=1

hi(t)ki(t)

ci(t) −n∑

j=m+1

hi(t)λi(t)

ci(t), (3.23)

where hi(t) = hi(t)/(t − ti)si (in points t = ti, i = 1, m, this equality is understood in the

limiting sense). We summarize received outcome in the form of the following assertion.

Theorem 3.3. Let the operator A(t) satisfy condition (i), and let its spectrum satisfy conditions (ii)–(iv). Then, for any vector function h(0)(t) ∈ V (0), system (3.5) has the unique solution y(0)(t) in spaceU(0).


For uniquely determination of functions αj(t), γiqi(t), consider system (3.4) withadditional conditions:

y(1)(0, 0, 0) = y∗, (3.24)⟨−∂y

(1)

∂t, νj(t, τ, σ)

⟩≡ 0 ∀t ∈ [0, T], j = 1, n, (3.25)

⟨−∂y

(1)

∂t, νiqi(t, τ, σ)

⟩≡ 0, i = 1, m, qi = 0, si − 1, (3.26)

where y∗ ∈ Cn is a constant vector.We have the following result.

Theorem 3.4. Let conditions of Theorem 3.1 hold. Then, the system (3.4) with additional conditions(3.24)-(3.25) has solutions of the form (3.11) in which all summands are uniquely determinate exceptfor γiqi(t)ci(t)σiqi(i = 1, m, qi = 0, si − 1). Functions γiqi(t) in the last summand are determined bythe formula

γiqi(t) = γ0iqi

· ePiqi (t) + fiqi(t), (3.27)

where Piqi(t), fiqi(t) are known functions, and γ0iqi arbitrary constants.

Proof. Denote in (3.11) that

gkj(t) =hkj(t)

λj(t) − λk(t) , gkiqi(t) =hkiqi(t)

λi(t) − λk(t) . (3.28)

Using (3.11) and condition (3.24), we obtain the equality

n∑

k=1

n∑

j=1

gkj(0)cj(0) +n∑

k=1

αk(0)ck(0) = y∗. (3.29)

Multiplying this equality scalarly by ds(0), we get

αs(0) =(y∗, ds(0)

) −n∑

k=1, k /= s

gks(0) ≡ α0s, s = 1, n. (3.30)

By (3.11) and conditions (3.25), we have

−αs(t) − (cs(t), ds(t))αs(t) −n∑

j=1, j /= s

gsj(t)(cj(t), ds(t)

)= 0, s = 1, n. (3.31)

Considering these equations with initial conditions (3.30), we can uniquely obtain functionsαs(t), s = 1, n.


Now, using (3.11) and conditions (3.26), we get

−γiqi(t) − (ci(t), di(t))γiqi(t) −n∑

k=1, k /= i

gkiqi(t)(ck(t), di(t)) = 0, i = 1, m, qi = 0, si − 1. (3.32)

This implies that γiqi(t) have the form (3.27) where

Piqi(t) = −∫ t

ti

(ci(s), di(s))ds,

fiqi(t) = ePiqi (t)

∫ t

ti

e−Piqi (s)n∑

k=1, k /= i

gkiqi(s)(ck(s), di(s))ds.

(3.33)

Theorem 3.4 is proved.

Remark 3.5. If conditions (3.6) hold for h(1)(t, τ, σ) ∈ U(1) and h(0)(t) ∈ U(0), then system (3.3)has a solution in the spaceU, representable in the form of

y(t, τ, σ) = y(1)(t, τ, σ) + y(0)(t), (3.34)

where y(1)(t, τ, σ) is a function in the form of (3.11), and y(0)(t) is a function in the form of(3.23); moreover, functions αk(t) ∈ C∞([0, T], C1) are found uniquely in (3.11), and functionsγiqi(t) are determined up to arbitrary constants γ0iqi in the form of (3.27).

Let us give the following result.

Theorem 3.6. Let h(0)(t) ∈ U(0), h(1)(t, τ, σ) ∈ U(1), and conditions (i)–(iv), (3.6), (3.24)–(3.26)hold. Then, there exist unique numbers γ0iqi involved in (3.27), such that the function (3.34) satisfiesthe condition

Py ≡ −∂y(0)

∂t−

m∑

i=1

si−1∑

qi=0

(t − ti)qiqi!

∂y(1)

∂σiqi+H(0)(t) ∈ V (0), (3.35)

whereH(0)(t) ∈ V (0) is a fixed vector function.

Proof. To determine functions uniquely, calculate

Py ≡ −m∑

i=1

[hi(t)ki(t)

ci(t)]′−

n∑

j=m+1

[hj(t)λj(t)

cj(t)

]′−

m∑

i=1

si−1∑

qi=0

(t − ti)qiqi!

γiqi(t)ci(t)

+n∑

k=1

m∑

i=1i /= k

si−1∑

qi=0

(t − ti)qiqi!

· hiqi(t)λi(t) − λk(t)ck(t) +H

(0)(t),


(Py, di(t)

) ≡ −[hi(t)ki(t)

]′−

m∑

i=1

hi(t)ki(t)

[ci(t), di(t)] −n∑

j=m+1

[hj(t)kj(t)

](cj(t), di(t)

)

−si−1∑

qi=0

m∑

i=1

(t − ti)qiqi!

γiqi(t) +[H(0)(t), di(t)

], i = 1, m.

(3.36)

Denote by ri(t) the known function

ri(t) ≡ −[hi(t)ki(t)

]′−

m∑

i=1

hi(t)ki(t)

[ci(t), di(t)] −n∑

j=m+1

[hj(t)kj(t)

](cj(t), di(t)

)+(H(0)(t), di(t)

),

(3.37)

andwrite the conditions (3.13) for (Py, di(t)). Taking into account expression (3.27) for γiqi(t),we get

si−1∑

qi=0

γ0iqi

[Dli

((t − ti)qiqi!

ePiqi (x))]

t=ti

+si−1∑

qi=0

[Dlifiqi(t)

]

t=ti=[Dliri(t)

]

t=ti, i = 1, m, li = 0, si − 1.

(3.38)

Using the Leibnitz formula, we obtain that

[Dli

((t − ti)qiqi!

ePiqi (t))]

t=ti

=

[li∑

ν=0

Cνli

((t − ti)qiqi!

)(ν)(ePiqi (t)

)(li−ν)]

t=ti

=

[qi∑

ν=0

Cνli

((t − ti)qiqi!

)(ν)(ePiqi (t)

)(li−ν)]

t=ti

= Cqili

(ePiqi (t)

)(li−qi)t=ti

,(3.39)

for li ≥ qi,

[Dli

((t − ti)qiqi!

ePiqi (t))]

t=ti

= 0, (3.40)

for 0 ≤ li ≤ qi.Therefore, previous equalities are written in the form of

si−1∑

qi=0

γ0iqiCqili

(ePiqi (t)

)(li−qi)t=ti

= r0ili(i = 1, m, li = 0, si − 1

), (3.41)


where

r0ili = −si−1∑

qi=0

[Dlifiqi(t)

]

t=ti−[Dliri(t)

]

t=ti,

for li = 0, we get γ0i0ePiqi (ti) = r0i0;

for li = 1, we get γ0i0c01

[ePiqi (t)

]′t=ti

+ γ0i1ePiqi (ti) = r0i1;

...

for li = si − 1, we get γ0i0c0si−1[ePiqi (t)

]si−1t=ti

+ · · · + γ0isi−1ePiqi (ti) = r0isi−1.

(3.42)

We obtain from here sequentially the numbers γ0i0, . . . , γ0isi−1. Theorem 3.6 is proved.

Thus, if conditions (3.24)–(3.26), (3.35) hold, all summands of solution (3.11) aredefined uniquely.

So, if h(0)(t) ∈ U(0), h(1)(t, τ, σ) ∈ U(1), and conditions (3.6), (3.24)–(3.26), and (3.35)are valid, then the systems (3.4), (3.5) (and (3.3) together with them) are solvable uniquelyin the class U = U(1) ⊕U(0). Two sequential problems (εk) and (εk+1) are connected uniquelyby conditions (3.23)–(3.25), (3.30); therefore, by virtue of Theorems 3.1–3.6, they are solvableuniquely in the spaceU.

4. Asymptotical Character of Formal Solutions

Let y−1(t, τ, σ), . . . , yk(t, τ, σ) be solutions of formal problems (ε−1), . . . , (εk) in the spaceU, respectively. Compose the partial sum for series (2.4):

Sn(t, τ, σ) =n∑

k=−1εkyk(t, τ, σ), (4.1)

and take its restriction yεn(t) = S n(t, ϕ(t)/ε, ψ(t, ε)).We have the following result.

Theorem 4.1. Let conditions (i)–(v) hold. Then, for sufficiently small ε (0 ≤ ε ≤ ε0), the estimates

∥∥y(t, τ) − yεn(t)∥∥C[0,T] ≤ Cnε

n+1, n = −1, 0, 1, . . . , (4.2)

hold. Here, y(t, ε) is the exact solution of problem (1.1), and yεn(t) is the states above restriction ofthe nth partial sum of series (2.4).

Proof. The restriction yεn(t) of series (2.4) satisfies the initial condition yεn(0) = y0 and system(1.1) up to terms containing εn+1, that is,

εdyεn(t)dt

= A(t)yεn(t) + εn+1Rn(t, ε) + h(t), (4.3)


where Rn(t, s) is a known function satisfying the estimate

‖R(t, ε)‖C[0,T] ≤ Rn, Rn—const. (4.4)

Under conditions of Theorem 4.1 on the spectrum of the operator A(t) for the fundamentalmatrix Y (t, s, ε) ≡ Y (t, ε)Y−1(t, ε) of the system εY = A(t)Y , the estimate

‖Y (t, s, ε)‖ ≤ const ∀(t, ε) ∈ Q ≡ {0 ≤ s ≤ t ≤ T}, ∀ε > 0 ∈ [0, ε0], (4.5)

is valid. Here, ε0 > 0− is sufficiently small. Now, write the equation

εdΔ(t, ε)dt

= A(t)Δ(t, ε) − εn+1Rn(t, ε), Δ(0, ε) = 0, (4.6)

for the remainder term Δ(t, ε) ≡ y(t, ε) − yεn(t). We obtain from this equation that

Δ(t, ε) = −εn∫ t

0Y (x, s, ε)Rn(s, ε)ds, (4.7)

whence we get the estimate

‖Δ(t, ε)‖C[0,T] ≤ −εnRn, (4.8)

where Rn = max(t,s)∈Q‖Y (t, s, ε)‖ · ‖Rn(t, s)‖ · T . So, we obtain the estimate

∥∥y(t, ε) − yεn(t)∥∥C[0,T] ≤ εnRn, n = −1, 0, 1, . . . . (4.9)

Taking instead of yεn(t) the partial sum

yε,n+1(t) ≡ yεn(t) + εn+1yn+1(t,ϕ(t)ε, ψ(t, ε)

), (4.10)

we get

∥∥∥∥(y(t, ε) − yεn(t)

) − εn+1yn+1(t,ϕ(t)ε, ψ(t, ε)

)∥∥∥∥ ≤ εn+1Rn+1, (4.11)

which implies the estimates (4.2). Theorem 4.1 is proved.


5. Example

Let it be required to construct the asymptotical solution for the Cauchy problem

ε

(y

z

)=

( −5t2 + 4 2t2 − 2

−10t2 + 10 4t2 − 5

)(y

z

)+

(t2h1(t)

0

), y(0, ε) = y0, z(0, ε) = z0, (5.1)

where h1(t) ∈ C∞[0, 2], ε > 0 is a small parameter. Eigenvalues of the matrix A(t) =(−5t2+4 2t2−2

−10t2+10 4t2−5

)are λ1(t) = −t2, λ2(t) = −1. Eigenvectors of matrices A(t) and A∗(t), are, res-

pectively,

ϕ1 =

(1

2

), ϕ2 =

(2

5

), ψ1 =

(5

−2

), ψ2 =

(−21

). (5.2)

We get (h(t), ψ1(t)) ≡ 5t2h1(t). Therefore,

(h(0), ψ1(0)

)= 0,

d

dt

(h(0), ψ1(0)

)= 0. (5.3)

Hence, we can apply to problem (5.1) the above developed algorithm of the regulari-zation method.

At first, obtain the basic Lagrange-Silvestre polynomials Kji(t). Since ψ(t) ≡ λ1(t) =−t2, there will be two such polynomials: K00(t) and K01(t).

Take the arbitrary numbers a00(t) and a01(t), and set the interpolation conditions forthe polynomial r(t),

r(t) = a00, r(1) = a01. (5.4)

Expand r(t) onto partial fractions

r(t)ψ(t)

=A

t2+B

t, (5.5)

from where

r(t) ≡ A + Bt. (5.6)

Use the interpolation polynomial (5.4). We get A = a00, B = a01. Hence, (5.6) takes the form

r(t) ≡ a00 + ta01. (5.7)

Since numbers a00 and a01 are arbitrary, basic Lagrange-Silvestre polynomials will be coeffici-ents standing before them, that is,

K00(t) ≡ 1, K01(t) ≡ t. (5.8)


Introduce the regularizing variables

σ00 = e(1/ε)∫ t0 λ1ds

∫ t

0e−(1/ε)

∫s0 λ1dx ·K00(s)ds = e−t

3/3ε∫ t

0es

3/3εds ≡ p00(t),

σ01 = e(1/ε)∫ t0 λ1ds

∫ t

0e−(1/ε)

∫s0 λ1dx ·K01(s)ds = e−t

3/3ε∫ t

0es

3/3ε · s ds ≡ p01(t),

τ1 =1ε

∫ t

0λ1ds = − t

3

3ε≡ q1(t), τ2 =

1ε

∫ t

0λ2ds = − t

ε≡ q2(t).

(5.9)

Construct the extended problem corresponding to problem (5.1):

ε∂w

∂t+ λ1(t)

∂w

∂τ1+ λ2(t)

∂w

∂τ2+ λ1(t)σ00

∂w

∂σ00+ λ1(t)σ01

∂w

∂σ01+ ε

∂w

∂σ00+ εt

∂w

∂σ01−A(t)w

= h(t), w(0, 0, 0, ε) = w0,

(5.10)

where τ ≡ (τ1, τ2), σ = (σ00, σ01), w = w(t, τ, σ, ε).Determining solutions of problem (5.10) in the form of a series

w(t, τ, σ, ε) =∞∑

k=0

εkwk(t, τ, σ), (5.11)

we obtain the following iteration problems:

Lw0 ≡ λ1(t)[∂w0

∂τ+∂w0

∂σ00+ t · σ01 ∂w0

∂σ01

]+ λ2(t)

∂w0

∂τ2−A(t)w0 = h(t), w0(0, 0, 0) = w0,

(5.12)

Lw1 = −∂w0

∂t− ∂w0

∂σ00− t ∂w0

∂σ01, w1(0, 0, 0) = 0,

...

(5.13)

We determine solutions of iteration problems (5.12), (5.13), and so on in the space Uof functions in the form of

w(t, τ, σ) = w1(t)eτ1 +w2(t)eτ2 +w00(t)σ00 +w01(t)σ01 +w0(t),

w0(t), w1(t), w2(t), w00(t), w01(t) ∈ C∞([0, 2], C2

).

(5.14)

Directly calculating, we obtain the solution of system (5.12) in the form of

w0(t, τ, σ) = α1(t)ϕ1eτ1 + α2(t)ϕ2e

τ2 + γ00(t)ϕ1σ00 + γ01(t)ϕ1σ01 + 5h1(t)ϕ1 − 2t2h1(t)ϕ2,

(5.15)

where αj(t), γji(t) ∈ C∞[0, 2] are for now arbitrary functions.


To calculate the functions αj(t) and γij(t), we pass to the following iteration problem(5.13). Taking into account (5.15), it will be written in the form of

Lw1 = −α1(t)ϕ1eτ1 − α2(t)ϕ2e

τ2 − γ00(t)ϕ1σ00 − γ01(t)ϕ1σ01

− 5h1(t)ϕ1 −(2t2h1(t)

)′ϕ2 − γ00(t)ϕ1 − tγ01(t)ϕ1.

(5.16)

For solvability of problem (5.13) in the space U, it is necessary and sufficient to fulfillthe conditions

−α1(t) = 0, −α2(t) = 0, −γ00(t) = 0, −γ01(t) = 0,

−5h1(0) − γ00(0) = 0, −5h1(0) − γ00(0) − γ01(0) = 0.(5.17)

Using solution (5.15) and the initial conditionw0(0, 0, 0) = w0, we obtain the equation

α1(0)ϕ1 + α2(0)ϕ2 + 5h1(0)ϕ1 = w0. (5.18)

Multiplying it (scalar) on ψ1 and ψ2, we obtain the values

α1(0) =(w0, ψ1

)− 5h1(0) ≡ 5y0 − 2z0 − 5h1(0),

α2(0) =(w0, ψ2

)= z0 − 2y0.

(5.19)

Using equalities (5.17), and also the initial data (5.19), we obtain uniquely the func-tions αj(t) and γji(t):

α1(t) = 5y0 − 2z0 − 5h1(0), α2(t) = z0 − 2y0.

γ00(t) = −5h1(0), γ01(t) = −5h1(0).(5.20)

Substituting these functions into (5.15), we obtain uniquely the solution of problem(5.12) in the spaceU,

w0(t, τ, σ) =(5y0 − 2z0 − 5h1(0)

)ϕ1e

τ1 +(z0 − 2y0

)ϕ2e

τ2

− 5h1(0)ϕ1σ00 − 5h1(0)ϕ1σ01 + 5h1(t)ϕ1 − 2t2h1(t)ϕ2.

(5.21)


Producing here restriction on the functions τ = q(t), σ = p(t), we obtain the principalterm of the asymptotics for the solution of problem (5.1):

w0ε(t) =(5y0 − 2z0 − 5h1(0)

)ϕ1e

−t3/3ε +(z0 − 2y0

)ϕ2e

−t/ε

− 5h1(0)ϕ1e−t3/3ε

∫ t

0es

3/3εds − 5h1(0)ϕ1e−t3/3ε

∫ t

0es

3/3εs ds + 5h1(t)ϕ1

− 2t2h1(t)ϕ2.

(5.22)

The zero-order asymptotical solution is obtained: it satisfies the estimate

‖w(t, ε) −w0ε(t)‖C[0,2] ≤ C1 · ε, (5.23)

where w(t, ε) is an exact solution of problem (1.1), and C1 > 0 is a constant independent ofε at sufficiently small ε (0 < ε ≤ ε0).

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Research ArticleInitial-Boundary Value Problem for FractionalPartial Differential Equations of Higher Order

Djumaklych Amanov1 and Allaberen Ashyralyev2, 3

1 Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences,29 Do’rmon yo’li street, Tashkent 100047, Uzbekistan

2 Department of Mathematics, Fatih University, 34500 Buyucekmece, Istanbul, Turkey3 ITTU, Ashgabat 74400, Turkmenistan

Correspondence should be addressed to Djumaklych Amanov, [email protected]

Received 30 March 2012; Revised 25 April 2012; Accepted 12 May 2012


Copyright q 2012 D. Amanov and A. Ashyralyev. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The initial-boundary value problem for partial differential equations of higher-order involvingthe Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solutionand its continuous dependence on the initial data and on the right-hand side of the equation areestablished.

1. Introduction

Many problems in viscoelasticity [1–3], dynamical processes in self-similar structures [4],biosciences [5], signal processing [6], system control theory [7], electrochemistry [8],diffusion processes [9], and linear time-invariant systems of any order with internal pointdelays [10] lead to differential equations of fractional order. For more details of fractionalcalculus, see [11–15].

The study of existence and uniqueness, periodicity, asymptotic behavior, stability, andmethods of analytic and numerical solutions of fractional differential equations have beenstudied extensively in a large cycle works (see, e.g., [16–42] and the references therein).

In the paper [43], Cauchy problem in a half-space {(x, y, t) : (x, y) ∈ R2, t > 0} for

partial pseudodifferential equations involving the Caputo fractional derivative was studied.The existence and uniqueness of a solution and its continuous dependence on the initial dataand on the right-hand side of the equation were established.

In the paper [44], the initial-boundary value problem for heat conduction equationwith the Caputo fractional derivative was studied. Moreover, in [45], the initial-boundary


value problem for partial differential equations of higher order with the Caputo fractionalderivative was studied in the case when the order of the fractional derivative belongs to theinterval (0,1).

In the paper [46], the initial-boundary value problem in plane domain for partialdifferential equations of fourth order with the fractional derivative in the sense of Caputo wasstudied in the case when the order of fractional derivative belongs to the interval (1,2). Thepresent paper generalizes results of [46] in the case of space domain for partial differentialequations of higher order with a fractional derivative in the sense of Caputo.

The organization of this paper is as follows. In Section 2, we provide the necessarybackground and formulation of problem. In Section 3, the formal solution of problem ispresented. In Sections 4 and 5, the solvability and the regular solvability of the problem arestudied. Theorems on existence and uniqueness of a solution and its continuous dependenceon the initial data and on the right-hand side of the equation are established. Finally, Section 6is conclusion.

2. Preliminaries

In this section, we present some basic definitions and preliminary facts which are usedthroughout the paper.

Definition 2.1. If g(t) ∈ C[a, b] and α > 0, then the Riemann-Liouville fractional integral isdefined by

Iαa+g(t) =1

Γ(α)

∫ t

a

g(s)

(t − s)1−αds, (2.1)

where Γ(·) is the Gamma function defined for any complex number z as

Γ(z) =∫∞

0tz−1e−tdt. (2.2)

Definition 2.2. The Caputo fractional derivative of order α > 0 of a continuous function g :(a, b) → R is defined by

cDαa+g(t) =

1Γ(n − α)

∫ t

a

g(n)(s)

(t − s)α−n+1ds, (2.3)

where n = [α] + 1, (the notation [α] stands for the largest integer not greater than α).

Lemma 2.3 (see [13]). Let p, q ≥ 0, f(t) ∈ L1[0, T]. Then,

Ip

0+Iq

0+f(t) = Ip+q0+ f(t) = Iq0+I

p

0+f(t) (2.4)

is satisfied almost everywhere on [0, T]. Moreover, if f(t) ∈ C[0, T], then (2.4) is true andcDα

0+Iα0+f(t) = f(t) for all t ∈ [0, T] and α > 0.


Theorem 2.4 (see [47, page 123]). Let f(t) ∈ L1(0, T). Then, the integral equation

z(t) = f(t) + λ∫ t

0

(t − τ)α−1Γ(α)

z(τ)dτ (2.5)

has a unique solution z(t) defined by the following formula:

z(t) = f(t) + λ∫ t

0(t − τ)α−1Eα,α

(λ(t − τ)α)f(τ)dτ, (2.6)

where Eα,β(z) =∑∞

k=0(zk/Γ(kα + β)) is a Mittag-Leffler type function.

For the convenience of the reader, we give the proof of Theorem 2.4, applying thefixed-point iteration method. We denote

Bz(t) = λ∫ t

0

(t − τ)α−1Γ(α)

z(τ)dτ. (2.7)

Then,

z(t) =m−1∑

k=0

Bkf(t) + Bmz(t), m = 1, . . . , n. (2.8)

The proof of this theorem is based on formula (2.8) and

Bmz(t) = λm∫ t

0

(t − τ)mα−1Γ(mα)

z(τ)dτ, (2.9)

for any m ∈ N. Let us prove (2.9) for any m ∈ N. For m = 1, it follows from (2.7) directly.Assume that (2.9) holds for some m − 1 ∈ N. Then, applying (2.7) and (2.9) for m − 1 ∈ N,we get

Bmz(t) = λm−1∫ t

0

(t − s)(m−1)α−1

Γ((m − 1)α)Bz(s)ds

= λm−1∫ t

0

(t − s)(m−1)α−1

Γ((m − 1)α)λ

∫ s

0(s − τ)α−1z(τ)dτ ds

=λm

Γ(α)Γ((m − 1)α)

∫ t

0

∫s

0(t − s)(m−1)α−1(s − τ)α−1z(τ)dτ ds

=λm

Γ(α)Γ((m − 1)α)

∫ t

0

∫ t

τ

(t − s)(m−1)α−1(s − τ)α−1dsz(τ)dτ.

(2.10)


Performing the change of variables s − τ = (t − τ)p, we get

∫ t

τ

(t − s)(m−1)α−1(s − τ)α−1ds = (t − τ)mα−1∫1

0

(1 − p)(m−1)α−1

pα−1dp

= (t − τ)mα−1B((m − 1)α, α)

=(t − τ)mα−1Γ(mα)

Γ((m − 1)α)Γ(α).

(2.11)

Then,

Bmz(t) = λm∫ t

0

(t − τ)mα−1Γ(mα)

z(τ)dτ. (2.12)

So, identity (2.9) holds form ∈N. Therefore, by induction identity (2.9) holds for anym ∈N.In the space domain, Ω = {(x, y, t) : 0 < x < p, 0 < y < q, 0 < t < T}, we consider the

initial-boundary value problem:

(−1)kcDα0+u +

∂2ku

∂x2k+∂2ku

∂y2k= f(x, y, t

), 0 < x < p, 0 < y < q, 0 < t < T,

∂2mu(0, y, t

)

∂x2m=∂2mu

(p, y, t

)

∂x2m= 0, m = 0, 1, . . . , k − 1, 0 ≤ y ≤ q, 0 ≤ t ≤ T,

∂2mu(x, 0, t)∂y2m

=∂2mu

(x, q, t

)

∂y2m= 0, m = 0, 1, . . . , k − 1, 0 ≤ x ≤ p, 0 ≤ t ≤ T,

u(x, y, 0

)= ϕ(x, y), ut

(x, y, 0

)= ψ(x, y), 0 ≤ x ≤ p, 0 ≤ y ≤ q

(2.13)

for partial differential equations of higher order with the fractional derivative order α ∈ (1, 2)in the sense of Caputo. Here, k(k ≥ 1) is a fixed positive integer number.

3. The Construction of the Formal Solution of (2.13)

We seek a solution of problem (2.13) in the form of Fourier series:

u(x, y, t

)=

∞∑

n,m=1

unm(t)vnm(x, y), (3.1)

expanded along a complete orthonormal system:

vnm(x, y)=

2√pq

sinnπ

px sin

mπ

qy, 1 ≤ n, m <∞. (3.2)


We denote

Ω0 = Ω ∩ (t = 0) ={(x, y, 0

): 0 ≤ x ≤ p, 0 ≤ y ≤ q},

nπ

p= νn,

mπ

q= μm, ν2kn + μ2k

m = λ2knm, 1 ≤ n,m <∞.(3.3)

We expand the given function f(x, y, t) in the form of a Fourier series along the functionsvnm(x, y), 1 ≤ n,m <∞:

f(x, y, t

)=

∞∑

n,m=1

fnm(t)vnm(x, y), (3.4)

where

fnm(t) =∫p

0

∫q

0f(x, y, t

)vnm(x, y)dy dx, 1 ≤ n,m <∞. (3.5)

Substituting (3.1) and (3.4) into (2.13), we obtain

(−1)k cDα0+unm(t) + (−1)kλ2knmunm(t) = fnm(t). (3.6)

By Lemma 2.3, we have that

cDα0+unm(t) = I

2−α0+ u′′nm(t), (3.7)

where

Iα0+f(t) =1

Γ(α)

∫ t

0(t − τ)α−1f(τ)dτ (3.8)

is Riemann-Liouville integral of fractional order α. Using (3.6) and (3.7), we get the followingequation:

I2−α0+ u′′nm(t) + λ2knmunm(t) = (−1)kfnm(t). (3.9)

Applying the operator Iα0+ to this equation, we get the following Volterra integral equation ofthe second kind:

unm(t) =−λ2knmΓ(α)

∫ t

0(t − τ)α−1unm(τ)dτ + unm(0) + tu′nm(0) + (−1)kIα0+fnm(t). (3.10)


According to the Theorem 2.4, (3.10) has a unique solution unm(t) defined by the followingformula:

unm(t) =(−1)kΓ(α)

∫ t

0(t − τ)α−1fnm(τ)dτ

+ unm(0)

[1 − λ2knm

∫ t


(−λ2knm(t − τ)α

)dτ

]

+ u′nm(0)

[t − λ2knm

∫ t



)τdτ

]

− λ2knmΓ(α)

∫ t

0

(t − η)α−1Eα,α

(−λ2knm

(t − η)α

)dη

∫η

0

(η − τ)α−1fnm(τ)dτ.

(3.11)

Using the formula (see, e.g., [27, page 118] and [47, page 120])

1Γ(β)∫z

0tμ−1Eα,μ(λtα)(z − t)β−1dt = zμ+β−1Eα,μ+β(λzα),

1Γ(μ) + zEα,α+μ(z) = Eα,μ(z),

(3.12)

we get

− λ2knmΓ(α)

∫ t

0


(−λ2knm

(t − η)α

)dη

∫η

0

(η − τ)α−1fnm(τ) dτ

=∫ t

0fnm(τ)

{− λ

2knm

Γ(α)

∫ t

τ


(−λ2knm

(t − η)α

)(η − τ)α−1dη

}dτ

=∫ t

0fnm(τ)

{− λ

2knm

Γ(α)

∫ t−τ

0zα−1Eα,α

(−λ2knmzα

)(t − τ − z)α−1dz

}dτ

= −∫ t

0fnm(τ)λ2knm(t − τ)2α−1Eα,2α


)dτ

=∫ t

0(t − τ)α−1fnm(τ)

{− 1Γ(α)

+ Eα,α(−λ2knm(t − τ)α

)}dτ,

− λ2knm∫ t



)dτ

= −λ2knm∫ t

0zα−1Eα,α

(−λ2knmzα

)(t − z)1−1dz


= Γ(1)λ2knmtαEα,α+1

(−λ2knmtα

)= Eα,1

(−λ2knmtα

)− 1,

− λ2knm∫ t



)τ dτ

= −λ2knm∫ t

0zα−1Eα,α

(−λ2knmzα

)(t − z)2−1dz

= Γ(2)λ2knmtα+1Eα,α+2

(−λ2knmtα

)= tEα,2

(−λ2knmtα

)− t.

(3.13)

From these three formulas and (3.11), it follows that

unm(t) = unm(0)Eα,1(−λ2knmtα

)+ tu′nm(0)Eα,2

(−λ2knmtα

)

+ (−1)k∫ t



)fnm(τ)dτ.

(3.14)

For unm(0) and u′nm(0), we expand the given functions ϕ(x, y) and ψ(x, y) in the form of aFourier series along the functions vnm(x, y), 1 ≤ n,m <∞:

ϕ(x, y)=

∞∑

n,m=1

ϕnmvnm(x, y),

ψ(x, y)=

∞∑

n,m=1

ψnmvnm(x, y),

(3.15)

where

ϕnm =∫p

0

∫q

0ϕ(x, y)vnm(x, y)dy dx,

ψnm =∫p

0

∫q

0ψ(x, y)vnm(x, y)dy dx.

(3.16)

Using (2.13), (3.14), (3.16), we obtain

unm(t) = Eα,1(−λ2knmtα

)ϕnm + tEα,2

(−λ2knmtα

)ψnm

+ (−1)k∫ t



)fnm(τ)dτ.

(3.17)

So, the unique solution of (3.10) is defined by (3.17). Consequently, the unique solution ofproblem (2.13) is defined by (3.1).


Applying the formula (3.17), the Cauchy-Schwarz inequality, and the estimate (see[13, page 136])

∣∣Eα,β(z)∣∣ ≤ M

1 + |z| , M = const > 0, Re z < 0, (3.18)

we get the following inequality:

|unm(t)| ≤ C0

⎛

⎝∣∣ϕnm∣∣ +∣∣ψnm

∣∣ +(∫ t

0

∣∣fnm(t)∣∣2dt)1/2

⎞

⎠ (3.19)

for the solution of (3.10) for any t, t ∈ [0, T]. Here, C0 = max{M,TM,M(Tα−1/2/√2α − 1)}.

4. Solvability of (2.13) in L2(Ω) Space

Now, we will prove that the solution u(x, y, t) of problem (2.13) continuously depends onϕ(x, y), ψ(x, y), and f(x, y, t).

Theorem 4.1. Suppose ϕ(x, y) ∈ L2(Ω0), ψ(x, y) ∈ L2(Ω0), and f(x, y, t) ∈ L2(Ω), then the series(3.1) converges in L2(Ω) to u ∈ L2(Ω) and for the solution of problem (2.13), the following stabilityinequality

‖u‖L2(Ω) ≤ C1

(∥∥ϕ∥∥L2(Ω0)

+∥∥ψ∥∥L2(Ω0)

+∥∥f∥∥L2(Ω)

)(4.1)

holds, where C1 does not depend on ϕ(x, y), ψ(x, y), and f(x, y, t).

Proof. We consider the sum:

uN(x, y, t

)=

N∑

n,m=1

unm(t)vnm(x, y), (4.2)

whereN is a natural number. For the positive integer number L, we have that

‖uN+L − uN‖2L2(Ω) =

∥∥∥∥∥

N+L∑

n,m=N+1

unm(·)vnm(·, ·)∥∥∥∥∥

2

L2(Ω)

=N+L∑

n,m=N+1

∫T

0|unm(t)|2dt.

(4.3)


Applying (3.19), we get

∞∑

n,m=1

∫T

0|unm(t)|2dt ≤ 3C2

0

( ∞∑

n,m=1

∣∣ϕnm∣∣2 +

∞∑

n,m=1

∣∣ψnm∣∣2 +

∞∑

n,m=1

∫T

0

∣∣fnm(t)∣∣2dt)

= C2(∥∥ϕ∥∥2L2(Ω0)

+∥∥ψ∥∥2L2(Ω0)

+∥∥f∥∥2L2(Ω)

),

(4.4)

where C2 = 3TC20. Therefore,

∑N+Ln,m=N+1

∫T0 |unm(t)|2dt → 0 as N → ∞. Consequently, the

series (3.1) converges in L2(Ω) to u(x, y, t) ∈ L2(Ω). Inequality (4.1) for the solution ofproblem (2.13) follows from the estimate (4.4). Theorem 4.1 is proved.

5. The Regular Solvability of (2.13)

In this section, we will study theregular solvability of problem (2.13).

Lemma 5.1. Suppose ϕ(x, y) ∈ C1(Ω0),ϕxy(x, y) ∈ L2(Ω0),ψ(x, y) ∈ C1(Ω0),ψxy(x, y) ∈L2(Ω0),ϕ(x, y) = 0 on ∂Ω0,ψ(x, y) = 0 on ∂Ω0, f(x, y, t) ∈ C2(Ω),fxxy(x, y, t) ∈C(Ω)fxyy(x, y, t) ∈ C(Ω0),fxxyy(x, y, t) ∈ C(Ω0), and f(x, y, t) = 0 on ∂Ω × [0, T]. Then, forany ε ∈ (0, 1), the following estimates

|unm(t)| ≤ C1

(∣∣ϕnm∣∣

νknμkm

+

∣∣ψnm∣∣

νknμkm

+1

νk+1n μk+1m

+1

νk+1−εn μk+1m

+1

νk+1n μk+1−εm

), (5.1)

λ2knm|unm(t)| ≤ C2

⎛

⎝

∣∣∣ϕ(1,1)nm

∣∣∣

νnμm+

∣∣∣ψ(1,1)nm

∣∣∣

νnμm+

1ν2nμ

2m

+1

ν2−εn μ2m

+1

ν2nμ2−εm

⎞

⎠ (5.2)

hold, where C1 and C2 do not depend on ϕ(x, y) and ψ(x, y).

Proof. Integrating by parts with respect to x and y in (3.5), (3.16), we get

ϕnm =1

νnμmϕ(1,1)nm , (5.3)

ψnm =1

νnμmψ(1,1)nm , (5.4)

fnm(t) =1

νnμmf(1,1,0)nm (t), (5.5)

fnm(t) =1

ν2nμ2m

f(2,2,0)nm (t), (5.6)


where

ϕ(1,1)nm =

∫p

0

∫q

0

∂2ϕ(x, y)

∂x∂yvnm(x, y)dy dx,

ψ(1,1)nm =

∫p

0

∫q

0

∂2ψ(x, y)


f(1,1,0)nm (t) =

∫p

0

∫q

0

∂2f(x, y, t

)


f(2,2,0)nm (t) =

∫p

0

∫q

0

∂4f(x, y, t

)

∂x2∂y2vnm(x, y)dy dx.

(5.7)

Under the assumptions of Lemma 5.1, it follows that the functions f (1,1,0)nm (t) and f (2,2,0)

nm (t) arebounded, that is,

∣∣∣f (1,1,0)nm (t)

∣∣∣ ≤N1,∣∣∣f (2,2,0)

nm (t)∣∣∣ ≤N2, (5.8)

where N1 = const > 0, N2 = const > 0. Let 0 < t0 ≤ t ≤ T , where t0 is a sufficiently smallnumber. For sufficiently large n andm, the following inequalities are true:

lnλεnm < λεnm < νεn + μεm, 0 < ε < 1,

1 + λ2knmTα < 2λ2knmT

α.(5.9)

Using (3.16), (5.8), (5.9), and (3.17), we get

|unm(t)| ≤ M

(12tα0

∣∣ϕnm∣∣

νknμkm

+1

2tα−10

∣∣ψnm∣∣

νknμkm

− N1

ανk+1n μk+1m

∫ t

0

d(1 + λ2knm(t − τ)α

)

1 + λ2knm(t − τ)α)

≤ M

(12tα0

∣∣ϕnm∣∣

νknμkm

+1

2tα−10

∣∣ψnm∣∣

νknμkm

+N1(ln 2Tα + (2k/ε) lnλεnm)

ανk+1n μk+1m

)

≤ C1

(∣∣ϕnm∣∣

νknμkm

+

∣∣ψnm∣∣

νknμkm

+1

νk+1n μk+1m

+1

νk+1−εn μk+1m

+1

νk+1n μk+1−εm

),

(5.10)


where C1 = max{M/2tα0 ,M/2tα−10 ,MN1 ln 2Tα/α, 2kMN1/αε}. Thus, inequality (5.1) isobtained. Now, we will prove inequality (5.2). Using (5.3), (5.4), (5.6), (5.8), (5.9), and (3.17),we get

λ2knm|unm(t)| ≤M(∣∣ϕnm

∣∣

tα+

∣∣ψnm∣∣

tα−1+ λ2knm

∫ t

0

(t − τ)α−1fnm(τ)1 + λ2knm(t − τ)α

dτ

)

≤M

⎛⎜⎝

1tα0

∣∣∣ϕ(1,1)nm

∣∣∣

νnμm+

∣∣∣ψ(1,1)nm

∣∣∣

tα−10 νnμm− N2

α

∫ t

0

d(1 + λ2knm(t − τ)α

)

ν2nμ2m

(1 + λ2knm(t − τ)α

)

⎞⎟⎠

≤ M

tα0

∣∣∣ϕ(1,1)nm

∣∣∣

νnμm+M

tα−10

∣∣∣ψ(1,1)nm

∣∣∣

νnμm+2MN2 ln Tα

αν2nμ2m

+2kMN2

αεν2−εn μ2m

+2kMN2

αεν2nμ2−εm

≤ C2

⎛

⎝

∣∣∣ϕ(1,1)nm

∣∣∣

νnμm+

∣∣∣ψ(1,1)nm

∣∣∣

νnμm+

1ν2nμ

2m

+1

ν2−εn μ2m

+1

ν2nμ2−εm

⎞

⎠,

(5.11)

where C2 = max{M/tα0 ,M/tα−10 ,MN2 ln 2Tα/α, 2kMN2/αε}. Lemma 5.1 is proved.

Theorem 5.2. Suppose that the assumptions of Lemma 5.1 hold. Then, there exists a regular solutionof problem (2.13).

Proof. We will prove uniform and absolute convergence of series (3.1) and

∂2ku(x, y, t

)

∂x2k=

∞∑

n,m=1

(−1)kν2kn unm(t)vnm(x, y), (5.12)

∂2ku(x, y, t

)

∂x2k=

∞∑

n,m=1

(−1)kμ2kn unm(t)vnm

(x, y), (5.13)

cDα0+u(x, y, t

)= −

∞∑

n,m=1

(−1)kλ2knmunm(t)vnm(x, y)+

∞∑

n,m=1

fnm(t)vnm(x, y). (5.14)

The series

∞∑

n,m=1

|unm(t)| (5.15)

is majorant for the series (3.1). From (5.1), it follows that the series (5.15) uniformlyconverges. Actually,

∞∑

n,m=1

|unm(t)| ≤ C∞∑

n,m=1

(∣∣ϕnm∣∣

νknμkm

+

∣∣ψnm∣∣

νknμkm

+1

νknμkn

+1

νk+1−εn μk+1n

+1

νk+1n μk+1−εn

). (5.16)


Applying the Cauchy-Schwarz inequality and the Parseval equality, we obtain

∞∑

n,m=1

∣∣ϕnm∣∣

νknμkm

≤( ∞∑

n,m=1

1

ν2kn μ2km

)1/2( ∞∑

n,m=1

∣∣ϕnm∣∣2)1/2

=pkqk

π2k

( ∞∑

n=1

1n2k

∞∑

m=1

1m2k

)1/2∥∥ϕ∥∥L2(Ω0)

.

(5.17)

Analogously, we get

∞∑

n,m=1

∣∣ψnm∣∣

νknμkm

≤ pkqk

π2k

( ∞∑

n=1

1n2k

∞∑

m=1

1m2k

)1/2∥∥ψ∥∥L2(Ω0)

. (5.18)

Since 2k ≥ 2, then the series∑∞

n=1(1/n2k),∑∞

m=1(1/m2k) converges by the integral test.

Further, k + 1 − ε > 1, then the series

∞∑

n,m=1

1

νk+1n μk+1m

,∞∑

n,m=1

1

νk+1−εn μk+1m

,∞∑

n,m=1

1

νk+1n μk+1−εm(5.19)

converges also by the integral test for any k ≥ 1 and ε ∈ (0, 1).Consequently, the series (3.1) absolutely and uniformly converges in the domainΩt0 =

Ω × [t0, T] for any t0 ∈ (0, T). At t = 0, the series (3.1) converges and has a sum equal toϕ(x, y). Since ν2kn < λ2knm,μ

2km < λ2knm, then the series

∞∑

n,m=1

λ2knm|unm| (5.20)

is majorant for the series (5.12), (5.13) and for the first series from (5.14). From (5.2), it followsthat the series (5.20) uniformly converges. Indeed, using the Parseval equality and Cauchy-Schwarz inequality, we get

∞∑

n,m=1

∣∣∣ϕ(1,1)nm

∣∣∣

νnμm≤( ∞∑

n=1

1ν2n

∞∑

m=1

1m2

)1/2( ∞∑

n,m=1

∣∣∣ϕ(1,1)nm

∣∣∣2)1/2

=pq

6

∥∥∥∥∥∂2ϕ

∂x∂y

∥∥∥∥∥L2(Ω0)

. (5.21)

Analogously, we conclude that

∞∑

n,m=1

∣∣∣ψ(1,1)nm

∣∣∣

νnμm≤ pq

6

∥∥∥∥∥∂2ψ

∂x∂y

∥∥∥∥∥L2(Ω0)

. (5.22)


The series

∞∑

n,m=1

(1

ν2nμ2m

+1

ν2−εn μ2m

+1

ν2nμ2−εm

)

(5.23)

converges for any ε ∈ (0, 1) according to the integral test. The series

∞∑

n,m=1

∣∣fnm(t)∣∣ (5.24)

is majorant for the second series from (5.14). From (5.6) and (5.8), it follows that the series(5.14) uniformly converges. Indeed,

∞∑

n,m=1

∣∣fnm(t)∣∣ =

∞∑

n,m=1

1ν2nμ

2m

∣∣∣f2,2,0nm (t)

∣∣∣ ≤N2

∞∑

n,m=1

1ν2nμ

2m

=N2p

2q2

36. (5.25)

Adding equality (5.12), (5.13), and (5.14), we note that the solution (3.1) satisfies equation(2.13). The solution (3.1) satisfies boundary conditions owing to properties of the functionsvnm(x, y). Simple computations show that

limt→ 0

Eα,1(−λ2knmtα

)= 1,

limt→ 0

d

dtEα,1(−λ2knmtα

)= 0,

limt→ 0

Eα,2(−λ2knmtα

)= 1,

limt→ 0

td

dtEα,2(−λ2knmtα

)= 0.

(5.26)

Consequently, limt→ 0unm(t) = ϕnm, limt→ 0u′nm(t) = ψnm. Hence, we conclude that the solution

(3.1) satisfies initial conditions. Theorem 5.2 is proved.

6. Conclusion

In this paper, the initial-boundary value problem (2.13) for partial differential equations ofhigher order involving the Caputo fractional derivative is studied. Theorems on existenceand uniqueness of a solution and its continuous dependence on the initial data and on the


right-hand side of the equation are established. Of course, such type of results have beenestablished for the initial-boundary value problem:

(−1)k cDα0+u +

∂2ku

∂x2k+∂2ku

∂y2k+ u = f

(x, y, t

), 0 < x < p, 0 < y < q, 0 < t < T,

∂2m+1u(0, y, t

)

∂x2m=∂2m+1u

(p, y, t

)

∂x2m= 0, m = 0, 1, . . . , k − 1, 0 ≤ y ≤ q, 0 ≤ t ≤ T,

∂2m+1u(x, 0, t)∂y2m

=∂2m+1u

(x, q, t

)

∂y2m= 0, m = 0, 1, . . . , k − 1, 0 ≤ x ≤ p, 0 ≤ t ≤ T,

u(x, y, 0

)= ϕ(x, y), ut

(x, y, 0

)= ψ(x, y), 0 ≤ x ≤ p, 0 ≤ y ≤ q

(6.1)

for partial differential equations of higher order with a fractional derivative of order α ∈ (1, 2)in the sense of Caputo. Here, k(k ≥ 1) is a fixed positive integer number.

Moreover, applying the result of the papers [15, 23], the first order of accuracydifference schemes for the numerical solution of nonlocal boundary value problems (2.13)and (6.1) can be presented. Of course, the stability inequalities for the solution of thesedifference schemes have been established without any assumptions about the grid steps τin t and h in the space variables.

Acknowledgment

The authors are grateful to Professor Valery Covachev (Sultan Qaboos University, Sultanateof Oman) for his insightful comments and suggestions.

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Research ArticleA Note on the Right-Hand Side IdentificationProblem Arising in Biofluid Mechanics

Abdullah Said Erdogan

Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey

Correspondence should be addressed to Abdullah Said Erdogan, [email protected]



Copyright q 2012 Abdullah Said Erdogan. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The inverse problem of reconstructing the right-hand side (RHS) of a mixed problem for one-dimensional diffusion equation with variable space operator is considered. The well-posedness ofthis problem in Holder spaces is established.

1. Introduction

It is known that many applied problems in fluid mechanics, other areas of physics, andmathematical biology were formulated as the mathematical model of partial differentialequations of the variable types [1–3]. A model for transport across microvessel endotheliumwas developed to determine the forces and bending moments acting on the structure of theflow over endothelial cells (ECs) [4]. Computational blood flow analysis through glycocalyxon the EC is performed as a direct problem previously under smooth and nonsmooth initialconditions (see [5–7]). But it is known that, due to the lack of some data and/or coefficients,many real-life problems are modeled as inverse problems [8–11].

In this paper, the well-posedness of the inverse problem of reconstructing the rightside of a parabolic equation arisen in computational blood flow analysis is investigated.The importance of well-posedness has been widely recognized by the researchers in thefield of partial differential equations [12–16]. Moreover, the well-posedness of the RHSidentification problems for a parabolic equation where the unknown function p is in spacevariable and in time variable is well investigated [17–27]. As it is known, well-posedness inthe sense of Hadamard means that there is existence and uniqueness of the solution and thesolution is stable. In this study, we deal with the stability analysis of the inverse problem ofreconstructing the right-hand side. The existence of a solution for two-phase flow in porousmedia has been studied previously (for instance, see [28]).


1.1. Problem Formulation

Blood flow over the EC inside the arteries is modeled in two regions (see [6]). Core region(0 < x < l) flow is defined through the center of capillary and porous region (l < x <L) flow is through the glycocalyx. RHS function includes the pressure difference alongthe microchannels under unsteady fluid flow conditions. When the pressure difference isan unknown function of t, we reach a new model, and, by overdetermined (additional)conditions derived from an observation point, the solution of this problem can be obtained.The model can be considered as the mixed problem for one-dimensional diffusion equationwith variable space operator:

∂u(t, x)∂t

= a(x)∂2u(t, x)∂x2

+ p(t)q(x) + f(t, x), x ∈ (0, l), t ∈ (0, T],

∂u(t, x)∂t

= a(x)∂2u(t, x)∂x2

+ b(t, x)u(t, x) + p(t)q(x) + g(t, x), x ∈ (l, L), t ∈ (0, T],

u(0, x) = ϕ(x), x ∈ [0, L],

ux(t, 0) = 0, u(t, L) = 0, t ∈ [0, T],

u(t, l+) = u(t, l−), ux(t, l+) = ux(t, l−), t ∈ [0, T],

u(t, x∗) = ρ(t), 0 ≤ x∗ ≤ l, 0 ≤ t ≤ T.

(1.1)

Here, u(t, x) and p(t) are unknown functions, a(x), b(t, x), f(t, x),g(t, x), ρ(t), and ϕ(x) aregiven sufficiently smooth functions, and a(x) � a > 0. Also, q(x) is a sufficiently smoothfunction assuming that q′(0) = q(L) = 0 and q(x∗)/= 0.

2. Main Results

2.1. Differential Case

To formulate our results, we introduce the Banach space◦Cα

[0, L], α ∈ (0, 1), of all continuousfunctions φ(x) defined on [0, L]with φ′(0) = φ(L) = 0 satisfying a Holder condition for whichthe following norm is finite:

∥∥φ∥∥ ◦Cα

[0,L]= max

0≤x≤L

∣∣φ(x)∣∣ + sup

0≤x<x+h≤L

∣∣φ(x + h) − φ(x)∣∣hα

. (2.1)

In a Banach space E, with the help of a positive operatorAwe introduce the fractionalspaces Eα, 0 < α < 1, consisting of all v ∈ E for which the following norm is finite:

‖v‖Eα = ‖v‖E + supλ>0

λ1−α∥∥A exp{−λA}v∥∥E. (2.2)

Positive constants will be indicated by M which can be differ in time. On the other handMi (α, β, . . .) is used to focus on the fact that the constant depends only on α, β, . . ., and thesubindex i is used to indicate a different constant.


Theorem 2.1. Let ϕ ∈◦C

2α+2[0, L], F1 ∈ C([0, T],

◦C

2α[0, L]), and ρ′ ∈ C[0, T]. Then for the

solution of problem (1.1), the following coercive stability estimates

‖ut‖C([0,T],

◦C

2α[0,L])

+ ‖u‖C([0,T],

◦C

2α+2[0,L])

≤M(x∗, q)∥∥ρ′∥∥C[0,T] +M

(a, δ, σ, α, x∗, q, T

)

×(∥∥ϕ

∥∥ ◦C

2α+2[0,L]

+ ‖F1‖C([0,T],

◦C

2α[0,L])

+∥∥ρ∥∥C[0,T]

),

∥∥p∥∥C[0,T] ≤ M

(x∗, q)∥∥ρ′∥∥C[0,T]

+M(a, δ, σ, α, x∗, q, T

)[∥∥ϕ∥∥ ◦C

2α+2[0,L]

+ ‖F1‖C([0,T],

◦C

2α[0,L])

+∥∥ρ∥∥C[0,T]

]

(2.3)

hold.

Proof. Let us search for the solution of inverse problem (1.1) in the following form (see [23]):

u(t, x) = η(t)q(x) +w(t, x), (2.4)

where

η(t) =∫ t

0p(s)ds. (2.5)

Taking derivatives from (2.4)with respect to t and x, we get

∂u(t, x)∂t

= p(t)q(x) +∂w(t, x)

∂t,

∂2u(t, x)∂x2

= η(t)d2q(x)dx2

+∂2w(t, x)∂x2

.

(2.6)

Moreover, substituting x by x∗ in (2.4), we obtain

u(t, x∗) = η(t)q(x∗) +w(t, x∗) = ρ(t),

η(t) =ρ(t) −w(t, x∗)

q(x∗).

(2.7)

Differentiating both sides of (2.7)with respect to t, we get

p(t) =ρ′(t) −wt(t, x∗)

q(x∗). (2.8)


From identity (2.8) and the triangle inequality, it follows that

∣∣p(t)∣∣ =∣∣∣∣ρ′(t) −wt(t, x∗)

q(x∗)

∣∣∣∣ ≤M(x∗, q)(∣∣ρ′(t)

∣∣ + |wt(t, x∗)|)

≤ M(x∗, q)(

max0≤t≤T

∣∣ρ′(t)∣∣ +max

0≤t≤Tmax0≤x≤L

|wt(t, x)|)

≤ M(x∗, q)(

max0≤t≤T

∣∣ρ′(t)∣∣ +max

0≤t≤T‖wt(t)‖ ◦

C2α[0,L]

),

(2.9)

for any t, t ∈ [0, T]. Using problem (1.1) and (2.4)–(2.7), one can show that w(t, x) is thesolution of the following problem:

∂w(t, x)∂t

= a(x)∂2w(t, x)∂x2

+ a(x)ρ(t) −w(t, x∗)

q(x∗)d2q(x)dx2

+ f(t, x), x ∈ (0, l), t ∈ (0, T],

∂w(t, x)∂t

= a(x)∂2w(t, x)∂x2

+ b(t, x)w(t, x)

+ρ(t) −w(t, x∗)

q(x∗)

(a(x)

d2q(x)dx2

+ b(t, x)q(x)

)+ g(t, x), x ∈ (l, L), t ∈ (0, T],

w(0, x) = ϕ(x), x ∈ [0, L],

wx(t, 0) = 0, w(t, L) = 0, t ∈ [0, T],

w(t, l+) = w(t, l−), wx(t, l+) = wx(t, l−), t ∈ [0, T],(2.10)

under the same assumptions on q(x). Estimate (2.9) and the following theorem conclude theproof of Theorem 2.1.


‖wt‖ ◦C

2α[0,L]

≤ M(a, δ, σ, α, x∗, q, T

)

×(∥∥ϕ

∥∥ ◦C

2α+2[0,L]

+ ‖F1‖C([0,T],

◦C

2α[0,L])

+∥∥ρ∥∥C[0,T]

) (2.11)

holds.

Proof. Let us rewrite problem (2.10) in the abstract form as an initial-value problem:

wt +Aw + Bw =(aq′′ − σq)ρ(t) −w(t, x∗)

q∗+ F1(t) + F2(t), 0 < t ≤ T,

w(0) = ϕ

(2.12)


in the Banach space E =◦C[0, L]. Here, the positive operator A is defined by

Au = −a(x)∂2u(t, x)∂x2

+ σu, (2.13)

with

D(A) ={u(x) : u, u′, u′′ ∈ C[0, L], ux(0) = u(L) = 0

}, (2.14)

and for every fixed t ∈ [0, T], the differential operator B is given by the formula

B(t)u =

{−σun, 0 ≤ x < l,−(σ − b(t))u, b(t) = b(t, x), l < x ≤ L. (2.15)

Here, σ is a positive constant. The right-hand side functions are defined by

F1(t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

f(t), 0 ≤ x < l,0,

g(t) +ρ(t)q∗

b(t)q, l < x ≤ L,

F2(t) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0, 0 ≤ x < l,0,

−w(t, x∗)q∗

b(t)q, l < x ≤ L,

(2.16)

where f(t) = f(t, x), g(t) = g(t, x), b(t) = b(t, x) are known, and w(t) = w(t, x) is unknown

abstract functions defined on [0, T] with values in E =◦C[0, L], w(t, x∗) is unknown scalar

function defined on [0, T], q = q(x), q′′ = q′′(x), ϕ = ϕ(x), and a = a(x) are elements of

E =◦C[0, L], and q∗ = q(x∗) is a number.

It is known that operator-A generates an analytic semigroup exp{−tA}(t > 0) and thefollowing estimate holds:

∥∥Aα exp{−tA}∥∥E→E ≤Me−δtt−α, 0 ≤ α ≤ 1, (2.17)

where t, δ,M > 0 [29].


By the Cauchy formula, the solution can be written as

w(t) = e−tAϕ −∫ t

0e−(t−s)A

aq′′ − σqq∗

w(s, x∗)ds

+∫ t

0e−(t−s)A

ρ(s)(aq′′ − σq)

q∗ds +

∫ t

0e−(t−s)AF1(s)ds

+∫ t

0e−(t−s)AF2(s)ds −

∫ t

0e−(t−s)AB(s)w(s)ds.

(2.18)

Then, the following presentation of the solution of abstract problem (2.12) exists:

Aw(t) = Ae−tAϕ −∫ t

0Ae−(t−s)A


w(s, x∗)ds

+∫ t

0Ae−(t−s)A


q∗ds +

∫ t

0Ae−(t−s)AF1(s)ds

+∫ t

0Ae−(t−s)AF2(s)ds +

∫ t

0Ae−(t−s)AB(s)w(s)ds =

6∑

k=1

Gk(t).

(2.19)

Here,

G1(t) = Ae−tAϕ,

G2(t) = −∫ t

0Ae−(t−s)A


w(s, x∗)ds,

G3(t) = −∫ t

0Ae−(t−s)A


q∗ds,

G4(t) =∫ t

0Ae−(t−s)AF1(s)ds,

G5(t) =∫ t

0Ae−(t−s)AF2(s)ds,

G6(t) =∫ t

0Ae−(t−s)AB(s)w(s)ds.

(2.20)

From the fact that the operators R, exp{−λA} and A commute, it follows that [29]

‖R‖Eα →Eα≤ ‖R‖E→E. (2.21)


Now, we estimate Gk(t) for k = 1, 2, . . . , 5 separately. Applying the definition of norm of thespaces Eα and estimate (2.21), we get

‖G1(t)‖Eα =∥∥∥Ae−tAϕ

∥∥∥Eα

≤∥∥∥e−tA

∥∥∥Eα →Eα

∥∥Aϕ∥∥Eα

≤∥∥∥e−tA

∥∥∥E→E

∥∥Aϕ∥∥Eα. (2.22)

Then, using estimate (2.17) for α = 0, we reach to

‖G1(t)‖Eα ≤M1∥∥Aϕ

∥∥Eα, (2.23)

for any t, t ∈ [0, T].Let us estimate G2(t):

‖G2(t)‖Eα =∥∥∥∥∥

∫ t

0Ae−(t−s)A


w(s, x∗)ds

∥∥∥∥∥

≤∫ t

0

∥∥∥∥Ae−(t−s)Aaq

′′ − σqq∗

∥∥∥∥Eα

|w(s, x∗)|ds.(2.24)

By the definition of norm of the spaces Eα, we have that

∫ t

0


′′ − σqq∗

∥∥∥∥Eα

ds =∫ t

0


′′ − σqq∗

∥∥∥∥E

ds

+ supλ>0

∫ t

0

∥∥∥∥λ1−αAe−λAAe−(t−s)A


∥∥∥∥E

ds.

(2.25)

Let us estimate the first term. From the definition of norm of the spaces Eα it follows that

∫ t

0


′′ − σqq∗

∥∥∥∥E

ds =∫ t

0(t − s)α−1

∥∥∥∥(t − s)1−αAe−(t−s)Aaq′′ − σq

q∗

∥∥∥∥E

ds

≤∫ t

0(t − s)α−1ds

∥∥∥∥aq′′ − σq

q∗

∥∥∥∥Eα

≤ Tα

α

∥∥∥∥aq′′ − σq

q∗

∥∥∥∥Eα

=M2(a, σ, α, x∗, q, T

).

(2.26)


Using estimate (2.17), we obtain

∫ t

0



∥∥∥∥E

ds ≤∫ t

0

22−αλ1−α

(λ + t − s)2−αds

∥∥∥∥λ + t − s

2Ae−((λ+t−s)/2)A

∥∥∥∥E→E

×∥∥∥∥∥

(λ + t − s

2

)1−αAe−((λ+t−s)/2)A


∥∥∥∥∥E

≤ M3(α)∥∥∥∥aq′′ − σq

q∗

∥∥∥∥Eα

∫ t

0

λ1−α

(λ + t − s)2−αds

≤ M3(α)∥∥∥∥aq′′ − σq

q∗

∥∥∥∥Eα

(λ1−α

(1 − α)(λ + t)1−α

),

(2.27)

for any λ > 0. From that it follows

supλ>0

∫ t

0



∥∥∥∥E

ds

≤M3(α)∥∥∥∥aq′′ − σq

q∗

∥∥∥∥Eα

1(1 − α) =M4

(a, σ, α, x∗, q

).

(2.28)

Then, we get

∫ t

0


′′ − σqq∗

∥∥∥∥Eα

ds ≤M5(a, σ, α, x∗, q, T

)(2.29)

‖G2(t)‖Eα ≤M6(a, σ, α, x∗, q, T

) ∫ t

0|w(s, x∗)|ds. (2.30)

Using definitions of norm of spaces E and Eα and estimate (2.21), we obtain that

|w(s, x∗)| ≤ ‖w‖E ≤ ‖w‖Eα =∥∥∥A−1Aw

∥∥∥Eα

≤∥∥∥A−1

∥∥∥E→E

‖Aw‖Eα ≤M‖Aw‖Eα , (2.31)

‖G2(t)‖Eα ≤M7(a, σ, α, x∗, q, T

)‖Aw‖Eα , (2.32)

for any t ∈ [0, T].


From estimate (2.29), the estimate of G3(t) is as follows:

‖G3(t)‖Eα =∥∥∥∥∥

∫ t

0Ae−(t−s)Aρ(s)


ds

∥∥∥∥∥Eα

≤∫ t

0


′′ − σqq∗

∥∥∥∥Eα

ds∥∥ρ∥∥C[0,T]

≤M8(a, σ, α, x∗, q, T

)∥∥ρ∥∥C[0,T].

(2.33)

Now, let us estimate G4(t). By the definition of the norm of the spaces Eα, we get

‖G4(t)‖Eα =∥∥∥∥∥

∫ t


∥∥∥∥∥Eα

=

∥∥∥∥∥

∫ t


∥∥∥∥∥E

+ supλ>0

λ1−α∥∥∥∥∥Ae

−λA∫ t


∥∥∥∥∥E

.

(2.34)

Equation (2.2) yields that

∥∥∥∥∥

∫ t


∥∥∥∥∥E

=∫ t

0(t − s)α−1

∥∥∥(t − s)1−αAe−(t−s)AF1(s)∥∥∥Eds

≤∫ t

0(t − s)α−1ds‖F1‖C(Eα) =

tα

α‖F1‖C(Eα) ≤M9(α, T)‖F1‖C(Eα).

(2.35)

Now, we consider the second term. Using (2.2), we get

λ1−α∥∥∥∥∥Ae

−λA∫ t


∥∥∥∥∥E

≤ λ1−α∫ t

0

(t − s + λ

2

)α−1( t − s + λ2

)−1

×∥∥∥∥t − s + λ

2Ae−((t−s+λ)/2)A

∥∥∥∥E→E

×∥∥∥∥∥

(t − s + λ

2

)1−αAe−((t−s+λ)/2)AF1(s)

∥∥∥∥∥E

ds

≤ M10λ1−α∫ t

0

(t − s + λ

2

)α−2‖F1‖Eαds

≤ M11λ1−α∫ t

0

(t − s + λ

2

)α−2ds‖F1‖C(Eα),

(2.36)


for any λ > 0. Then,

supλ>0

λ1−α∥∥∥∥∥Ae

−λA∫ t


∥∥∥∥∥E

≤ M921−α

1 − α ‖F1‖C(Eα) =M11(α)‖F1‖C(Eα).(2.37)

Combining estimates (2.35) and (2.37), we obtain

‖G4(t)‖Eα ≤M12(α, T)‖F1‖C(Eα). (2.38)

The estimate of G5(t) is as follows. Since operators A and e−tA commute, we can write that

‖G5(t)‖Eα ≤∥∥∥∥∥

∫ t


∥∥∥∥∥Eα

≤M13(α, T)‖Aw‖Eα . (2.39)

Let us estimate G6(t) :

‖G6(t)‖Eα =∥∥∥∥∥

∫ t

0Ae−(t−s)AB(s)w(s)ds

∥∥∥∥∥Eα

=

∥∥∥∥∥

∫ t

0Ae−(t−s)AB(s)A−1Aw(s)ds

∥∥∥∥∥Eα

≤∫ t

0

∥∥∥Ae−(t−s)AB(s)A−1∥∥∥Eα →Eα

‖Aw(s)‖Eαds.

(2.40)

Since

∥∥∥e−tA∥∥∥Eα →Eα

≤∥∥∥e−tA

∥∥∥E→E

≤Me−δt,

∥∥∥AB(s)A−1∥∥∥Eα →Eα

≤M,

(2.41)

we get

‖G6(t)‖Eα ≤M14

∫ t

0‖Aw(s)‖Eαds. (2.42)


Finally combining estimates (2.23), (2.32), (2.33), (2.38), (2.39), and (2.42), we get

‖Aw‖Eα ≤ M1∥∥Aϕ

∥∥Eα

+M8(a, σ, α, x∗, q, T

)∥∥ρ∥∥C[0,T]

+M12(α, T)‖F1‖C(Eα) +M15

∫ t

0‖Aw(s)‖Eαds,

(2.43)

whereM15 =M7 +M13 +M14.Using Gronwall’s inequality, we can write

‖Aw‖Eα ≤ eM15T[M1∥∥Aϕ

∥∥Eα

+M8(a, σ, α, x∗, q, T

)∥∥ρ∥∥C[0,T]

+M12(α, T)‖F1‖C(Eα)].

(2.44)

Applying the formulas

w(t, x∗) = w(0, x∗) +∫ t

0wz(z, x∗)dz = ϕ(x∗) +

∫ t

0wz(z, x∗)dz,

∣∣ϕ(x∗)∣∣ ≤ max

0≤x≤L

∣∣ϕ(x)∣∣ =∥∥ϕ∥∥E ≤ ∥∥ϕ∥∥Eα

≤∥∥∥A−1

∥∥∥Eα →Eα

∥∥Aϕ∥∥Eα

≤M∥∥Aϕ∥∥Eα

(2.45)

and the triangle inequality, we can write

∥∥∥∥∥

(aq′′ − σq)

q∗(ρ(t) −w(t, x∗)

)∥∥∥∥∥Eα

≤∥∥∥∥∥

(aq′′ − σq)

q∗

∥∥∥∥∥Eα

(∥∥ρ∥∥C[0,T] +M13

∥∥Aϕ∥∥Eα

+∫ t

0‖wz‖Eαdz

).

(2.46)

Using boundedness of B, problem (2.12), and estimate (2.46), we have

‖wt‖Eα ≤ ‖Aw‖Eα + ‖Bw‖Eα + ‖F1‖C(Eα)

+

∥∥∥∥∥

(aq′′ − σq)

q∗

∥∥∥∥∥Eα

(∥∥ρ∥∥C[0,T] +M13

∥∥Aϕ∥∥Eα

+∫ t

0‖wz‖Eαdz

).

(2.47)

So, Gronwall’s inequality and the following theorem finish the proof of Theorem 2.3.

Theorem 2.3 (see [29]). For 0 < α < 1/2, the spaces Eα (C[0, L], A) and C2α[0, L] coincide andtheir norms are equivalent.


2.2. Difference Case

For the approximate solution of problem (1.1), the Rothe difference scheme

ukn − uk−1n

τ= a(xn)


h2+ pkqn + f(tk, xn),

1 ≤ k ≤N, 1 ≤ n ≤Ml − 1, Mlh = l, Nτ = T,

ukn − uk−1n

τ= a(xn)


h2+ b(tk, xn)ukn + p

kqn + g(tk, xn),

1 ≤ k ≤N, Ml + 1 ≤ n ≤M, Mh = L, Nτ = T,

u0n = ϕ(xn), 0 ≤ n ≤M,

uk1 − uk0 = ukM = 0, 0 ≤ k ≤N,

ukMl+1 − ukMl= ukMl

− ukMl−1, 0 ≤ k ≤N,

ukx∗/h = uks = ρ(tk), 0 ≤ k ≤N, 0 ≤ s ≤ M,

(2.48)

where pk = p(tk), qn = q(xn), xn = nh, and tk = kτ is constructed. Here, qs /= 0 and q1 − q0 =qM = 0 are assumed. x represents the floor function of x.

With the help of a positive operatorA, we introduce the fractional spaces E′α, 0 < α < 1,

consisting of all v ∈ E for which the following norm is finite:

‖v‖E′α= ‖v‖E + sup

λ>0λα∥∥∥A(λ +A)−1v

∥∥∥E. (2.49)

To formulate our results, we introduce the Banach space◦Ch

α

=◦Cα

[0, L]h, α ∈ (0, 1), of all gridfunctions φh = {φn}M−1

n=1 defined on

[0, L]h = {xn = nh, 0 ≤ n ≤M,Mh = L}, (2.50)

with φ1 − φ0 = φM = 0 equipped with the norm

∥∥φh∥∥ ◦Cα

h

=∥∥φh∥∥Ch

+ sup1≤n<n+r≤M

∣∣φn+r − φn∣∣(rh)−α,

∥∥φh∥∥Ch

= max1≤n≤M

∣∣φn∣∣.

(2.51)


Moreover, Cτ(E) = C([0,T]τ , E) is the Banach space of all grid functions φτ = {φ(tk)}N−1k=1

defined on [0, T]τ = {tk = kτ, 0 ≤ k ≤N,Nh = T}with values in E equipped with the norm

∥∥φτ∥∥Cτ (E)

= max1≤k≤N

∥∥φ(tk)∥∥E. (2.52)

Then, the following theorem on well-posedness of problem (2.48) is established.

Theorem 2.4. For the solution of problem (2.48), the following coercive stability estimates

∥∥∥∥∥∥

{uhk − uhk−1τ

}N

k=1

∥∥∥∥∥∥Cτ (

◦C

2α

h )

+∥∥∥∥{D2hu

hk

}Nk=1

∥∥∥∥Cτ (

◦C

2α

h )

≤M(q, s)∥∥∥∥∥

{ρ(tk) − ρ(tk−1)

τ

}N

k=1

∥∥∥∥∥C[0,T]τ

+M(a, φ, α, T

)(∥∥∥D2

hϕh∥∥∥ ◦C

2α

h

+∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (

◦C

2α

h )+∥∥ρτ∥∥C[0,T]τ

),

∥∥pτ∥∥C[0,T]τ

≤ M(q, s)∥∥∥∥∥

{ρ(tk) − ρ(tk−1)

τ

}N

k=1

∥∥∥∥∥C[0,T]τ

+M(a, φ, α, T

)[∥∥∥D2

hϕh∥∥∥ ◦C

2α

h

+∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (

◦C

2α

h )+∥∥ρτ∥∥C[0,T]τ

]

(2.53)

hold. Here,

Fh1 (tk) =

⎧⎪⎪⎨

⎪⎪⎩

f(tk, xn)0

b(tk, xn)ρ(tk)qs

qn + g(tk, xn)

⎫⎪⎪⎬

⎪⎪⎭

M−1

n=1

, ϕh ={ϕ(xn)

}M−1n=1 ,

ρτ ={ρ(tk)

}Nk=0, D2

huh ={un+1 − 2un + un−1

h2

}M−1

n=1

a =1qs

(aD2

hqh − σqh

).

(2.54)

Proof. The solution of problem (2.48) is searched in the following form:



where

ηk =k∑

i=1

piτ, 1 ≤ k ≤N, η0 = 0. (2.56)

Difference derivatives of (2.55) can be written as

ukn − uk−1n

τ=ηk − ηk−1

τqn +

wkn −wk−1

n

τ= pkqn +

wkn −wk−1

n

τ,


h2= ηk

qn+1 − 2qn + qn−1h2

+wkn+1 − 2wk

n +wkn−1

h2,

(2.57)

for any n, 1 ≤ n ≤M − 1. At the interior grid point s = x∗/h, we have that

uks = ηkqs +wks = ρ(tk),

ηk =ρ(tk) −wk

s

qs.

(2.58)

Taking the difference derivative of the last equality and using the triangle inequality, weobtain

pk =1qs

(ρ(tk) − ρ(tk−1)

τ− wk

s −wk−1s

τ

), (2.59)

∣∣∣pk∣∣∣ ≤ M

(q, s)(∣∣∣∣

ρ(tk) − ρ(tk−1)τ

∣∣∣∣ +

∣∣∣∣∣wks −wk−1

s

τ

∣∣∣∣∣

)

≤ M(q, s)(

max1≤k≤N

∣∣∣∣ρ(tk) − ρ(tk−1)

τ

∣∣∣∣ + max1≤k≤N

max0≤s≤M

∣∣∣∣∣wks −wk−1

s

τ

∣∣∣∣∣

)

≤ M(q, s)⎛

⎝max1≤k≤N

∣∣∣∣ρ(tk) − ρ(tk−1)

τ

∣∣∣∣ + max1≤k≤N

∥∥∥∥∥whk −wh

k−1τ

∥∥∥∥∥ ◦C

2α

h

⎞

⎠,

(2.60)

for any k, 1 ≤ k ≤N.


In estimate (2.60), {whk}

N

k=0 is the solution of the following difference scheme:

wkn −wk−1

n

τ= a(xn)

wkn+1 − 2wk

n +wkn−1

h2+ a(xn)

ρ(tk) −wks

qs

qn+1 − 2qn + qn−1h2

+ f(tk, xn), 1 ≤ k ≤N, 1 ≤ n ≤Ml − 1, Mlh = l, Nτ = T,

wkn −wk−1

n

τ= a(xn)

wkn+1 − 2wk

n +wkn−1

h2+ a(xn)

ρ(tk) −wks

qs

qn+1 − 2qn + qn−1h2

b(tk, xn)wkn + b(tk, xn)

ρ(tk) −wks

qsqn + g(tk, xn),

1 ≤ k ≤N, Ml + 1 ≤ n ≤M − 1, Mh = L, Nτ = T,

w0n = ϕ(xn), 0 ≤ n ≤M,

wk1 −wk

0 = wkM = 0, 0 ≤ k ≤N,

wkMl+1 −wk

Ml= wk

Ml−wk

Ml−1, 0 ≤ k ≤N,

(2.61)

where xn = nh, tk = kτ . Therefore, estimate (2.60) and the following theorem finish the proofof Theorem 2.5.


∥∥∥∥∥∥

{whk−wh

k−1τ

}N

k=1

∥∥∥∥∥∥Cτ (

◦C

2α

h )

≤ M(a, φ, α, T

)

×(∥∥∥ϕh

∥∥∥ ◦C

2α

h

+∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (

◦C

2α

h )+∥∥ρτ∥∥C[0,T]τ

) (2.62)

holds.

Proof. We can rewrite difference scheme (2.61) in the abstract form:

whk−wh

k−1τ

+Axhw

hk + B

xhw

hk =(aqn+1 − 2qn + qn−1

h2− σq

)ρ(tk) −wk

s

qs

+ Fh1 (tk) + Fh2 (tk), tk = kτ, 1 ≤ k ≤N, Nτ = T,

wh0 = ϕh,

(2.63)

in a Banach space E =◦C[0, l]h with the positive operator Ax

hdefined by

Axhu

h ={−a(xn)un+1 − 2un + un−1

h2+ σu

}M−1

n=1, (2.64)


acting on grid functions uh such that it satisfies the condition

u1 − u0 = uM = 0. (2.65)

For every fixed t ∈ [0, T], the difference operators Bxh(t) are given by the formula

Bxh(t)uh =

⎧⎪⎪⎨

⎪⎪⎩

−σun, 1 ≤ n ≤Ml,

−(σ − bn(t))un, bn(t) = b(t, xn),xn = nh, Ml + 1 ≤ n ≤M − 1,

⎫⎪⎪⎬

⎪⎪⎭

M−1

n=1

, (2.66)

where σ is a positive constant and the right-hand side functions are

Fh1 (tk) =

⎧⎪⎨

⎪⎩

f(tk), 1 ≤ n ≤Ml,

b(tk)qnρ(tk)qs

+ g(tk), Ml + 1 ≤ n ≤M − 1,

⎫⎪⎬

⎪⎭

M−1

n=1

,

Fh2 (tk) =

⎧⎪⎨

⎪⎩

0, 1 ≤ n ≤Ml,

−b(tk)qnwks

qs, Ml + 1 ≤ n ≤M − 1,

⎫⎪⎬

⎪⎭

M−1

n=1

.

(2.67)

Let us denote R = (I + τAxh)−1. In problem (2.63), we have that

whk = Rwh

k−1 + Rτ

(aqn+1 − 2qn + qn−1

h2ρ(tk) −wk

s

qs− Bxh(t)wh

k + Fh1 (tk) + F

h2 (tk)

), (2.68)

for all k, 1 ≤ k ≤N. By recurrence relations, we get

whk = Rkϕh +

k∑

m=1

Rk−m+1 τ

qsaqn+1 − 2qn + qn−1

h2ρ(tm)

−k∑

m=1

Rk−m+1 τ


h2wms

−k∑

m=1

Rk−m+1Bxh(t)τwms +

k∑

m=1

Rk−m+1τFh1 (tm) +k∑

m=1

Rk−m+1τFh2 (tm).

(2.69)


Then, the following presentation of the solution of problem (2.63)

Axhw

hk = Ax

hRkϕh +

k∑

m=1

AxhR

k−m+1 τ


h2ρ(tm)

−k∑

m=1

AxhR

k−m+1 τ


h2wms

−k∑

m=1

AxhR

k−m+1Bxh(t)τwms +

k∑

m=1

AxhR

k−m+1τFh1 (tm)

+k∑

m=1

AxhR

k−m+1τFh2 (tm) =6∑

k=1

Jk

(2.70)

is obtained. Here,

Jk1 = AxhR

kϕh,

Jk2 =k∑

m=1

AxhR

k−m+1 τ


h2ρ(tm),

Jk3 = −k∑

m=1

AxhR

k−m+1 τ


h2wms ,

Jk4 = −k∑

m=1

AxhR

k−m+1Bxh(t)τwms ,

Jk5 =k∑

m=1

AxhR

k−m+1τFh1 (tm),

Jk6 =k∑

m=1

AxhR

k−m+1τFh2 (tm).

(2.71)

Now, let us estimate Jkr for r = 1, 2, . . . , 6 separately. We start with Jk1 . Applying thedefinition of norm of the spaces E

′α, we get

∥∥∥Jk1∥∥∥E′α

=∥∥∥RkAx

hϕh∥∥∥E′α

≤∥∥∥Rk∥∥∥E′α →E

′α

∥∥∥Axhϕ

h∥∥∥E′α

≤∥∥∥Rk∥∥∥E→E

∥∥∥Axhϕ

h∥∥∥E′α

.

(2.72)

Using estimate

∥∥∥Rk∥∥∥E→E

≤M, (2.73)


we get

∥∥∥Jk1∥∥∥E′α

≤M1

∥∥∥Axhϕ

h∥∥∥E′α

, (2.74)

for any k, 1 ≤ k ≤N.Let us estimate Jk2 :

∥∥∥Jk2∥∥∥E′α

=

∥∥∥∥∥

k∑

m=1

AxhR

k−m+1 τ


h2ρ(tm)

∥∥∥∥∥E′α

≤ max1≤m≤N

ρ(tm)k∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E′α

,

(2.75)

where

a = aqn+1 − 2qn + qn−1

qsh2. (2.76)

From the definition of norm of the spaces E′α, it follows that

∥∥∥∥∥

k∑

m=1

AxhR

k−m+1τa

∥∥∥∥∥E′α

≤k∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E

+ supλ>0

k∑

m=1

∥∥∥λαAxh

(λ +Ax

h

)−1AxhR

k−m+1τa∥∥∥E.

(2.77)

Let us estimate each term separately. We divide first term into two parts:

k∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E=

k−1∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E+∥∥Ax

hRτa∥∥E. (2.78)

In the first part, by the definition of norm of the spaces E′α and the identity (see [29])

(I + τA)−k =1

(k − 1)!

∫∞

0tk−1e−t exp{−τtA}dt, k ≥ 2, (2.79)

we deduce that

k−1∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E≤

k−1∑

m=1

τ

(k −m)!

∫∞

0

tk−m

(τt)1−αe−t∥∥∥(τt)1−αAx

he−τtAa

∥∥∥Edt

≤ ‖a‖E′α

k−1∑

m=1

τ

(k −m)!

∫∞

0

tk−m

(τt)1−αe−tdt


= ‖a‖E′α

k−1∑

m=1

τα

(k −m)!

∫∞

0tk−m−1+αe−tdt

= ‖a‖E′α

k−1∑

m=1

τα

(k −m)!

∫∞

0t(k−m−1)α+αe−αtt(k−m−1)(1−α)e−(1−α)tdt.

(2.80)

The Holder inequality with p = 1/α, q = 1/(1 − α) and the definition of the gamma functionyield that

k−1∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E≤ ‖a‖E′

α

k−1∑

m=1

τα

(k −m)!

(∫∞

0

(t(k−m−1)α+αe−αt

)1/αdt

)α

×(∫∞

0

(t(k−m−1)(1−α)e−(1−α)t

)1/(1−α)dt

)1−α

= ‖a‖E′α

k−1∑

m=1

τα

(k −m)!

(∫∞

0tk−me−tdt

)α(∫∞

0tk−m−1e−tdt

)1−α

= ‖a‖E′α

k−1∑

m=1

τα

(k −m)!(Γ(k −m + 1))α(Γ(k −m))1−α.

(2.81)

By the fact that Γ(n) = (n − 1)! and Γ(n) = (n − 1)Γ(n − 1), we get

k−1∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E≤ ‖a‖E′

α

k−1∑

m=1

τα

(k −m)!(k −m)αΓ(k −m)

= ‖a‖E′α

k−1∑

m=1

τα

(k −m)1−α= ‖a‖E′

α

k−1∑

m=1

τ

((k −m)τ)1−α

≤ M2‖a‖E′α

∫kτ

0

1

(kτ − s)1−αds =M2‖a‖E′

α

([− (kτ − s)α

α

]kτ

0

).

(2.82)

So, we have that

k−1∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E≤M2‖a‖E′

α

(kτ)α

α≤M3(α, T)‖a‖E′

α. (2.83)

In the second part, we have that

∥∥AxhRτa

∥∥E≤ ∥∥Ax

hRτ∥∥E→E

‖a‖E ≤M4‖a‖E′α. (2.84)


Combining estimates (2.83) and (2.84), we obtain

k∑

m=1

∥∥∥AxhR

k−m+1τa∥∥∥E≤M5(α, T)‖a‖E′

α. (2.85)

Let us estimate the second term. From the Cauchy-Riesz formula (see [29])

f(A) =1

2πi

∫

Γf(z)(z −A)−1dz, (2.86)

it follows that

k∑

m=1

λαAxh

(λ +Ax

h

)−1Rk−m+1Ax

hτa

=1

2πi

∫

S1∪S2

k∑

m=1

z

(1 + z)k−m+1

λα

λ + zτ−1Axh

(z − τAx

h

)−1adz

=1

2πi

∫

S1∪S2

k∑

m=1

(z/τ)−α

(1 + z)k−m+1

λα

λτ + z

(zτ

)αAxh

(zτ−Ax

h

)−1adz.

(2.87)

Since z = ρe±iφ, with |φ| ≤ π/2, the estimate (see [29])

∥∥∥(λ −A)−1∥∥∥E→E

≤ M(φ)

1 + |λ| (2.88)

yields

∥∥∥∥(zτ

)αAxh

(zτ−Ax

h

)−1a

∥∥∥∥E

≤M6

∥∥∥∥(ρτ

)αAxh

(ρτ+Ax

h

)−1a

∥∥∥∥E

,

1|λτ + z| ≤

M6

λτ + ρ.

(2.89)

Hence,

∥∥∥∥∥

k∑

m=1

λαAxh

(λ +Ax

h

)−1Rk−m+1Ax

hτa

∥∥∥∥∥E

≤M6

∫∞

0

k∑

m=1

ρ1−α[1 + 2ρ cosφ + ρ2

](k−m+1)/2

(λτ)αdρλτ + ρ

‖a‖E′α.

(2.90)


Summing the geometric progression, we get

∥∥∥∥∥

k∑

m=1

λαAxh

(λ +Ax

h

)−1Rk−m+1Ax

hτa

∥∥∥∥∥E

≤ M6

∫∞

0

k∑

m=1

ρ1−α[1 + 2ρ cosφ + ρ2

]1/2

×(1 − 1[1 + 2ρ cosφ + ρ2

]1/2

)−1(λτ)αdρλτ + ρ

‖a‖E′α

≤ M6

∫∞

0

(λτ)ακ(ρ)dρ

(λτ + ρ

)ρα

‖a‖E′α.

(2.91)

Since the function

κ(ρ)=

ρ[1 + 2ρ cosφ + ρ2

]1/2 − 1=

1 +[1 + 2ρ cosφ + ρ2

]1/2

2 cosφ + ρ(2.92)

does not increase for ρ ≥ 0, we have κ(0) = 1/ cosφ ≥ κ(ρ) for all ρ > 0. Consequently,

∥∥∥∥∥

k∑

m=1

λαAxh

(λ +Ax

h

)−1Rk−m+1Ax

hτa

∥∥∥∥∥E

≤ M6

cosφ

∫∞

0

(λτ)αdρ(λτ + ρ

)ρα

‖a‖E′α

(2.93)

for any λ > 0. Hence,

supλ>0

∥∥∥∥∥

k∑

m=1

λαAxh

(λ +Ax

h

)−1Rk−m+1Ax

hτa

∥∥∥∥∥E

≤M7(φ, α)‖a‖E′

α. (2.94)

Then, using estimates (2.85) and (2.94), we get

∥∥∥∥∥

k∑

m=1

(Rk−m+1 − Rk−m

)a

∥∥∥∥∥E′α

≤M8(φ, α, T

)‖a‖E′α, (2.95)

∥∥∥Jk2∥∥∥E′α

≤ max1≤m≤N

ρ(tm)M8(φ, α, T

)‖a‖E′α. (2.96)

Now, let us estimate Jk3 :

∥∥∥Jk3∥∥∥E′α

=

∥∥∥∥∥−k∑

m=1

AxhR

k−m+1 τ


h2wms

∥∥∥∥∥E′α

≤k∑

m=1

∥∥∥∥AxhR

k−m+1 τ

qs

(aqn+1 − 2qn + qn−1

h2− σq

)∥∥∥∥E′α

|wms |.

(2.97)


Since

|wms | ≤ max

0≤s≤M|wm

s | =∥∥∥wh∥∥∥E≤∥∥∥wh∥∥∥E′α

≤∥∥∥(Axh

)−1∥∥∥E′α →E′

α

∥∥∥Axhw

h∥∥∥E′α

≤M∥∥∥Ax

hwh∥∥∥E′α

,

(2.98)

and using estimate (2.95), we obtain

‖J3‖E′α≤M9

(φ, α, T, τ

)‖a‖E′α

k∑

m=1

∥∥∥Axhw

h∥∥∥E′α

τ. (2.99)

Jk4 can be estimated as follows:

∥∥∥Jk4∥∥∥E′α

=

∥∥∥∥∥−k∑

m=1

AxhR

k−m+1Bxh(t)τwms

∥∥∥∥∥E′α

=

∥∥∥∥∥

k∑

m=1

AxhR

k−m+1Bxh(t)(Axh

)−1Axhτw

ms

∥∥∥∥∥E′α

≤k∑

m=1

∥∥∥AxhR

k−m+1Bxh(t)(Axh

)−1∥∥∥E′α →E′

α

∥∥Axhw

ms

∥∥E′ατ.

(2.100)

From

∥∥∥Rk∥∥∥E′α →E

′α

≤∥∥∥Rk∥∥∥E→E

≤M,

∥∥∥AxhB

xh(t)(Axh

)−1∥∥∥E′α →E′

α

≤M(2.101)

it follows that

∥∥∥Jk4∥∥∥E′α

≤M10

k∑

m=1

∥∥Axhw

ms

∥∥E′ατ. (2.102)

The estimations of Jk5 and Jk6 are as follows. By the definition of the norm of the spaces E′α and

(2.95), we get

∥∥∥Jk5∥∥∥E′α

≤∥∥∥∥∥

k∑

m=1

AxhR

k−m+1τFh1 (tm)

∥∥∥∥∥E′α

+M11(φ, α, T

)∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (E′

α),

(2.103)


‖J6‖E′α≤M12

(φ, α) k∑

m=1

∥∥∥Axhw

h∥∥∥E′α

τ. (2.104)

Combining estimates (2.74), (2.96), (2.99), and (2.102)–(2.104), we get

∥∥∥Axhw

hk

∥∥∥E′α

≤ M1

∥∥∥Axhϕ

h∥∥∥E′α

+ max1≤m≤N

ρ(tm)M8(φ, α, T

)‖a‖E′α

+(M9(φ, α, T, τ

)‖a‖E′α+M10 +M12

(φ, α)) k∑

m=1

∥∥∥Axhw

hk

∥∥∥Eατ

+M11(φ, α, T

)∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (E′

α).

(2.105)

Using the discrete analogue of Gronwall’s inequality, we get

∥∥∥Axhw

hk

∥∥∥E′α

≤ eM13(a,φ,α,T,τ)

×[M1

∥∥∥Axhϕ

h∥∥∥E′α

+M14(a, φ, α, T

)∥∥ρτ∥∥C[0,T]τ

+M11(φ, α, T

)∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (E′

α)

].

(2.106)

It follows from (2.63) and the triangle inequality that

∥∥∥∥∥whk−wh

k−1τ

∥∥∥∥∥E′α

≤ eM12(a,φ,α,T)

×[M1

∥∥∥Axhϕ

h∥∥∥E′α

+M13(a, φ, α, T

)∥∥ρτ∥∥C[0,T]τ

+M11(φ, α, T

)∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (E′

α)

],

(2.107)

for every k, 1 ≤ k ≤N. Then, we have that

∥∥∥∥∥∥

{whk−wh

k−1τ

}N

k=1

∥∥∥∥∥∥Cτ (E′

α)

≤ M14(a, φ, α, T

)

×(∥∥∥Ax

hϕh∥∥∥E′α

+∥∥∥∥{Fh1 (tk)

}Nk=1

∥∥∥∥Cτ (E′

α)+∥∥ρτ∥∥C[0,T]τ

).

(2.108)

The following theorem finishes the proof of Theorem 2.6.


Theorem 2.6 (see [30]). For 0 < α < 1/2, the spaces E′α(C[0, L]h,A

xh) and C

2α[0, L]h coincide andtheir norms are equivalent.

3. Conclusion

Since artery disease caused by atherosclerosis is one of the most important causes of the deathin the world, investigation of the effect of flow over the glycocalyx takes an important place.The flow equations can be formulated as an inverse problem. Here, our aim is to give moredetailed understanding of the flow phenomena. Therefore, the well-posedness of the inverseproblem of reconstructing the right side of a parabolic equation was investigated. Further, anew computer code regarding the flow analysis for the unknown pressure difference will bewritten.

Acknowledgment

The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for hiscomments and suggestions to improve the quality of the paper.

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Research ArticleOn the Second Order of Accuracy Stable ImplicitDifference Scheme for Elliptic-Parabolic Equations

Allaberen Ashyralyev and Okan Gercek


Correspondence should be addressed to Okan Gercek, [email protected]

Received 7 April 2012; Accepted 24 April 2012


Copyright q 2012 A. Ashyralyev and O. Gercek. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We are interested in studying a second order of accuracy implicit difference scheme for the solutionof the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of thisdifference scheme is established. In an application, coercivity estimates in Holder norms forapproximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolicdifferential equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for mixed-type differential equa-tions have been studied extensively by various researchers (see, e.g., [1–19] and the referencestherein).

In [20], we considered the well-posedness of the followingmultipoint nonlocal bound-ary value problem:

−d2u(t)dt2

+Au(t) = g(t), (0 ≤ t ≤ 1),

du(t)dt

−Au(t) = f(t), (−1 ≤ t ≤ 0),

u(1) =J∑

i=1

αiu(λi) + ϕ,

−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,

(1.1)


in a Hilbert spaceH with the self-adjoint positive definite operator A under assumption

J∑

i=1

|αi| ≤ 1. (1.2)

The well-posedness of multipoint nonlocal boundary value problem (1.1) in Holderspaces with a weight was established. Moreover, coercivity estimates in Holder norms forthe solutions of nonlocal boundary value problems for elliptic-parabolic equations wereobtained.

In [21], we studied the well-posedness of the first order of accuracy difference schemefor the approximate solution of boundary value problem (1.1) under assumption (1.2).

Throughout this work, we consider the following second order of accuracy differencescheme:

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk,

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

τ−1(uk − uk−1) −(I +

τ

2A)Auk−1 =

(I +

τ

2A)fk, fk = f(tk−1/2),

tk−1/2 =(k − 1

2

)τ, −(N − 1) ≤ k ≤ 0,

u2 − 4u1 + 3u0 = −3u0 + 4u−1 − u−2,

uN =J∑

k=1

αi

(u[λi/τ] +

(λi −

[λiτ

]τ

)(f[λi/τ] +Au[λi/τ]

))+ ϕ,

(1.3)

for the approximate solution of boundary value problem (1.1) under assumption (1.2).The well-posedness of difference scheme (1.3) in Holder spaces with a weight is

established. As an application, the stability, almost coercivity stability, and coercivity stabilityestimates for solutions of second order of accuracy difference scheme for the approximatesolution of the nonlocal boundary elliptic-parabolic problem are obtained.

2. Main Theorems

Throughout the paper, H is a Hilbert space and we denote B = (1/2)(τA +√A(4 + τ2A)),

where A is a self-adjoint positive definite operator. Then, it is clear that B is the self-adjointpositive definite operator and B ≥ δ1/2I where δ > δ0 > 0, andR = (I+τB)−1, which is defined


on the whole space H, is a bounded operator. Here, I is the identity operator. The followingoperators

D =

(I + τA +

(τA)2

2

), G =

(I − τ2A

2

), P =

(I +

τ

2A), R = (I + τB)−1,

Tτ =

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1

−GKP−2(2I + τB)RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D −[λi/τ]u0

])−1

(2.1)

exist and are bounded for a self-adjoint positive operator A. Here,

B =12

(τA +

√A(4 + τ2A)

), K =

(I + 2τA +

54(τA)2

)−1. (2.2)

Furthermore, positive constants will be indicated by M which can differ in time. Onthe other hand Mi(α, β, . . .) is used to focus on the fact that the constant depends only onα, β, . . . and the subindex i is used to indicate a different constant.

First of all, let us start with some auxiliary lemmas from [16, 22–24] that are essentialbelow.

Lemma 2.1. For a self-adjoint positive operator A, the following estimates are satisfied:

∥∥∥Rk∥∥∥H→H

≤M1(δ)(1 + δτ)−k,

∥∥∥Dk∥∥∥H→H

≤M1(δ),

∥∥∥BRk∥∥∥H→H

≤ M1(δ)kτ

,∥∥∥P−1

∥∥∥H→H

≤M1(δ),∥∥∥ADk

∥∥∥H→H

≤ M1(δ)kτ

,

∥∥∥Dk − e−kτA∥∥∥H→H

≤ M1(δ)k2

,

∥∥∥∥(I − R2N

)−1∥∥∥∥H→H

≤M1(δ),

∥∥∥Rk − e−kτA1/2∥∥∥H→H

≤ M1(δ)k

, k ≥ 1, δ > 0.

(2.3)

From these estimates, it follows that

∥∥∥∥∥

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1 −GKP−2(2I + τB)

×RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D−[λi/τ]

])−1∥∥∥∥∥∥H→H

≤M2(δ).

(2.4)


Lemma 2.2. For any gk, 1 ≤ k ≤N −1 and fk,−N +1 ≤ k ≤ 0, the solution of problem (1.3) exists,and the following formulas hold:

uk =(I − R2N

)−1

×{[Rk − R2N−k

]u0 +

[RN−k − RN+k

]

×⎡

⎣n∑

i=1

αi

⎡

⎣(I +

(λi −

[λiτ

]τ

)A

)⎛

⎝D −[λi/τ]u0 − τ0∑

s=[λi/τ]+1

PDs−[λi/τ]fs

⎞

⎠

+(λi −

[λiτ

]τ

)f[λi/τ]

]+ ϕ

]

−[RN−k − RN+k

](I + τB)(2I + τB)−1B−1

N−1∑

s=1

[RN−s − RN+s

]gsτ

}

+ (I + τB)(2I + τB)−1B−1N−1∑

s=1

[R|k−s| − Rk+s

]gsτ, 1 ≤ k ≤N,

uk = D−ku0 − τ0∑

s=k+1

PDs−kfs, −N ≤ k ≤ −1,

u0 =12TτKP

−2

×{(

2I − τ2A)

×{(2 + τB)RN

×[

n∑

i=1

αi

[(I +

(λi −

[λiτ

]τ

)A

)

×⎛

⎝D −[λi/τ]u0 − τ0∑

s=[λi/τ]+1

PDs−[λi/τ]fs

⎞

⎠

+(λi −

[λiτ

]τ

)f[λi/τ]

]+ ϕ

]

−RN−1B−1N−1∑

s=1

[RN−s − RN+s

]gsτ +

(I − R2N

)B−1

N−1∑

s=1

Rs−1gsτ

}

+(I − R2N

)(I + τB)

(τB−1g1 − 4PB−1f0 + PDB−1f0 + PB−1f−1

)},

Tτ =

(I + B−1A

(I + τA +

τ

2P−2

)K(I − R2N−1

)+GKP−2R2N−1

− GKP−2(2I + τB)RN

[n∑

i=1

αi

(I +

(λi −

[λiτ

]τ

)A

)D −[λi/τ]u0

])−1.

(2.5)


Now, we study well-posedness of problem (1.3). Let Fτ(H) = F([a, b]τ ,H) be the

linear space of mesh functions ϕτ = {ϕk}ÑN

defined on [a, b]τ = {tk = kh, N ≤k ≤ Ñ, Nτ = a, Ñτ = b} with values in the Hilbert space H. Next, on Fτ(H)we denote C([a, b]τ ,H), Cα

0,1([−1, 1]τ ,H), Cα0,1([−1, 0]τ ,H), Cα

0 ([0, 1]τ ,H), Cα0,1([−1, 1]τ ,H),

and Cα0 ([−1, 0]τ ,H), 0 < α < 1 Banach spaces with the following norms:

∥∥ϕτ∥∥C([a,b]τ ,H) = max

Na≤k≤Nb

∥∥ϕk∥∥H,

∥∥ϕτ∥∥Cα

0,1([−1,1]τ ,H) =∥∥ϕτ

∥∥C([−1,1]τ ,H) + sup

−N≤k<k+r≤0

∥∥ϕk+r − ϕk∥∥E(−k)αr−α


∥∥ϕk+r − ϕk∥∥E((k + r)τ)α(N − k)αr−α,

∥∥ϕτ∥∥Cα

0 ([−1,0]τ ,H) =∥∥ϕτ

∥∥C([−1,0]τ ,H) + sup

−N≤k<k+r≤0

∥∥ϕk+r − ϕk∥∥E(−k)αr−α,

∥∥ϕτ∥∥Cα

0,1([0,1]τ ,H) =∥∥ϕτ

∥∥C([0,1]τ ,H)


‖ϕk+r − ϕk‖E((k + r)τ)α(N − k)αr−α,∥∥ϕτ

∥∥Cα

0,1([−1,1]τ ,H) =∥∥ϕτ

∥∥C([−1,1]τ ,H) + sup

−N≤k<k+2r≤0

∥∥ϕk+2r − ϕk∥∥E(−k)α(2r)−α


∥∥ϕk+r − ϕk∥∥E((k + r)τ)α(N − k)αr−α,

∥∥ϕτ∥∥Cα

0 ([−1,0]τ ,H) =∥∥ϕτ

∥∥C([−1,0]τ ,H)

+ sup−N≤k<k+2r≤0

∥∥ϕk+2r − ϕk∥∥E(−k)α(2r)−α, respectively.

(2.6)

Theorem 2.3. Nonlocal boundary value problem (1.3) is stable in C([−1, 1]τ ,H) space.

Proof. By [22], we have

∥∥∥{uk}N−11

∥∥∥C([0,1]τ ,H)

≤M3(δ)[∥∥gτ

∥∥C([0,1]τ ,H) + ‖ξ‖H +

∥∥ψ∥∥H

], (2.7)

for the solution of the following boundary value problem:

−τ−2(uk+1 − 2uk + uk−1) +Auk = gk,

gk = g(tk), tk = kτ, 1 ≤ k ≤N − 1,

u0 = ξ, uN = ψ.

(2.8)

By [24], we have

∥∥∥{uk}0−N∥∥∥C([−1,0]τ ,H)

≤M4(δ)[∥∥fτ

∥∥C([−1,0]τ ,H) + ‖ξ‖H

](2.9)


for the solution of an inverse Cauchy difference problem:

τ−1(uk − uk−1) −(I +

τ

2A)Auk−1 =

(I +

τ

2A)fk,

−(N − 1) ≤ k ≤ 0, u0 = ξ.(2.10)

Then, the proof of Theorem 2.3 is based on stability inequalities (2.7) and (2.9) and on thefollowing estimates:

‖ξ‖H ≤M5(δ)[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H) +

∥∥ϕ∥∥H

],

∥∥ψ∥∥H ≤M6(δ)

[∥∥fτ∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H) +

∥∥ϕ∥∥H

],

(2.11)

for the solution of boundary value problem (1.3). Estimates (2.11) follow from estimates (2.3)and (2.4) and formula (2.5). This finishes the proof of Theorem 2.3.

Theorem 2.4. Assume that ϕ ∈ D(A) and f0, f−1, g1 ∈ D(I+τB). Then, for the solution of differenceproblem (1.3), the following almost coercivity inequality holds:

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥C([0,1]τ ,H)

+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥C([0,1]τ ,H)

+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

≤M7(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}[∥∥fτ∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]

+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

].

(2.12)

Proof. We have

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥C([0,1]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥C([0,1]τ ,H)

≤M8(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}∥∥gτ∥∥C([0,1]τ ,H) + ‖Aξ‖H +

∥∥Aψ∥∥H

],

(2.13)

for the solution of boundary value problem (2.8) (see [22]), and we get

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,H)

≤M9(δ)[min

{ln

1τ, 1 + |ln ‖A‖H→H |

}∥∥fτ∥∥C([−1,0]τ ,H) + ‖Aξ‖H

],

(2.14)


for the solution of inverse Cauchy difference problem (2.10) (see [24]). Then, the proof ofTheorem 2.4 is based on almost coercivity inequalities (2.13) and (2.14) and on the followingestimates:

‖Aξ‖H ≤ M10(δ)[∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +min

{ln

1τ, 1 + |ln ‖A‖H→H |

}

×[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]],

∥∥Aψ∥∥H ≤ M11(δ)

[∥∥Aϕ∥∥H +

∥∥(I + τB)f0∥∥H +min

{ln

1τ, 1 + |ln ‖A‖H→H |

}

×[∥∥fτ

∥∥C([−1,0]τ ,H) +

∥∥gτ∥∥C([0,1]τ ,H)

]]

(2.15)

for the solution of boundary value problem (1.3). Proofs of these estimates follow thescheme of the papers [23, 24] and rely on both formula (2.5) and estimates (2.3) and (2.4).Theorem 2.4 is proved.

Theorem 2.5. Let assumptions of Theorem 2.5 be satisfied. Then, boundary value problem (1.3) iswell-posed in Holder spaces Cα

0,1([−1, 1]τ ,H), and Cα0,1([−1, 1]τ ,H), and the following coercivity

inequalities hold:

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M12(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H

+∥∥(I + τB)g1

∥∥H +

∥∥(I + τB)f−1∥∥H

],

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M13(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]+∥∥Aϕ

∥∥H

+∥∥(I + τB)f0

∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

].

(2.16)


Proof. By [22, 24], we have

∥∥∥∥{τ−2(uk+1 − 2uk + uk−1)

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,H)+∥∥∥{Auk}N−1

1

∥∥∥Cα

0,1([0,1]τ ,H)

≤M14(δ)[

1α(1 − α)

∥∥gτ∥∥Cα

0,1([0,1]τ ,H) + ‖Aξ‖H +∥∥Aψ

∥∥H

],

(2.17)

for the solution of boundary value problem (2.8), and

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M15(δ)[

1α(1 − α)

∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) + ‖Aξ‖H],

(2.18)

∥∥∥∥{τ−1(uk − uk−1)

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)+∥∥∥∥{(I +

τ

2A)Auk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,H)

≤M16(δ)[

1α(1 − α)

∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) + ‖Aξ‖H] (2.19)

for the solution of inverse Cauchy difference problem (2.10), respectively. Then, the proof ofTheorem 2.5 is based on coercivity inequalities (2.17)–(2.19) and the following estimates:

‖Aξ‖H ≤ M17(δ)[

1α(1 − α)

[∥∥fτ∥∥Cα

0 ([−1,0]τ ,H) +∥∥gτ

∥∥Cα

0,1([0,1]τ ,H)

]

+∥∥Aϕ

∥∥H +

∥∥(I + τB)f0∥∥H +

∥∥(I + τB)g1∥∥H +

∥∥(I + τB)f−1∥∥H

],

∥∥Aψ∥∥H ≤ M18(δ)

[1

α(1 − α)[‖fτ‖Cα

0 ([−1,0]τ ,H) + ‖gτ‖Cα0,1([0,1]τ ,H)

]

+‖Aϕ‖H + ‖(I + τB)f0‖H + ‖(I + τB)g1‖H + ‖(I + τB)f−1‖H]

(2.20)

for the solution of difference scheme (1.3). Proofs of these estimates follow the scheme of thepapers [22, 24] and rely on both estimates (2.3) and (2.4) and formula (2.5). This concludesthe proof of Theorem 2.5.


3. An Application

In this section, an application of these abstract Theorems 2.3, 2.4, and 2.5 is considered.In [−1, 1] × Ω, let us consider the following boundary value problem for multidimensionalelliptic-parabolic equation:

−utt −n∑

r=1

(ar(x)uxr )xr = g(t, x), 0 < t < 1, x ∈ Ω,

ut +n∑

r=1

(ar(x)uxr )xr = f(t, x), −1 < t < 0, x ∈ Ω,

u(t, x) = 0, x ∈ S, −1 ≤ t ≤ 1,

u(1, x) =J∑

i=1

αiu(λi, x) + ϕ(x),J∑

i=1

|αi| ≤ 1,

−1 ≤ λ1 < λ2 < · · · < λi < · · · < λJ ≤ 0,

u(0+, x) = u(0−, x), ut(0+, x) = ut(0−, x), x ∈ Ω,

(3.1)

where ar(x) (x ∈ Ω), ϕ(x) (ϕ(x) = 0, x ∈ S), g(t, x) (t ∈ (0, 1), x ∈ Ω), and f(t, x) (t ∈(−1, 0), x ∈ Ω) are given smooth functions. Here,Ω is the unit open cube in the n-dimensionalEuclidean space R

n (0 < xk < 1, 1 ≤ k ≤ n)with boundary S, Ω = Ω ∪ S, and ar(x) � a > 0.The discretization of problem (3.1) is carried out in two steps. In the first step, let us

define the following grid sets:

Ωh = {x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn),

0 ≤ mr ≤Nr, hrNr = 1, r = 1, . . . , n},


(3.2)


We introduce the Hilbert spaces L2h = L2(Ωh),W12h = W1

2 (Ωh), and W22h = W2

2 (Ωh) ofthe grid functions ϕh(x) = {ϕ(h1m1, . . . , hnmn)} defined on Ωh, equipped with the followingnorms:

∥∥∥ϕh∥∥∥L2h

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W1

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xr∣∣∣2h1 · · ·hn

⎞

⎠1/2

,

∥∥∥ϕh∥∥∥W2

2h

=∥∥∥ϕh

∥∥∥L2h

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xr∣∣∣2h1 · · ·hn

⎞

⎠1/2

+

⎛

⎝∑

x∈Ωh

n∑

r=1

∣∣∣(ϕh)xrxr ,mr

∣∣∣2h1 · · ·hn

⎞

⎠1/2

.

(3.3)


h by formula

Axhu

h = −n∑

r=1

(ar(x)uhxr

)

xr ,mr

(3.4)

acting in the space of grid functions uh(x), satisfying the conditions uh(x) = 0 for all x ∈ Sh.With the help of Ax

h, we arrive at the following nonlocal boundary value problem:

−d2uh(t, x)dt2

+Axhu

h(t, x) = gh(t, x), 0 < t < 1, x ∈ Ωh,

duh(t, x)dt

−Axhu

h(t, x) = fh(t, x), −1 < t < 0, x ∈ Ωh,

uh(1, x) =n∑

k=1

αkuh(λk, x) + ϕh(x),

n∑

k=1

|αk| ≤ 1, x ∈ Ωh,

uh(0+, x) = uh(0−, x), duh(0+, x)dt

=duh(0−, x)

dt, x ∈ Ωh,

(3.5)

for an infinite system of ordinary differential equations.


In the second step, we replace problem (3.5) by difference scheme (1.3) accurate to thefollowing second order (see [22, 24]):

−uhk+1(x) − 2uhk(x) + u

hk−1(x)

τ2+Ax

huhk(x) = g

hk (x),

ghk (x) = gh(tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,


−(Axh +

τ

2(Axh

)2)uhk−1(x) =

(I +

τ

2Axh

)fhk (x),

fhk (x) = fh(tk−1/2, x), tk−1/2 =

(k − 1

2

)τ, −N + 1 ≤ k ≤ 0, x ∈ Ωh,

−uh2(x) + 4uh1(x) − 3uh0(x) = 3uh0(x) − 4uh−1(x) + uh−2(x), x ∈ Ωh,

uhN(x) =J∑

k=1

αi

(uh

[λi/τ](x) +

(λk −

[λiτ

]τ

)(fh

[λi/τ]+Ax

huh[λi/τ]

(x)))

+ ϕh(x), x ∈ Ωh.

(3.6)

Theorem 3.1. Let τ and |h| =√h21 + · · · + h2n be sufficiently small positive numbers. Then, solutions

of difference scheme (3.6) satisfy the following stability and almost coercivity estimates:

∥∥∥∥{uhk

}N−1

−N

∥∥∥∥C([−1,1]τ ,L2h)

≤M19(δ)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{ghk

}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

+∥∥∥ϕh

∥∥∥L2h

],

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

+∥∥∥∥{uhk

}N−1

1

∥∥∥∥C([0,1]τ ,W

22h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥C([−1,0]τ ,W2

2h)

≤M20(δ)

⎡

⎣∥∥∥fh0

∥∥∥L2h

+∥∥∥fh−1

∥∥∥L2h

+∥∥∥gh1

∥∥∥L2h

+∥∥∥ϕh

∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+τ∥∥∥gh1

∥∥∥W1

2h

+ ln1

τ + |h|

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥C([−1,0]τ ,L2h)

+∥∥∥∥{ghk

}N−1

1

∥∥∥∥C([0,1]τ ,L2h)

]].

(3.7)

The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetryproperty of the difference operator Ax

hdefined by formula (3.4), the estimate

min{ln

1τ, 1 +

∣∣∣ln∥∥Ax

h

∥∥L2h →L2h

∣∣∣}

≤M21(δ) ln1

τ + |h| , (3.8)

and the following theorem on the coercivity inequality for the solution of elliptic differenceequation in L2h.


Theorem 3.2. For the solution of the following elliptic difference problem:

Axhu

h(x) = ωh(x), x ∈ Ωh, uh(x) = 0, x ∈ Sh, (3.9)


n∑

r=1

∥∥∥∥(uh)

xrxr ,mr

∥∥∥∥L2h

≤M22(δ)∥∥∥ωh

∥∥∥L2h. (3.10)

Theorem 3.3. Let τ and |h| be sufficiently small positive numbers. Then, solutions of differencescheme (3.6) satisfy the following coercivity stability estimates:

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,W22h)

+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,W22h)

≤M23(δ)

[∥∥∥ϕh∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+ τ∥∥∥gh1

∥∥∥W1

2h

+1

α(1 − α)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{ghk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

]],

∥∥∥∥{τ−2(uhk+1 − 2uhk + u

hk−1)}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)+∥∥∥∥{uhk−1

}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,W22h)

+∥∥∥∥{τ−1(uhk − uhk−1

)}0−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{uhk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,W22h)

≤M24(δ)

[∥∥∥ϕh∥∥∥W2

2h

+ τ∥∥∥fh0

∥∥∥W1

2h

+ τ∥∥∥fh−1

∥∥∥W1

2h

+ τ∥∥∥gh1

∥∥∥W1

2h

+1

α(1 − α)

[∥∥∥∥{fhk

}−1−N+1

∥∥∥∥Cα

0 ([−1,0]τ ,L2h)+∥∥∥∥{ghk

}N−1

1

∥∥∥∥Cα

0,1([0,1]τ ,L2h)

]].

(3.11)

The proof of Theorem 3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and thesymmetry property of the difference operator Ax

h defined by formula (3.4).

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[25] P. E. Sobolevskii, On Difference Methods for the Approximate Solution of Differential Equations, VoronezhState University Press, Voronezh, Russia, 1975.


Research ArticleOn Global Solutions for the Cauchy Problem of aBoussinesq-Type Equation

Hatice Taskesen, Necat Polat, and Abdulkadir Ertas

Department of Mathematics, Dicle University, 21280 Diyarbakir, Turkey

Correspondence should be addressed to Necat Polat, [email protected]



Copyright q 2012 Hatice Taskesen et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We will give conditions which will guarantee the existence of global weak solutions of theBoussinesq-type equation with power-type nonlinearity γ |u|p and supercritical initial energy. Bydefining new functionals and using potential well method, we readdressed the initial valueproblem of the Boussinesq-type equation for the supercritical initial energy case.

1. Introduction

This paper is devoted to the initial value problem of a Boussinesq-type equation:

utt − uxx + uxxxx + uxxxxtt =(f(u)

)xx, x ∈ R, t > 0, (1.1)

u(x, 0) = ϕ(x), ut(x, 0) = ψ(x), x ∈ R, (1.2)

where f(u) = γ |u|p, γ > 0. Equation (1.1) is of fundamental physical interest because itarises in the study of shallow water theory, nonlinear lattice theory, and some other physicalphenomena. In the absence of the sixth-order term, (1.1) becomes the good Boussinesqequation and has been intensively studied from a mathematical viewpoint with variousadditional terms [1–9].

Concerning the initial value problem (1.1) and (1.2), it is important to cite the worksof Xu et al. [10], Y. Z. Wang and Y. X. Wang, [11] and Wang and Xu [12].

Wang and Xu [12] investigated the Cauchy problem (1.1) and (1.2). They first provedthe existence and nonexistence of global solutions of the problem for a general nonlinearfunction f(u) provided that the antiderivative F of f satisfies F(u) ≥ 0 or f ′(u) is boundedbelow. Then by the potential well method, they proved the global existence of solutions for aspecial case of the nonlinear term f(u) = −β|u|puwith 0 < E(0) ≤ d.


In [11], the initial value problem for a class of nonlinear wave equations of higherorder:

utt − uxx + uxxxx + uxxxxtt =(ϕ(u)x

)x (1.3)

was considered. The global well-posedness of the initial value problem of (1.3) with ϕ(s) =α|s|p, α /= 0 was studied making use of the potential well method.

Very recently, Xu et al. [10] studied the multidimensional Boussinesq-type equation:

utt −Δu + Δ2u + Δ2utt = Δ(f(u)

), (1.4)

with the nonlinear terms ±a|u|p, −a|u|p−1u, ±au2k, and −au2k+1. When E(0) < d, the globalexistence and finite-time blowup of solutions were proved by the aid of the potential wellmethod.

All of the above-mentioned three studies considered the cases E(0) < d or 0 <E(0) ≤ d. The case E(0) > 0 was also investigated in [12], but the conditions imposedon the nonlinear term are only valid for the odd-degree nonlinearities, that is, positivitycondition is not enough for the nonlinear term f(u) = γ |u|p. In the present paper, we againinvestigate problem (1.1) and (1.2) and give the global existence of solutions for f(u) = γ |u|pwith supercritical initial energy. We emphasize that our main results would seem to bethe first contribution to global well-posedness of the sixth-order Boussinesq equation withsupercritical initial energy and this type of nonlinearity.

The plan of the paper is as follows. Section 2 contains some definitions and anabbreviated description of the global existence theory given in [10, 12]. Also it contains ourfirst new functional and some theorems about sign preserving property of this functional. Theglobal existence theory for supercritical initial energy is presented in Section 3, following thelines laid down by Kutev et al. [13]. For this purpose, we introduce the second new functionaland give the invariance of this functional under the flow of (1.1) and (1.2). A few concludingremarks and an example are to be found in the last section.

The notation employed is standard and may be found explained in detail in [10].

2. Preliminaries

In this section, we give some definitions and some theorems from the papersmentioned in thefirst section. We also introduce a new functional, and we try to prove global well-posednessfor supercritical energy by the aid of this functional.

Let s ≥ 1. For the Cauchy problem (1.1) and (1.2), we define

E(t) = E(u(t), ut(t)) =12

∥∥∥∥(−∂2x)−1/2

ut

∥∥∥∥2

+ ‖uxt‖2 + ‖u‖2H1 +γ

p + 1

∫

R

|u|pudx = E(0), (2.1)

J(u) =12‖u‖2H1 +

γ

p + 1

∫

R

|u|pudx, (2.2)

I(u) = ‖u‖2H1 + γ∫

R

|u|pudx, (2.3)

d = infu∈N

J(u), N ={u ∈ H1 : I(u) = 0, ‖u‖H1 /= 0

}, (2.4)

which are all well defined.


In the following, we show one more characterization of d used before in the literature

d =p − 1

2(p + 1

)(γS

p+1p

)−2/(p−1), (2.5)

where Sp is the imbedding constant fromH1(R) into Lp+1(R) given by

Sp = supu∈H1

‖u‖p+1‖u‖H1

. (2.6)

By the use of (2.1) and definition of Sp, (2.5) can be easily obtained as follows:

12‖u‖2H1 −

γ

p + 1Sp+1p ‖u‖(p+1)/2

H1 ≤ J(u) ≤ E(0). (2.7)

The function

h(k) =12k − γ

p + 1Sp+1p k(p+1)/2 (2.8)

is strictly increasing in [0, k1) and strictly decreasing in (k1,∞), where k1 = γ−2/(p+1)

S−2(p+1)/(p−1)p . By (2.4), we get

d = maxk∈[0,∞)

h(k) = h(k1) =p − 1

2(p + 1

)(γS

p+1p

)−2/(p−1). (2.9)

The following theorem is a generalization of Theorem 3.4 in [12] and Theorem 6.2 in [10].

Theorem 2.1. Assume that ϕ ∈ H1, ψ ∈ H1 and (−∂2x)−1/2ϕ, (−∂2x)−1/2ψ ∈ L2(R).

(i) If E(0) < 0, then all weak solutions of (1.1) and (1.2) blow up in finite time;

(ii) if E(0) = 0, then all weak solutions of (1.1) and (1.2), except the trivial one blow up infinite time;

(iii) let 0 < E(0) < d. If I(ϕ) > 0, then the weak solution of (1.1) and (1.2) is globally definedfor every t ∈ [0,∞), if I(ϕ) < 0, then the weak solution of (1.1) and (1.2) blows up infinite time.

Remark 2.2. For the critical and subcritical initial energy case, the functional I(ϕ) determinesthe behavior of solutions of (1.1) and (1.2). But, the numerical results in [13] showed thatin the supercritical case the sign of the functional I(ϕ) cannot guarantee the global well-posedness of the problem. Due to the choice of initial data, for I(ϕ) > 0, there are somesolutions that blow up in finite time, and for I(ϕ) < 0, there are some solutions that existglobally. A new functional Iδ was considered in [6]. In the rest of this section, we prove thateven if we take a more general functional than Iδ, it will be insufficient in determining thebehavior of problem (1.1) and (1.2) in supercritical case. For a satisfactory result, we have totake into account the initial velocity in a new functional. This will be done in the next section.


Now, we define the first new functional:

Iσ(u) = (1 − σ)‖u‖2H1 + γ∫

R

|u|pudx = I(u) − σ‖u‖2H1 , (2.10)

for σ > −(p − 1)/2. The depth Dσ andNσ are as follows:

Dσ = infu∈Nσ

J(u), Nσ ={u ∈ H1 : Iσ(u) = 0, ‖u‖H1 /= 0

}. (2.11)

Obviously, taking σ = 0 corresponds to the functional I(u). Moreover in the case of σ <−(p − 1)/2 we have Dσ < 0. From Theorem 2.1, we know that in this case all weak solutionsof (1.1) and (1.2) blow up in a finite time.

For σ > −(p − 1)/2, we have the following lemmas.

Lemma 2.3. Assume that u ∈ H1(R). If Iσ(u) < 0, then ‖u‖H1 > r(σ), and if Iσ(u) = 0, then

‖u‖H1 ≥ r(σ) or ‖u‖H1 = 0, where r(σ) = ((1 − σ)/γSp+1p )1/(p−1)

.

Proof. First, from Iσ(u) < 0, we have ‖u‖H1 /= 0. Hence, by

(1 − σ)‖u‖2H1 < −γ∫

R

|u|pudx ≤ γSp+1p ‖u‖p+1H1 , (2.12)

we have ‖u‖H1 > r(σ).If ‖u‖H1 = 0, then Iσ(u) = 0, if Iσ(u) = 0 and ‖u‖H1 /= 0, then from

(1 − σ)‖u‖2H1 = −γ∫

R

|u|pudx ≤ γSp+1p ‖u‖p+1H1 , (2.13)

we have ‖u‖H1 ≥ r(σ).

Lemma 2.4. If ‖u‖H1 < r(σ), then Iσ(u) ≥ 0.

Proof. From ‖u‖H1 < r(σ), we obtain

−γ∫

R

|u|pudx ≤ γSp+1p ‖u‖p+1H1 < (1 − σ)‖u‖2H1 (2.14)

from which follows Iσ(u) ≥ 0.

The properties of I(u) have been studied in detail. Particularly, analogous results toabove lemmas are obtained in [10] (Lemma 3.3), for σ = 0.

Theorem 2.5. Let Dσ be defined as above. Then for σ > −(p − 1)/2, one has

Dσ = b(σ)r2(σ), (2.15)


where b(σ) = (1/2) − ((1 − σ)/(p + 1)). If one writes Dσ in terms of d, one obtains the followingstatement:

Dσ = b(σ)(1 − σ)2/(p−1) 2(p + 1

)

p − 1d. (2.16)

Proof. If u ∈ Nσ , we have by Lemma 2.3 that ‖u‖H1 ≥ r(σ). In the proof of Lemma 2.3, theinequality (2.12) is an equality if and only if u is a minimizer of the imbeddingH1 into Lp+1.Since ‖u‖p+1 = Sp‖u‖H1 is attained only for u = (cosh((p − 1)/2)x)−2/(p−1) [14], and it has aconstant sign, we have

infu∈Nσ

‖u‖H1 = r(σ). (2.17)

Hence from

infu∈Nσ

J(u) = infu∈Nσ

(12‖u‖2H1 +

γ

p + 1

∫

R

|u|pudx)

= infu∈Nσ

[(12− (1 − σ)

p + 1

)‖u‖2H1 +

1p + 1

Iσ(u)]

=(12− (1 − σ)

p + 1

)infu∈Nσ

‖u‖2H1 ,

(2.18)

and by definition of Dσ , we obtain Dσ = b(σ)r2(σ).

As properties ofDσ , the following corollary can be obtained by a simple computation.

Corollary 2.6. Dσ is strictly increasing on σ ∈ (−(p − 1)/2, 0) ∪ (1,∞) and strictly decreasing on(0, 1). Moreover limσ→ 1Dσ = 0, and Dσ0 = 0, where σ0 = −(p − 1)/2.

The following theorems show the invariance of Iσ under the flow of (1.1) and (1.2) inthe framework of weak solutions for 0 < E(0) < d and E(0) = d, respectively.

Theorem 2.7. Assume that ϕ,ψ ∈ H1(R), (−∂2x)−1/2ψ ∈ L2(R). Let 0 < E(0) < d. Then the signof Iσ is invariant under the flow of (1.1) and (1.2) for σ ∈ (σ1, σ2], where (σ1, σ2] is the maximalinterval such that Dσ = E(0).

Proof. In consequence of assumption E(0) > 0, we have ‖u‖H1 > 0. If for σ ∈ (σ1, σ2], thesign of Iσ is changeable, then we must have a σ ∈ (σ1, σ2] such that Iσ = 0. First, we provethe theorem for σ ∈ (σ1, σ2). Notice from (2.16) that σ1 < 0 and σ2 ∈ (0, 1). Thus, we haveDσ2 ≥ E(0) ≥ J(u) ≥ Dσ . Since Dσ is strictly decreasing on (0, 1), from Corollary 2.6 wehave Dσ > Dσ2 , which contradicts with the previous inequality. So, the theorem is provedfor σ ∈ (σ1, σ2). For σ = σ2, assume that Iσ2 = 0 and there exists some σ ∈ (0, σ2) such thatIσ = Iσ2 +(σ2 −σ)‖u‖2H1 . Hence Iσ = (σ2 −σ)‖u‖2H1 > 0 and for some t′ > 0, we get Iσ(u(t′)) = 0.Then, by the similar argument used for σ ∈ (σ1, σ2), we obtain a contradiction. Thus, theproof is finished.


Theorem 2.8. If on top of all the assumptions of Theorem 2.7, we suppose that E(0) = d. Then thesign of I0 (recall that when E(0) = d, we have σ1 = σ2 = 0) is invariant with respect to (1.1) and (1.2)for every t ∈ [0,∞).

Proof. The theorem states that if I0(ϕ) ≥ 0, then I0(u(t)) ≥ 0, contrary if I0(ϕ) < 0, thenI0(u(t)) < 0. We only give the proof of the first statement, the second is similar. To check thatI0 does not change sign, we proceed as follows. Let u(t) be any weak solution of problem(1.1),(1.2) with E(0) = d. If the first statement is false, then there must exist a t′ > 0 such thatI0(u(t′)) < 0. It follows from Lemma 2.3 that ‖u(t′)‖H1 > 0. From the energy identity, we get

d = E(0) =12

∥∥∥∥(−∂2x)−1/2

ut(t′)∥∥∥∥

2

+∥∥uxt

(t′)∥∥2 + J

(u(t′))

≥ J(u(t′)) ≥ infu∈N0

J(u) = d.(2.19)

If u(t′) ∈ N0, then u(t′) must be a minimizer of J(u) for u ∈ N0, and we have I0(u(t′)) = 0.This, however, is impossible, since it violates I0(u(t′)) < 0. Thus the lemma is proved.

Now, we give a lemma for σ > 1, which states similar results to Lemmas 2.3 and 2.4,and can be proved similarly.

Lemma 2.9. Assume that u ∈ H1(R). For σ > 1, if Iσ(u) > 0, then ‖u‖H1 > s(σ), and if Iσ(u) = 0,

then ‖u‖H1 ≥ s(σ) or ‖u‖H1 = 0, where s(σ) = ((σ − 1)/γSp+1p )1/(p−1)

. Moreover, if ‖u‖H1 < s(σ),then Iσ(u) ≤ 0 and Iσ(u) = 0 if and only if ‖u‖H1 = 0.

Theorem 2.10. Assume that ϕ, ψ ∈ H1(R), (−∂2x)−1/2ψ ∈ L2(R). If E(0) > 0, then Iσ(u(t)) ≤ 0 forevery t > 0 and σ ≥ σm, where σm is the maximal positive root of Dσ = E(0).

Proof. We give the proof of the theorem for σ = σm and σ > σm separately. First we prove thetheorem for σ = σm. Proceeding by contradiction, assume that there exists some t′ > 0 suchthat Iσm(u(t

′)) > 0. By Lemma 2.3, we have ‖u‖H1 > 0, and there exists a value σ, σ > σm suchthat Iσ(u(t′)) = 0. Then by (2.1), Dσm = E(0) ≥ J(u(t′)) ≥ infu∈NσJ(u) = Dσ . From definition ofDσ for σ > σm > 1, we have Dσ > Dσm . A contradiction occurs, which proves the theorem forσ = σm. For σ ≥ σm, Iσm(u(t)) ≥ Iσ(u(t)) implies that the theorem is true for every σ ≥ σm.

The following corollary is a direct consequence of Theorems 2.1 and 2.10.

Corollary 2.11. Suppose ϕ, ψ ∈ H1, (−∂2x)−1/2ψ ∈ L2(R). Let 0 < E(0) < d and I0(ϕ) > 0. Then

0 < I0(u(t)) < σm‖u‖2H1 , (2.20)

for every t > 0.

Remark 2.12. We tried to characterize the behavior of solutions for E(0) > 0 in terms of initialdisplacement. We constituted a new functional Iσ(u) and proved the sign invariance of Iσ(u)for 0 < E(0) < d and E(0) = d. But the case E(0) > 0 is still an open question, because fromTheorem 2.10, we concluded that in this case Iσ(u) is always nonpositive. Due to numerical


results of [13], we know that such a functional to prove global existence must include theinitial velocity too. We will introduce this new functional in the next section.

3. Global Existence for Supercritical Initial Energy

In this section, we state the main result of the paper. The functional we introduce here, whichwas used before in [13] in a similar form, permits us to establish global existence of solutionsfor (1.1) and (1.2) in the supercritical initial energy case

H(υ,ω) = ‖υ‖2H1 + γ∫

R

|υ|pυdx −∥∥∥∥(−∂2x)−1/2

ω

∥∥∥∥2

− ‖ωx‖2

= I0(v) −∥∥∥∥(−∂2x)−1/2

ω

∥∥∥∥2

− ‖ωx‖2.(3.1)

In order to simplify notation, we rewrite H(u(·, t), ut(·, t)) as

H(u, t) = H(u(·, t), ut(·, t)). (3.2)

Once we have proved the invariance of the above functional with respect to (1.1) and (1.2),then global existence can be proved by the aid of invariance of this functional.

Theorem 3.1. Assume that ϕ, ψ ∈ H1(R), (−∂2x)−1/2ϕ, (−∂2x)−1/2ψ ∈ L2(R) and E(0) > 0. Forsome σ > σm, σm defined as above, one have

((−∂2x)−1/2

ψ,(−∂2x)−1/2

ϕ

)+(ψx, ϕx

)+12

∥∥∥∥(−∂2x)−1/2

ϕ

∥∥∥∥2

+12∥∥ϕx∥∥2 +

(p + 1

)σ

p − 1 +(p + 3

)σE(0) ≤ 0.

(3.3)

Moreover,H(u, t) is positive provided thatH(u, 0) is positive, for every t ∈ [0,∞).

Proof. Looking for the global solution is equivalent to showing that there is no blow up. So,we modify a blow up technique for the proof [15]. To this end, we define

θ(t) =∥∥∥∥(−∂2x)−1/2

u

∥∥∥∥2

+ ‖ux‖2. (3.4)


Direct computations yield

θ′(t) = 2((

−∂2x)−1/2

ut,(−∂2x)−1/2

u

)+ 2(uxt, ux),

θ′′(t) = 2∥∥∥(−∂2x

)−1/2ut∥∥∥2+ 2((−∂2x

)−1/2utt,(−∂2x

)−1/2u)+ 2(uxtt, ux) + 2‖uxt(t)‖2

= 2∥∥∥(−∂2x

)−1/2ut∥∥∥2+ 2‖uxt(t)‖2 + 2

((−∂2x)−1

utt, u)− 2(uxxtt, u)

= 2∥∥∥(−∂2x

)−1/2ut∥∥∥2+ 2‖uxt(t)‖2 − 2I0(u)

= −2H(u, t).

(3.5)

For contradiction, assume that there exists some t′ > 0 such thatH(u, t′) = 0. Since θ′′(t) < 0,we conclude that θ′(t) is strictly decreasing on [0, t′). Moreover, (3.3) implies θ′(0) < 0 andtherefore θ′(t) < 0 in [0, t′], from which follows that θ(t) is strictly decreasing on [0, t′]. By theenergy identity andH(u, t′) = 0, we have

E(0) =12

(∥∥∥∥(−∂2x)−1/2

ut(t′)∥∥∥∥

2

+∥∥uxt

(t′)∥∥2)

+p − 1

2(p + 1

)∥∥u(t′)∥∥2

H1 +1

p + 1I(u(t′))

=(12+

1p + 1

)(∥∥∥∥(−∂2x)−1/2

ut(t′)∥∥∥∥

2

+∥∥uxt

(t′)∥∥2)

+p − 1

2(p + 1

)∥∥u(t′)∥∥2

H1 .

(3.6)

Theorem 2.10, Corollary 2.11, andH(u, t′) = 0 yield

‖u‖2H1 ≥ σ−1m I0(u(t′)) ≥ σ−1

(∥∥∥∥(−∂2x)−1/2

ut(t′)∥∥∥∥

2

+∥∥uxt

(t′)∥∥2). (3.7)

The use of the above inequality in (3.6) shows that

E(0) ≥(

12+

1p + 1

+p − 1

2(p + 1

)σ

)(∥∥∥∥(−∂2x)−1/2

ut(t′)∥∥∥∥

2

+∥∥uxt

(t′)∥∥2). (3.8)

This can be rephrased in terms of θ(t) and θ′(t) as

E(0) ≥(p + 3

)σ + p − 1

2(p + 1

)σ

[∥∥∥∥(−∂2x)−1/2(

ut(t′)+ u(t′))∥∥∥∥

2

+∥∥uxt

(t′)+ u(t′)∥∥2

− 2((

−∂2x)−1/2

ut(t′),(−∂2x)−1/2

u(t′)) − 2

(uxt(t′), ux(t′))

−

∥∥∥∥∥∥∥

(−∂2x)−

12u(t′)∥∥∥∥∥∥∥

2

− ∥∥ux(t′)∥∥2⎤⎥⎦.

(3.9)


From the monotonicity of θ(t) and θ′(t), we get

E(0) >

(p + 3

)σ + p − 1

(p + 1

)σ

[−((

−∂2x)−1/2

ψ,(−∂2x)−1/2

ϕ

)− (ψx, ϕx

)

−12

∥∥∥∥(−∂2x)−1/2

ϕ

∥∥∥∥2

− 12∥∥ϕx∥∥2],

(3.10)

which contradicts (3.3). Thus, the theorem is proved.

Theorem 3.2. Assume that ϕ, ψ ∈ H1(R), (−∂2x)−1/2ϕ, (−∂2x)−1/2ψ ∈ L2(R). Suppose that E(0) >0, H(u, 0) > 0 and (3.3) holds for some σ > σm. Then, the weak solution of (1.1) and (1.2) is globallydefined for every t ∈ [0,∞).

Proof. The proof of this theorem follows from adding some arguments to the local existenceresult of Corollary 2.10 of [10]. If (−∂2x)−1/2ψ ∈ L2, then (−∂2x)−1/2ut ∈ L2 (Lemma 2.8 of [10]).H(u, 0) > 0 implies from the sign preserving property of H(u, t) that H(u, t) > 0, therebyI0(u) > 0 for every t > 0. From the energy identity we have

E(0) =12

(∥∥∥∥(−∂2x)−1/2

ut

∥∥∥∥2

+ ‖uxt‖2)

+p − 1

2(p + 1

)‖u‖2H1 +1

p + 1I(u)

≥ 12

∥∥∥∥(−∂2x)−1/2

ut

∥∥∥∥2

+12‖uxt‖2 +

p − 12(p + 1

)‖u‖2H1 .

(3.11)

Therefore ‖u‖H1 and ‖ut‖H1 are bounded for every t > 0. The previously mentioned localexistence theory completes the proof.

4. Final Remarks

Remark 4.1. In a section of the paper of Y. Z. Wang and Y. X. Wang [11], problem (1.1) and(1.2) was studied with the supercritical initial energy. A global existence result was obtainedunder the assumption F(u) ≥ 0 or f ′(u) is bounded below, that is, f ′(u) ≥ A0. As the authorshave mentioned in an example, this condition is valid only for odd-degree nonlinearities,namely for f(u) = βu2p+1, u ∈ R, β > 0, p is a nonnegative integer. For f(u) = β|u|p, thiscondition cannot guarantee the global existence. For example, if we take f(u) = (3/2)u2, thenthe antiderivative and derivative of f are F(u) = (1/2)u3 and f ′(u) = 3u, respectively. Forf(u) = β|u|p, we have F(u) = (β/(p + 1))|u|pu and f ′(u) = ap|u|p−2u, which do not alwayssatisfy the positivity condition. To remedy this, we generate a new functional for potentialwell method, which also contains the initial velocity different from the previous ones, anduse the invariance of this functional with respect to problem (1.1) and (1.2).

Remark 4.2. In Section 2, we introduce the first new functional Iσ(u), which is more generalthan the ones introduced before in some papers for the fourth-order Boussinesq equation.However, in the case of E(0) > 0 we are not able to prove the sign invariance of Iσ(u), becausewe see that for E(0) > 0, Iσ(u) is always nonpositive. Eventually, a satisfactory result comes


from H(u, t) which includes not only the initial displacement ϕ, but also the initial velocityψ.

References

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[2] R. Xue, “Local and global existence of solutions for the Cauchy problem of a generalized Boussinesqequation,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 307–327, 2006.

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Research ArticleInverse Scattering from a Sound-Hard Crack viaTwo-Step Method

Kuo-Ming Lee

Department of Mathematics, National Cheng Kung University, Tainan City 70101, Taiwan

Correspondence should be addressed to Kuo-Ming Lee, [email protected]

Received 14 February 2012; Accepted 18 April 2012


Copyright q 2012 Kuo-Ming Lee. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We present a two-step method for recovering an unknown sound-hard crack in R2 from the

measured far-field pattern. This method, based on a two-by-two system of nonlinear integralequations, splits the reconstruction into two consecutive steps which consists of a forward andan inverse problems. In this spirit, only the latter needs to be regularized.

1. Introduction

In this paper we consider an acoustic scattering problem from a sound-hard crack. Thisproblem is modeled by an exterior boundary value problem governed by Helmholtzequation for an open arc with Neumann boundary conditions on both sides of the arc.Our major concern is the inverse problem which aim is to reconstruct the crack from somemeasurements. This kind of problem is of fundamental importance, for example, in materialinvestigation, nondestructive testing, or in seismic exploration.

The inverse scattering problem for an open arc was first investigated by Kress [1]. Inhis article, integral equation method was used to solve both the direct and inverse problemsfor a sound-soft crack. The scattering problem in the unbounded domain was then convertedinto a boundary integral equation. Monch [2] extended this approach to aNeumann problem.Their integral-equation-based method was to solve the so-called far-field equation.

F(Γ) = u∞, (1.1)

for the unknown crack Γ from the measured far-field data. Because of the nonlinearity andthe compactness of the far-field operator, both linearization and regularization are needed to


keep the solving of this equation in the framework of linear regularization theory. One of themajor drawbacks of the regularized Newton’s method used in [1, 2] is that at every iteration,a direct problem with different boundary data must be solved for the Frechet derivative ofthis far-field operator which is an essential part of this method.

The nonlinear integral equations method, proposed by Kress and Rundell [3], can beseen as a remedy for the Newton’s method. They proposed a two-by-two system of nonlinearintegral equations which was shown to be equivalent to the original inverse problem. TheFrechet derivatives were obtained simply through the solving of this system. This methodwas extended to other boundary conditions and cracks in [4, 5] and to obstacle scattering in[6].

Another group of method which is not iterative is the decomposition method (see [7–9]). The idea is to split the nonlinearity and the ill-posedness of the original inverse problem.The scattered field us is computed in the first linear ill-posed step from the far-field patternu∞. In the second step, which is nonlinear, the unknown crack Γ is reconstructed as thelocation where the boundary condition fits in the least square sense. This method is thuscarried out without computation of the direct problem. One of its disadvantages is that thereconstruction is less accurate than that of the Newton’s method.

In [10], a method consisting of two steps was proposed for a sound-soft crack. It isa mixture of the above mentioned methods. In this paper, we’d like to extend this methodto the case of a sound-hard crack and to the case of limited aperture. The plan of thepaper is as follows. For the sake of completeness and also for the introduction of notations,we will briefly summarize the main results of the direct problem in Section 2. In Section 3we will consider the inverse scattering problem in its equivalence form of a two-by-twosystem of integral equations. In Section 4 we will discuss an iteration scheme for numericalcomputation of the inverse problem. This will be followed by some numerical examples inthe final section.

2. Direct Neumann Problem

Given a regular nonintersecting C3-smooth open arc Γ ⊂ R2 which can be represented as

Γ ={z(s) : s ∈ [−1, 1], z ∈ C3[−1, 1], ∣∣z′(s)∣∣/= 0, ∀s ∈ [−1, 1]

}. (2.1)

The two end points of the crack are denoted by z∗−1, z∗1 respectively. The left hand side and the

right-hand side of the crack are written by Γ+ and Γ−, respectively. The unit outward normalto Γ+ is denoted by ν. Further we set Γ0 := Γ \ {z∗−1, z∗1}.

The direct scattering problem for a sound-hard crack that we are considering is asfollows.

Problem 1 (DP). Given an incident plane wave ui(x, d) := eik〈x,d〉 with a wave number k > 0and a unit vector d giving the direction of propagation, we find a solution us ∈ C2(R2 \ Γ) ∩C(R2 \ Γ0) to the Helmholtz equation

Δus + k2us = 0, in R2 \ Γ, (2.2)


which satisfies the Neumann boundary conditions:

∂us±∂ν

= −∂ui

∂νon Γ0, (2.3)

on both sides of the crack and the Sommerfeld radiation condition:

limr→∞

√r

(∂us

∂r− ikus

)= 0, r := |x| (2.4)

uniformly for all directions x := x/|x|.

In (2.3), the limits

∂us±(x)∂ν

:= limh→ 0

⟨ν(x), gradus(x ± hν(x))⟩, x ∈ Γ0 (2.5)

are required to exist in the sense of locally uniform convergence.Note that the boundary conditions (2.3) can be reformulated as homogeneous

Neumann conditions for the total field u := ui + us, that is, ∂u±/∂ν = 0.Using boundary integral equations, this direct problem can be solved via the layer

approach.We refer to themonograph [11] for the details. In terms of the fundamental solutionto the Helmholtz equation in R

2

Φ(x, y

):=

i

4H

(1)0

(k∣∣x − y∣∣), x /=y, (2.6)

the following theorem, which was proved in [12], ensures the unique solvability of the directscattering problem.

Theorem 2.1. The direct Neumann Problem 1 has a unique solution given by

us(x) =∫

Γ

∂Φ(x, y

)

∂ν(y) ϕ

(y)ds

(y), x ∈ R

2 \ Γ, (2.7)

where ϕ ∈ C1,α,∗(Γ) is the (unique) solution to the following integral equation

∂

∂ν(x)

∫

Γ

∂Φ(x, y

)

∂ν(y) ϕ

(y)ds

(y)= −∂u

i(x)∂ν(x)

. (2.8)


The function space C1,α,∗(Γ) is defined by

C1,α,∗(Γ) :={ϕ | ϕ(z∗−1

)= ϕ

(z∗1)= 0,

dϕ(z(s))ds

=ϕ(arccos s)√

1 − s2, ϕ ∈ C0,α[0, π]

}, (2.9)

for 0 < α < 1.

At this place, we introduce the far-field pattern u∞ of the scattered field us. The far-field pattern describes the behavior of the scattered wave at the infinity

us(x) =eik|x|√|x|

{u∞(x) +O

(1|x|

)}|x| −→ ∞ (2.10)

uniformly for all directions x ∈ Ω := {x ∈ R2 | |x| = 1}. The one-to-one correspondence

between radiating wave and its far-field pattern is established by the Rellich’s lemma. In thecase of a sound-hard crack, the far-field pattern is given by

u∞(x) = ρ∞

∫

Γ

⟨ν(y), x

⟩e−ik〈x,y〉ϕ

(y)ds

(y), (2.11)

with the density function ϕ from Theorem 2.1 and the constant ρ∞ =√(k/8π)e−i(π/4). We

note that from the viewpoint of the inverse scattering problems, the far-field pattern is ofparticular interest since usually one has no access to the close neighborhood of the crack.

It is convenient for our further treatment to rewrite our integral equation (2.8) inoperator form. For this purpose, we introduce the following integral operators:

(Sϕ

)(x) :=

∫

ΓΦ(x, y

)ϕ(y)ds

(y), x ∈ Γ,

(Tϕ

)(x) :=

∂

∂ν(x)

∫

Γ

∂Φ(x, y

)

∂ν(y) ϕ

(y)ds

(y), x ∈ Γ,

F∞(ϕ)(x) := ρ∞

∫

Γ

⟨ν(y), x

⟩e−ik〈x,y〉ϕ

(y)ds

(y), x ∈ Ω.

(2.12)

The operator F∞(ϕ), which is termed the far-field operator in the literatures, plays animportant role in the scattering theory. In terms of these operators, the solving of the directscattering problem amounts to the solving of the following system of two operator equations:

Tϕ = −∂ui

∂ν,

u∞ = F∞ϕ.

(2.13)

In words, given the crack, the unknown density function ϕ is firstly solved from the firstequation. Inserting ϕ into the very definition of the far-field operator, one obtains the far-field


pattern u∞ from the second equation of the system (2.13) straight forward. This reasoningforms the basic idea of our numerical scheme for the inverse problem. At this place, we wantto reduce the hypersingularity of the operator T . Using Maue’s identity, the hypersingularoperator T is splitted into two parts (Theorem 7.29 in [11]):

Tϕ =∂

∂ϑS∂ϕ

∂ϑ+ k2

⟨ν, Sϕν

⟩, (2.14)

where ϑ is the unit tangent vector.

3. Inverse Scattering Problem

After introducing the notations in the last section, we consider the following inverse problem.

Problem 2 (IP). Determine the crack Γ if the far-field pattern u∞ is known for one incidentwave.

For the uniqueness of this inverse problem, that is, for the identifiability of the arc, werefer to [2]where infinitely many incident waves are used. Although it is not yet proven, it iswidely believed that the scatterer can be uniquely determined from one incident wave. Sinceour numerical method is based on Newton’s method which has the advantage of dealingwith only one incident wave well, this leads to our problem setting. For the general solutiontheory of the inverse scattering problem, we refer to the monograph [13].

Motivated by the method of nonlinear integral equations in [3], a system of nonlinearintegral equations which solution is the solution to the inverse problem was presented in [5]for a Neumann crack. In [5], the following system was proposed:

B(Γ, ϕ

)= −∂u

i

∂ν

∣∣∣∣∣Γ

,

F∞(Γ, ϕ

)= u∞,

(3.1)

where the operators B and F∞ are no more than the operators T and F∞ in (2.13), respectively.The additional parameter Γ refers to the dependence on the crack. In [5, Theorem 2], theequivalence of the inverse problem and the solving of the system (3.1) was proven. Thebasic idea of this system is that both the direct and inverse problem share the same integralequations, with slightly different interpretations. This makes the inverse problem reallyinverse to the original physical problem.

In [5], (3.1) was treated as a coupled system with two nonlinear ill-posed equations.This means that the two equations in (3.1) have to be linearized and regularized at the sametime, and at every iterative step. The Frechet derivatives for every operator both w.r.t. theunknown boundary and the unknown density have to be calculated at every step. Besides,two regularization parameters have to be selected for the scheme.

Start the inverse problem with the same system (3.1), we suggest a different schemein the next section in details. Before formulating our algorithm, we need to parametrize thesystem.


As in Section 2, we use the parametrization Γ = {z(t) : t ∈ [−1, 1]} for the sound-hardcrack. To incorporate the square root singularities of the solution us at the crack tips, we usethe cosine substitution t = cos τ , as suggested in [14]. Keeping the notations in [5], we writeγ(τ) := z(cos τ) for the crack Γ and ψ(τ) = ϕ(cos τ) for the unknown density.

The system of integral equation (3.1), using Maue’s identity, is now parametrized asfollows:

A0(γ, ψ ′) −A1

(γ, ψ

)= a

(γ), (3.2)

A∞(γ, ψ

)= u∞, (3.3)

where

A0(γ, ψ ′) :=

∫π

0K0(τ, σ)ψ ′(σ)dσ,

A1(γ, ψ

):=

∫π

0K1(τ, σ)ψ(σ) sinσ dσ,

A∞(γ, ψ

):= ρ∞

∫π

0K∞(x, σ)ψ(σ) sinσ dσ,

a(γ):= −2ik〈n(τ), d〉ui(γ(τ), d),

(3.4)

with the kernels

K0(τ, σ) :=ik

4H

(1)′

0

(k∣∣γ(τ) − γ(σ)∣∣)

⟨γ(τ) − γ(σ), γ ′(τ)⟩∣∣γ(τ) − γ(σ)∣∣ ,

K1(τ, σ) :=ik2

4⟨γ ′(τ), γ ′(σ)

⟩H

(1)0

(k∣∣γ(τ) − γ(σ)∣∣),

K∞(x, σ) := 〈n(σ), x〉e−ik〈γ(σ),x〉

(3.5)

and the parametrized outward normal n = (ν ◦ γ) · |γ ′|.

4. A Two-Step Method

Now we want to introduce our two-step algorithm. The parametrized systems (3.2), (3.3),or the original system (3.1) can be interpreted differently as in [5]. The first equation of thesystem can be seen as the equation for the direct problem. To be more precise, given the crackΓ, which is equivalent to fixing the variable γ , (3.2) is solved for ψ. Hence this equation iswell posed according to Theorem 2.1. Moreover, it is a linear integral equation in ψ. Anotheradvantage is numerically as it can be just solved by the direct solver. With this ψ at hand,we can solve (3.3) for γ . This is a nonlinear ill-posed equation which we will linearize andregularize. Based on Newton’s method, the Frechet derivative of A∞ w.r.t. γ is needed for


the linearization. The Frechet derivative of the integral operator is simply given by the Frechetderivative of its kernel (cf. [15]). For brevity, we set a⊥ = (a2,−a1)t if a = (a1, a2)

t. We have

A′∞(γ, ψ; q

):= ρ∞

∫π

0K′

∞(γ, ψ; q

)ψ(σ) sinσ dσ, (4.1)

where

K′∞(γ, ψ; q

)={⟨

q′(σ)⊥, x⟩− ik⟨q(σ), x⟩〈n(σ), x〉

}e−ik〈γ(σ), x〉. (4.2)

The two-step scheme reads

A0(γ, ψ ′) −A1

(γ, ψ

)= a

(γ), (4.3)

A′∞(γ, ψ

)(q)+A∞

(γ, ψ

)= u∞. (4.4)

Because of the compactness of the operatorA′∞, (4.4) still has to be regularized. To accomplish

this, we apply the Tikhonov regularization, that is, instead of (4.4), the equation

(αI +A′∗

∞(γ, ψ

)A′

∞(γ, ψ

))q = A′∗

∞(γ, ψ

)(u∞ −A∞

(γ, ψ

))(4.5)

has to be solved with a regularization parameter α > 0.At this point, wewant to discuss the uniqueness and the solvability of the system (4.3),

(4.5). From the solution theory of the direct problem in Section 2, (4.3) is uniquely solvable.However, the Frechet derivative A′

∞ is not injective as pointed out in [1]. The null space ofthis operator is given by

N(A′

∞)={q : ν(τ) · q(τ) = 0, τ ∈ [0, π]

}. (4.6)

This reflects the fact that different parametrizations of the arc leading to the same set ofpoints turn out to have the same far-field pattern.We can avoid this ambiguity by limiting oursolution space to the set of arcs representable as the graph of a function as suggested in [1].Once this restriction is made, the operator A′

∞ is then an injective linear compact operator.As a consequence of the regularization theory, (4.5) is uniquely solvable, see [11]. Again,(4.3) is as a direct problem uniquely solvable as stated in Theorem 2.1. Thus, we can have thefollowing theorem.

Theorem 4.1. If the pairs (ψ, q) and (ψ, q) are solutions to the system (4.3), (4.5), then ψ = ψ andq = q .

Numerically, we solve the systems (4.3), (4.5) in two steps: At the start, given an initialguess γ0 for the unknown crack Γ. The first equation is treated as the corresponding directproblem and is solved for the density ψ. Then the second equation updates the crack bymeans of the regularized Newton’s method.


The algorithm for our method can be formulated as follows:

(i) given an initial guess γ0 for the unknown crack;

(ii) iterative steps, for n = 0, 1, 2, . . ..

(1) Step 1: solve

A0(γn, ψ

′) −A1(γn, ψ

)= a

(γn), (4.7)

for ψ.(2) Step 2: solve

(αI +A′∗

∞(γn, ψ

)A′

∞(γn, ψ

))qn = A′∗

∞(γn, ψ

)(u∞ −A∞

(γn, ψ

)), (4.8)

for the update qn of γn.

(iii)

γn+1 = γn + qn. (4.9)

At this place we comment that from the numerical point of view our scheme is very attractive.As a variant of the nonlinear integral equations method, the Frechet derivative of the integraloperator is computed directly by solving the integral equation (4.8). Besides, the derivativeis very simple as compared to those in [5] since only the far-field operator which has asmooth kernel is differentiated. Numerically it can be easily solved via the rectangular rule,for example. Another advantage of this method is that it splits the problem into two smallerparts and thus makes the computation cheaper. However, being a Newton’s method, theconvergence of our numerical scheme is still open.


In this section we will demonstrate the applicability of our method via some examples. Wereconstruct the unknown crack from the knowledge of the far-field pattern at a number ofpoints resulted from just one incident wave. For the direct problem, the forward solver isappliedwith 63 collocation points. In order to avoid committing an inverse crime, the numberof collocation points used in the inverse problem is chosen to be different from that of theforward solver. The point is, in the inverse problem, (4.3) is solved with the same forwardsolver as for the direct problem. We therefore choose 31 collocation points in the inversealgorithm. In all our examples, we take the incident wave coming from the direction d =(1/

√2)(−1, 1). The far-field pattern is measured at 15 points evenly distributed on the unit

circle.The basis functions for the parametrization are taken from the space:

Vm = span{T0, T1, . . . , Tm−1}, (5.1)


0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−0.2−1 −0.8 −0.6 −0.4

(a) Exact data,N = 47

0 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−0.2−0.5−1 0.5

(b) 3% error,N = 31

Figure 1: k = 1.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−0.2−1 −0.8 −0.6 −0.4 −0.2


0 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

−0.2−0.5−1 0.5

(b) 3% error,N = 44

Figure 2: k = 3.

where T ′ks are the kth Chebyshev monomials Tk(x) = cos(kcos−1x). The selection of the

Chebyshev polynomials is based on the fact that they can take care of the square rootsingularity in their own right. Thus the updates for the crack can be represented in the form:

q(x) =

(m−1∑

k=0

b1kTk(x),m−1∑

k=0

b2kTk(x)

), (5.2)

with the unknown coefficients b1k, b2k, k = 0, . . . , m − 1 which have to be determined from

(4.5). We test our scheme for two different wave numbers, k = 1, 3. In the case where k = 1,m is taken to be 4. For k = 3, we choose m = 5. The starting curve, that is, the initial guess


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0

(b) 3% error,N = 43

Figure 3: k = 1.

1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0

(b) 3% error,N = 25

Figure 4: k = 3.

for the regularized Newton’s method, is taken to be the straight line y = 0 in all examples.According to the discrepancy principle, the stopping criterion for the iterative scheme is givenby the relative error:

‖u∞ − u∞,n‖2‖u∞‖2

≤ ε, (5.3)

which is taken to be 0.003 in case of exact data and 0.03 in the case where 3% data error arepresent. In all our figures, the dotted line (blue) represents the initial guess. We denote by thedashed line (red) the true solution and by the solid line (black) the reconstruction.


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0

(b) 3% error,N = 34

Figure 5: d ± π/2.

1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0


1.5

1

1

0.5

0.5

−0.5

−0.5

−1

−1−1.5

−1.5 1.5 2 2.5

0

0

(b) 3% error,N = 50

Figure 6: d ± π/4.

Example 5.1. For the first example, we take the arc

Γ =

(x, exp

(x3

2

)), x ∈ [−1, 1], (5.4)

which is representable of a graph of a function. The numerical results are given in the Figures1, 2 for k = 1 and k = 3, respectively.

From the numerical results, we see that the reconstructions for exact data are verygood. The reconstructions in the case where random are present, the reconstructions are notbad.


Example 5.2. To demonstrate the benefit of our method, we choose a curve which does notbelong to our solution space. To this end, we take the arc

Γ =(2 sin

(3π8

(x +

43

)),− sin

(3π4

(x +

43

))), x ∈ [−1, 1], (5.5)

which is not representable as a function.The results are shown in Figures 3 and 4.

We see that the reconstructions are very good, even in the case of erroneous data.

Example 5.3. In this final example, we choose the same curve as in the last example for thecase of limited aperture. We assume that the data are only measurable within a certain rangeapart from the incident angle. Figure 5 shows the results for uniformly distributed far-fieldmeasurements within 90 degrees from both sides of the incident wave.We see that in this case,the reconstructions are very accurate both for the exact and erroneous data. In Figure 6, theresults are shown for measurements taken within 45 degrees from both sides of the incidentwave. We see that the reconstructions are convincible for only 15 data points.

We conclude this paper with some remarks. First of all, our examples above show thefeasibility of the proposed numerical method. On one hand, being a Newton-type method,our method is conceptually simple and numerically more accurate than the traditionaldecompositionmethod. On the other hand, being a variant of the nonlinear integral equationsmethod, the derivative is directly computable from the algorithm itself which makes themethod more easily accessible than the classical Newton’s method. The splitting of theproblem into two smaller parts makes our method more competitive as the computationalcost concerns. Finally we want to point out that our method can be carried over to other typeof boundary conditions and also to other type of scattering problems.

Acknowledgment

This work is partially supported by the NSC Grant NSC-100-2115-M-006-003-MY2.

References

[1] R. Kress, “Inverse scattering from an open arc,” Mathematical Methods in the Applied Sciences, vol. 18,no. 4, pp. 267–293, 1995.

[2] L. Monch, “On the inverse acoustic scattering problem by an open arc: the sound-hard case,” InverseProblems, vol. 13, no. 5, pp. 1379–1392, 1997.

[3] R. Kress and W. Rundell, “Nonlinear integral equations and the iterative solution for an inverseboundary value problem,” Inverse Problems, vol. 21, no. 4, pp. 1207–1223, 2005.

[4] O. Ivanyshyn and R. Kress, “Nonlinear integral equations for solving inverse boundary valueproblems for inclusions and cracks,” Journal of Integral Equations and Applications, vol. 18, no. 1, pp.13–38, 2006.

[5] K.-M. Lee, “Inverse scattering via nonlinear integral equations for a Neumann crack,” InverseProblems, vol. 22, no. 6, pp. 1989–2000, 2006.

[6] O. Ivanyshyn and R. Kress, “Nonlinear integral equations in inverse obstacle scattering,” inProceedings of the 7th International Workshop on Mathematical Methods in Scattering Theory and BiomedicalEngineering, Nymphaio, Greece, 2005.


[7] A. Kirsch and R. Kress, “An optimizationmethod in inverse acoustic scattering,” in Boundary ElementsIX, Vol. 3: Fluid Flow and Potential Applications, C. A. Brebbia, W. L. Wendland, and G. Kuhn, Eds., pp.3–18, Springer, Berlin, Germany, 1987.

[8] R. Kress and P. Serranho, “A hybrid method for two-dimensional crack reconstruction,” InverseProblems, vol. 21, no. 2, pp. 773–784, 2005.

[9] R. Kress and P. Serranho, “A hybrid method for sound-hard obstacle reconstruction,” Journal ofComputational and Applied Mathematics, vol. 204, no. 2, pp. 418–427, 2007.

[10] K.-M. Lee, “A two step method in inverse scattering problem for a crack,” Journal of MathematicalPhysics, vol. 51, no. 2, Article ID 023529, 10 pages, 2010.

[11] R. Kress, Linear Integral Equations, Springer, Berlin, Germany, 2nd edition, 1999.[12] L. Monch, “On the numerical solution of the direct scattering problem for an open sound-hard arc,”

Journal of Computational and Applied Mathematics, vol. 71, no. 2, pp. 343–356, 1996.[13] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin,

Germany, 2nd edition, 1998.[14] Y. Yan and I. H. Sloan, “On integral equations of the first kind with logarithmic kernels,” Journal of

Integral Equations and Applications, vol. 1, no. 4, pp. 549–579, 1988.[15] R. Potthast, “Frechet differentiability of boundary integral operators in inverse acoustic scattering,”

Inverse Problems, vol. 10, no. 2, pp. 431–447, 1994.


Research ArticleExistence and Uniqueness ofSolutions for the System of Nonlinear FractionalDifferential Equations with Nonlocal and IntegralBoundary Conditions

Allaberen Ashyralyev1, 2 and Yagub A. Sharifov3

1 Department of Mathematics, Fatih University, 34500 Buyucekmece, Turkey2 ITTU, Ashgabat, Turkmenistan3 Institute of Cybernetics, ANAS, and Baku State University, 1141 Baku, Azerbaijan

Correspondence should be addressed to Yagub A. Sharifov, [email protected]



Copyright q 2012 A. Ashyralyev and Y. A. Sharifov. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

In the present study, the nonlocal and integral boundary value problems for the system of non-linear fractional differential equations involving the Caputo fractional derivative are investigated.Theorems on existence and uniqueness of a solution are established under some sufficient con-ditions on nonlinear terms. A simple example of application of the main result of this paper ispresented.

1. Introduction

Differential equations of fractional order have been proved to be valuable tools in the model-ing of many phenomena of various fields of science and engineering. Indeed, we can obtainnumerous applications in viscoelasticity [1–3], dynamical processes in self-similar structures[4], biosciences [5], signal processing [6], system control theory [7], electrochemistry [8],diffusion processes [9], and linear time-invariant systems of any order with internal pointdelays [10]. Furthermore, fractional calculus has been found many applications in classicalmechanics [11], and the calculus of variations [12], and is a very useful means for obtainingsolutions of nonhomogenous linear ordinary and partial differential equations. For moredetails, we refer the reader to [13].

There are several approaches to fractional derivatives such as Riemann-Liouville,Caputo, Weyl, Hadamar and Grunwald-Letnikov, and so forth. Applied problems requirethose definitions of a fractional derivative that allow the utilization of physically interpretable


initial and boundary conditions. The Caputo fractional derivative satisfies these demands,while the Riemann-Liouville derivative is not suitable for mixed boundary conditions. Thedetails can be found in [14–17].

The study of existence and uniqueness, periodicity, asymptotic behavior, stability, andmethods of analytic and numerical solutions of fractional differential equations have beenstudied extensively in a large cycle works (see, e.g., [10, 18–37] and the references therein).However, many of the physical systems can better be described by integral boundaryconditions. Integral boundary conditions are encountered in various applications suchas population dynamics, blood flow models, chemical engineering, and cellular systems.Moreover, boundary value problems with integral boundary conditions constitute a veryinteresting and important class of problems. They include two-point, three-point, multipoint,and nonlocal boundary value problems as special cases, see [38–41].

In the present paper, we study existence and uniqueness of the problem for the systemof nonlinear fractional differential equations of form

cDα0+x(t) = f(t, x(t)), t ∈ [0, T], (1.1)

with the nonlocal and integral boundary condition

Ex(0) + Bx(T) =∫T

0g(s, x(s))ds, (1.2)

where E ∈ Rn×n is an identity matrix, B ∈ Rn×n is the given matrix, and

‖B‖ < 1. (1.3)

Here, f(t, x(t)) and g(t, x(t)) ∈ Rn are smooth vector functions, cDα0+ is the Caputo fractional

derivative of order α, 0 < α ≤ 1.The organization of this paper is as follows. In Section 2, we provide necessary

background. In Section 3, theorems on existence and uniqueness of a solution are establishedunder some sufficient conditions on nonlinear terms. Finally, in Section 4, a simple exampleof application of the main result of this paper is presented.

2. Preliminaries

In this section, we present some basic definitions and preliminary facts which are usedthroughout the paper. By C([0, T], Rn), we denote the Banach space of all vector continuousfunctions x(t) from [0, T] into Rn with the norm

‖x‖ = max{|x(t)| : t ∈ [0, T]}. (2.1)

Definition 2.1. If g(t) ∈ C[a, b] and α > 0, then the operator Iαa+, defined by

Iαa+g(t) =1

Γ(α)

∫ t

a

g(s)

(t − s)1−αds, (2.2)


for a ≤ t ≤ b, is called the Riemann-Liouville fractional integral operator of order α. Here Γ(·)is the Gamma function defined for any complex number z as

Γ(z) =∫∞

0tz−1e−tdt. (2.3)

Definition 2.2. The Caputo fractional derivative of order α > 0 of a continuous function g :(a, b) → R is defined by

cDαa+g(t) =

1Γ(n − α)

∫ t

a

g(n)(s)

(t − s)α−n+1ds, (2.4)

where n = [α] + 1, (the notation [α] stands for the largest integer not greater than α).

Remark 2.3. Under natural conditions on g(t), the Caputo fractional derivative becomes theconventional integer order derivative of the function g(t) as α → n.

Remark 2.4. Let α, β > 0 and n = [α] + 1, then the following relations hold:

cDα0+t

β =Γ(β)

Γ(β − α) t

β−1, β > n, cDα0+t

k = 0, k = 0, 1, . . . , n − 1. (2.5)

Lemma 2.5 (see, [42]). For α > 0, g(t) ∈ C(0, 1) ∩ L1(0, 1), the homogenous fractional differentialequation

cDα0+g(t) = 0, (2.6)

has a solution

g(t) = c0 + c1t + c2t2 + · · · + cn−1tn−1, (2.7)

where, ci ∈ R, i = 0, 1, . . . , n − 1, and n = [α] + 1.

Lemma 2.6 (see, [42]). Assume that g(t) ∈ C(0, 1)∩ L1(0, 1), with derivative of order n that belongsto C(0, 1) ∩ L1(0, 1), then

Iα0+cDα

0+g(t) = g(t) + c0 + c1t + c2t2 + · · · + cn−1tn−1, (2.8)

where ci ∈ R, i = 0, 1, . . . , n − 1, and n = [α] + 1.

Lemma 2.7 (see, [42]). Let p, q ≥ 0, f(t) ∈ L1[0, T]. Then

Ip

0+Iq

0+f(t) = Ip+q0+ f(t) = Iq0+I

p

0+f(t) (2.9)

is satisfied almost everywhere on [0, T]. Moreover, if f(t) ∈ C[0, T], then identity (2.9) is true for allt ∈ [0, T].


Lemma 2.8 (see, [42]). If α > 0, f(t) ∈ C[0, T], then cDα0+I

α0+f(t) = f(t) for all t ∈ [0, T].

3. Main Results

Lemma 3.1. Let 0 < α ≤ 1, y(t) and g(t) ∈ C([0, T], Rn). Then, nonlocal boundary value problem

cDα0+x(t) = y(t), t ∈ [0, T], (3.1)

Ex(0) + Bx(T) =∫T

0g(s)ds (3.2)

has a unique solution x(t) ∈ C([0, T], Rn) given by

x(t) =∫T

0G(t, s)y(s)ds + C, (3.3)

where

G(t, s) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

E(t − s)α−1Γ(α)

− (E + B)−1B(T − s)α−1

Γ(α), 0 ≤ s ≤ t,

−(E + B)−1B(T − s)α−1

Γ(α), t ≤ s ≤ T,

(3.4)

C = (E + B)−1∫T

0g(s)ds. (3.5)

Proof. Assume that x(t) is a solution of nonlocal boundary value problem (3.1) and (3.2), thenusing Lemma 2.6, we get

x(t) = Iα0+y(t) + c1, c1 ∈ Rn. (3.6)

Applying condition (3.2) and identity (3.6), we get

c1 + B(Iα0+y(T) + c1

)=∫T

0g(s)ds. (3.7)

From condition (1.3) it follows that the inverse of the matrix E + B exists. Therefore, we canwrite

c1 = (E + B)−1∫T

0g(s)ds − (E + B)−1BIα0+y(T). (3.8)


Using formulas (3.6) and (3.8), we obtain

x(t) = Iα0+y(t) + (E + B)−1∫T

0g(s)ds − (E + B)−1BIα0+y(T), (3.9)

which can be written as (3.3). Lemma 3.1 is proved.

Lemma 3.2. Assume that f and g ∈ C ([0, T]×Rn, Rn). Then, the vector function x(t) ∈ C ([0, T],Rn) is a solution of the boundary value problem (1.1) and (1.2) if and only if it is a solution of theintegral equation

x(t) =∫T

0G(t, s)f(s, x(s))ds + (E + B)−1

∫T

0g(s, x(s))ds. (3.10)

Proof. If x(t) solves boundary value problem (1.1) and (1.2). Then, by the same manner as inLemma 3.1, we can prove that x(t) is solution of integral equation (3.10). Conversely, let x(t)is solution of integral equation (3.10). We denote that

v(t) =∫T

0G(t, s)f(s, x(s))ds + (E + B)−1

∫T

0g(s, x(s))ds. (3.11)

Then, by Lemmas 2.7 and 2.8, we obtain

v(t) = Iα0+f(t, x(t)) − (E + B)−1B

(∫T

0

(T − s)α−1Γ(α)

f(s, x(s))ds

)

+ (E + B)−1∫T

0g(s, x(s))ds.

(3.12)

The application of the fractional differential operator cDα0+ to both sides of (3.12) yields

cDα0+v(t) =

cDα0+I

α0+f(t, x(t)) − (E + B)−1BcDα

o+

(∫T

0

(T − s)α−1Γ(α)

f(s, x(s))ds

)

+cDαo+C = f(t, x(t)).

(3.13)

Hence, x(t) solves fractional differential equation (1.1). Also, it is easy to see that x(t) satisfiesnonlocal boundary condition (1.2).

The first main statement of this paper is an existence and uniqueness of boundaryvalue problem (1.1) and (1.2) result that it is based on a Banach fixed point theorem.


Theorem 3.3. Assume that:

(H1) There exists a constant L > 0 such that

∣∣f(t, x) − f(t, y)∣∣ ≤ L∣∣x − y∣∣, (3.14)

for each t ∈ [0, T] and all x, y ∈ Rn.

(H2) There exists a constantM > 0 such that

∣∣g(t, x) − g(t, y)∣∣ ≤M∣∣x − y∣∣, (3.15)

for each t ∈ [0, T] and all x, y ∈ Rn.

If

1Γ(α + 1)

[LTα(1 + (1 − ‖B‖)−1‖B‖

)]+ (1 − ‖B‖)−1MT < 1, (3.16)

then boundary value problem (1.1) and (1.2) has unique solution on [0, T].

Proof . Transform problem (1.1) and (1.2) into a fixed point problem. Consider the operator

P : C([0, T], Rn) −→ C([0, T], Rn), (3.17)

defined by

P(x)(t) =∫T

0G(t, s)f(s, x(s))ds + (E + B)−1

∫T

0g(s, x(s))ds. (3.18)

Clearly, the fixed points of the operator P are solution of problem (1.1) and (1.2). We will usethe Banach contraction principle to prove that under assumption (3.16) operator P has a fixedpoint. It is clear that the operator P maps into itself and that

∣∣P(x)(t) − P(y)(t)∣∣

≤∫T

0|G(t, s)|∣∣f(s, x(s)) − f(s, y(s))∣∣ds +

∥∥∥(E + B)−1∥∥∥∫T

0

∣∣g(s, x(s)) − g(s, y(s))∣∣ds

≤ 1Γ(α)

[∫ t

0(t − s)α−1∣∣f(s, x(s)) − f(s, y(s))∣∣ds +

∥∥∥(E + B)−1B∥∥∥

×∫T

0(T − s)α−1∣∣f(s, x(s)) − f(s, y(s))∣∣ds

]+∥∥∥(E + B)−1

∥∥∥MT∥∥x − y∥∥

≤{

1Γ(α + 1)

[LTα(1 +∥∥∥(E + B)−1B

∥∥∥)]

+∥∥∥(E + B)−1

∥∥∥MT

}∥∥x − y∥∥,(3.19)


for any x, y ∈ C([0, T], Rn) and t ∈ [0, T]. From condition (1.3) it follows that

∥∥∥(E + B)−1∥∥∥ ≤ 1

1 − ‖B‖ . (3.20)

Then, using estimates (3.19) and (3.20), we get

∥∥P(x)(·) − P(y)(·)∥∥

≤{

1Γ(α + 1)

[LTα(1 + (1 − ‖B‖)−1‖B‖

)]+ (1 − ‖B‖)−1MT

}∥∥x − y∥∥.(3.21)

Consequently, by assumption (3.16) operator P is a contraction. As a consequence of Banach’sfixed point theorem, we deduce that operator P has a fixed point which is a solution ofproblem (1.1) and (1.2). Theorem 3.3 is proved.

The second main statement of this paper is an existence of boundary value problem(1.1) and (1.2) result that it is based on Schaefer’s fixed point theorem.

Theorem 3.4. Assume that:

(H3) The function f : [0, T] × Rn → Rn is continuous.

(H4) There exists a constantN1 > 0 such that |f(t, x)| ≤N1 for each t ∈ [0, T] and all x ∈ Rn.

(H5) The function g : [0, T] × Rn → Rn is continuous.

(H6) There exists a constantN2 > 0 such that |g(t, x)| ≤N2 for each t ∈ [0, T] and all x ∈ Rn.

Then, boundary value problem (1.1) and (1.2) has at least one solution on [0, T].

Proof. We will divide the proof into four main steps in which we will show that under theassumptions of theorem operator P has a fixed point.


Step 1. Operator P under the assumptions of theorem is continuous. Let {xn} be a sequencesuch that xn → x in C([0, T], Rn). Then, for each t ∈ [0, T]

|P(xn)(t) − P(x)(t)|

≤∫T

0|G(t, s)|∣∣f(s, xn(s)) − f(s, x(s))

∣∣ds +∥∥∥(E + B)−1

∥∥∥∫T

0

∣∣g(s, xn(s)) − g(s, x(s))∣∣ds

≤ 1Γ(α)

[∫ t

0(t − s)α−1 max

∣∣f(s, xn(s)) − f(s, x(s))∣∣ds

+(1 − ‖B‖)−1‖B‖∫T

0(T − s)α−1 max

∣∣f(s, xn(s)) − f(s, x(s))∣∣ds]

+ (1 − ‖B‖)−1MT max∣∣g(s, xn(s)) − g(s, x(s))

∣∣

≤ 1Γ(α + 1)

[LTα(1 + (1 − ‖B‖)−1‖B‖

)]max

∣∣f(s, xn(s)) − f(s, xn(s))∣∣


∣∣.(3.22)

Since f and g are continuous functions, we have

‖P(xn)(·) − P(x)(·)‖

≤ 1Γ(α + 1)

[LTα(1 + (1 − ‖B‖)−1‖B‖

)]max

∣∣f(s, xn(s)) − f(s, xn(s))∣∣


∣∣ −→ 0

(3.23)

as n → ∞.

Step 2. Operator P maps bounded sets in bounded sets in C([0, T], Rn). Indeed, it is enoughto show that for any η > 0, there exists a positive constant l such that for each x ∈ Bη = {x ∈C([0, T], Rn) : ‖x‖ ≤ η}, we have ‖P(x(·))‖ ≤ l. By assumptions (H4) and (H6), we have foreach t ∈ [0, T],

|P(x)(t)| ≤∫T

0|G(t, s)|∣∣f(s, x(s))∣∣ds +

∥∥∥(E + B)−1∥∥∥∫T

0

∣∣g(s, x(s))∣∣ds. (3.24)

Hence,

|P(x)(t)| ≤ N1Tα

Γ(α + 1)

[1 + (1 − ‖B‖)−1‖B‖

]+N2(1 − ‖B‖)−1T. (3.25)


Thus,

‖P(x)(·)‖ ≤ N1Tα

Γ(α + 1)

[1 + (1 − ‖B‖)−1‖B‖

]+N2(1 − ‖B‖)−1T = l. (3.26)

Step 3. Operator P maps bounded sets into equicontinuous sets of ([0, T], Rn).Let t1, t2 ∈ (0, T], t1 < t2, Bη be a bounded set ofC([0, T], Rn) as in Step 2, and let x ∈ Bη.

Then,

|P(x)(t2) − P(x)(t1)| =∣∣∣∣∣

1Γ(α)

∫ t1

0

[(t2 − s)α−1 − (t1 − s)α−1

]f(s, x(s))ds

+1

Γ(α)

∫ t2

t1

(t2 − s)α−1f(s, x(s))ds∣∣∣∣∣ ≤

N1

Γ(α + 1)[2(t2 − t1)α +

(tα2 − tα1

)].

(3.27)

As t1 → t2, the right-hand side of the above inequality tends to zero. As a consequenceof Steps 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that the operatorP : C([0, T], Rn) → C([0, T], Rn) is completely continuous.

Step 4. A priori bounds. Now, it remains to show that the set

Δ = {x ∈ C([0, T], Rn) : x = λP(x) for some 0 < λ < 1} (3.28)

is bounded.Let x = λ(Px) for some 0 < λ < 1. Then, for each t ∈ [0, T] we have

x(t) = λ

[∫T

0G(t, s)f(s, x(s))ds + (E + B)−1

∫T

0g(s, x(s))ds

]. (3.29)

This implies by assumptions (H4) and (H6) (as in Step 2) that for each t ∈ [0, T]we have

|P(x)(t)| ≤ N1Tα

Γ(α + 1)

[1 + (1 − ‖B‖)−1‖B‖

]+N2(1 − ‖B‖)−1T. (3.30)

Thus, for every t ∈ [0, T], we have

|x(t)| ≤ N1Tα

Γ(α + 1)

[1 + (1 − ‖B‖)−1‖B‖

]+N2(1 − ‖B‖)−1T = R. (3.31)

Therefore,

‖x‖ ≤ R. (3.32)


This shows that the setΔ is bounded. As a consequence of Schaefer’s fixed point theorem, wededuce that P has a fixed point which is a solution of problem (1.1) and (1.2). Theorem 3.4 isproved.

Moreover, we will give an existence result for problem (1.1) and (1.2) by means of anapplication of a Leray-Schauder type nonlinear alternative, where conditions (H4) and (H6)are weakened.

Theorem 3.5. Assume that (H3), (H5) and the following conditions hold.

(H7) There exist θf ∈ L1([0, T], R+) and continuous and nondecreasing

ψf : [0,∞) −→ [0,∞), (3.33)

such that |f(t, x)| ≤ θf(t)ψf(|x|) for each t ∈ [0, T] and all x ∈ Rn.

(H8) There exist θg ∈ L1([0, T], R+) and continuous nondecreasing

ψg : [0,∞) −→ [0,∞), (3.34)

such that |g(t, x)| ≤ θg(t)ψg(|x|) for each t ∈ [0, T] and all x ∈ Rn.

(H9) There exists a number K > 0 such that

K

ψf(K)[∥∥Iαθf

∥∥L1

+ (1 − ‖B‖)−1‖B‖(Iαθf)(T)]+ ψg(K)(1 − ‖B‖)−1∥∥θg

∥∥L1

> 1. (3.35)

Then, boundary value problem (1.1) and (1.2) has at least one solution on [0, T].

Proof. Consider the operator P defined in Theorems 3.3 and 3.4. It can be easily shown thatoperator P is continuous and completely continuous. Let x(t) = (Px)(t) for each t ∈ [0, T].Then, from assumptions (H7) and (H8) if follows that for each t ∈ [0, T]

|x(t)| ≤ 1Γ(α)

∫ t

0(t−s)α−1θf(s)ψf(|x(s)|)ds + 1

Γ(α)

∥∥∥(E+B)−1B∥∥∥∫T

0(T−s)α−1θf(s)ψf(|x(s)|)ds

+∥∥∥(E + B)−1

∥∥∥∫T

0θg(s)ψg(|x(s)|)ds ≤ ψf(‖x‖) 1

Γ(α)

∫ t

0(t − s)α−1θf(s)ds

+ ψf(‖x‖) 1Γ(α)

∥∥∥(E+B)−1B∥∥∥∫T

0(T−s)α−1θf(s)ds+ψg(‖x‖)

∥∥∥(E+B)−1∥∥∥

×∫T

0θg(s)ds≤ψf(‖x‖)

[∥∥Iαθf∥∥L1

+ (1 − ‖B‖)−1‖B‖(Iαθf)(T)]

+ ψg(‖x‖)(1 − ‖B‖)−1∥∥θg∥∥L1.

(3.36)


Thus,

‖x‖ψf(‖x‖)

[∥∥Iαθf∥∥L1

+ (1 − ‖B‖)−1‖B‖(Iαθf)(T)]+ ψg(‖x‖)(1 − ‖B‖)−1∥∥θg

∥∥L1

≤ 1. (3.37)

Then, by condition (H9), there exists K such that ‖x‖/=K.Let

U = {x ∈ C([0, T], R) : ‖x‖ < K}. (3.38)

The operator P : U → C([0, T], R) is continuous and completely continuous. By the choiceof U, there exists x ∈ ∂U such that x = λP(x) for some λ ∈ (0, 1). As a consequence of thenonlinear alternative of Leray-Schauder type [43], we deduce that P has a fixed point x inU,which is a solution of problem (1.1) and (1.2). This completes of proof of Theorem 3.5.

4. An Example

In this section, we give an example to illustrate the usefulness of our main results. Letus consider the following nonlocal boundary value problem for the system of fractionaldifferential equation

cDαx1(t) =110

sinx2, t ∈ [0, 1], 0 < α < 1,

cDαx2(t) =|x1|

(9 + et)(1 + |x1|),

x1 =∫1

0sin 0.1x2(t)dt, x2(0) + 0.5x1(1) = 1.

(4.1)

Evidently,

E + B =(

1 00.5 1

), B =

(0 00.5 0

), ‖B‖ = 0.5, (1 − ‖B‖)−1 = 2. (4.2)

Hence, conditions (H1) and (H2) hold with L =M = 0, 1. We will check that condition (3.16)is satisfied for appropriate values of 0 < α ≤ 1 with T = 1. Indeed,

0.2Γ(α + 1)

+ 0.2 < 1. (4.3)

Then, by Theorem 3.3 boundary value problem (4.1) has a unique solution on [0, 1] for valuesof α satisfying condition (4.3). For example, if α = 0.2 then

Γ(α + 1) = Γ(1.2) = 0.92,0.2

Γ(α + 1)+ 0.2 = 0.418 < 1. (4.4)


5. Conclusion

In this work, some existence and uniqueness of a solution results have been established for thesystem of nonlinear fractional differential equations under the some sufficient conditions onnonlinear terms. Of course, such type existence and uniqueness results hold under the samesufficient conditions on nonlinear terms for the system of nonlinear fractional differentialequations (1.1), subject to multipoint nonlocal and integral boundary conditions

Ex(0) +J∑

j=1

Bjx(tj)=∫T

0g(s, x(s))ds, (5.1)

where Bj ∈ Rn×n are given matrices and∑J

j=1 ‖Bj‖ < 1. Here, 0 < t1 < · · · < tJ ≤ T .Moreover, applying the result of the paper [44] the first order of accuracy difference

scheme for the numerical solution of nonlocal boundary value problem (1.1) and (5.1) can bepresented. Of course, such type existence and uniqueness results hold under the some suf-ficient conditions on nonlinear terms for the solution system of this difference scheme.

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Research ArticleEfficient Variational Approaches for DeformableRegistration of Images

Mehmet Ali Akinlar,1 Muhammet Kurulay,2 Aydin Secer,2and Mustafa Bayram2

1 Department of Mathematics, Bilecik Seyh Edebali University, 11210 Bilecik, Turkey2 Department of Mathematics, Yildiz Technical University, 34220 Istanbul, Turkey

Correspondence should be addressed to Muhammet Kurulay, [email protected]

Received 15 March 2012; Revised 14 April 2012; Accepted 10 May 2012


Copyright q 2012 Mehmet Ali Akinlar et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Dirichlet, anisotropic, and Huber regularization terms are presented for efficient registrationof deformable images. Image registration, an ill-posed optimization problem, is solved usinga gradient-descent-based method and some fundamental theorems in calculus of variations.Euler-Lagrange equations with homogeneous Neumann boundary conditions are obtained. Theseequations are discretized by multigrid and finite difference numerical techniques. The method isapplied to the registration of brain MR images of size 65 × 65. Computational results indicate thatthe presented method is quite fast and efficient in the registration of deformable medical images.

1. Introduction

The purpose of image registration is to match two or more images of the same scene or objectto each other, which can be obtained at different times, perspectives, and sensors such asMRI,X-ray, CT, PET, and tomography. In the present day, correct matching of similar images arequite useful and still a challenging problem. Although there exist some efficient imageregistration methods (for instance, [1–4]) in the literature, finding reliable and efficient imageregistration techniques are still significantly important and active research area among imageprocessing problems.

Mathematically, variational techniques are based on minimizing an energy or costfunction that usually consist of the sum of a data term and a regularization term (e.g., see[5]). In this paper, we express the image registration algorithm as a variational optimizationproblem, which consists of a sum of the similarity measure and a regularization term. Our


method incorporates L2-norm sense similarity measures with several different regularizationterms such as Dirichlet, anisotropic, and Huber.

Organization of the paper is as follows. In the first section, we present optimal controlformulation of the image registration problem. We use the sum of squared differences in theL2-norm sense as the similarity measure and introduce three different regularization terms.We are mostly motivated by the efficient performance of these regularization terms in the sta-tistical approach of image restoration problems (for instance, [6, 7] apply these regularizationterms to image denoising problems). It is plausible to explore the utility of these statisticallyderived regularization terms on the deformation field. In the same section, we solve theexisting optimization problem in a systematic way by the techniques in variational calculussuch as gradient-descent-based methods and numerical methods such as finite differences.Computational results given in the last section indicate that new regularization terms providefast, stable, and efficient image registration techniques.

2. Optimal Control Formulation of Image Registration Problem

The state of the art of the image registration problem can be expressed in the following way.Assume that both the template T and reference R images are defined on the same domain Ω.Then, the image registration problem can be formulated as the optimization problem

minφ∈Γ

J[R,T;φu

](2.1)

for the functional

J[R,T;φu

]= Csim

[R,T;φu

]+ λCreg[u], (2.2)

where Csim[R,T;φu] denotes a similarity measure between the template image T and thereference image R, φu(x) := x + u(x) is the deformation field, u is displacement field, Γ is theset of all possible admissible transformations, Creg[u] is a regularization term, and λ is a reg-ularization constant. Because reference and template images are obtained from different dis-tances, angles, times, and sometimes even by different individuals, a deformation field mayoccur between these images. A deformation field is a vector field that maps pixels (orcoordinates) of reference image to the corresponding ones of the template image. One ofthe major goals of this paper is to compute the deformation field in a systematic way.

We choose the L2-norm type similarity measure defined as

Csim[R(x),T(x);φ(x)] =12

∫

Ω(T(x + u(x)) − R(x))2dx. (2.3)

This similarity measure is often referred to as the “sum of squared differences” (SSD)measure. Note that other similarity measures can be selected depending on the problem. Wechoose the similarity measure (2.3) due to its well-known effectiveness (e.g., see [8]), for theconvenience in computations and for easily adapting the regularization terms in numericalsolutions.

Without the regularizing term in functional (2.2), the image registration problem (2.1)is ill-posed; furthermore, imaging data usually is not smooth due to edges, folding, or other


unwanted deformations. Ill-posed problems are widely used in PDE-based image processingproblems and inverse problems. For instance, in [9] the authors solve ill-posed equationswithtransformed argument as an inverse problem. An optimization problem is said to be well-posed if the solution of the problem uniquely exists and the solution depends continuouslyon the data of the problem. If one of these three conditions are not satisfied, it is called anill-posed problem. Image registration is an ill-posed optimal control problem. In order toovercome the ill-posedness of the optimization problem (2.1) and to assure smooth solutions,we introduce the additional regularization terms. The main idea behind adding a regulariza-tion term is to smoothen the problem with respect to both the functional and the solutionso that well-posedness is assured and efficient computational methods can be defined todetermine minimizers. Typical regularization terms associated with image registrationproblems include curvature, diffusion, elasticity, and fluid. Details about each of theseregularization approaches can be seen, for example, in [1].

In this paper, we shall introduce several different types of regularization terms,Creg[u(x)], mostly in the form of

Creg[u(x)] :=∫

ΩF(|∇u(x)|)dx, (2.4)

where F : R+ → R is a smooth function,∇u is the gradient vector of u, and |∇u| is the L2-norm

of ∇u. For the convenience of the computations, we define Tu(x) := Toφu(x) = T(x + u(x)).Using formulas (2.3) and (2.4) in functional (2.2), we can express (2.2) as

J[R(x),Tu(x);φ(x)] =12

∫

Ω(Tu(x) − R(x))2dx + λ

∫

ΩF(|∇u(x)|)dx. (2.5)

Next we review some fundamental mathematical lemmas and theorems that characterize theexistence of a unique solution of a minimization problem to which we will fit our optimi-zation problem.

Lemma 2.1. If a Gateux differentiable function, sayΨ, defined on an open setO has a local extremumat y0 ∈ O, then DΨ(y0) = 0.

The solutions of DΨ(y0) = 0 are known as Euler-Lagrange equations of the problem. If Ψ isconvex, then a solution of DΨ(y0) = 0 is a solution of a minimization problem. This Lemma and bothof the following Theorems are well known in calculus of variations (e.g., see [10] or [11]).

Theorem 2.2. Assume O is a nonempty, closed, convex subset of a Hilbert spaceH. Let b(·, ·) : H×H → R be bilinear, symmetric, bounded,H elliptic function, and E[u] = (1/2)b(u, u)−μ(u), u ∈ Hwith μ ∈ L1(H,R). Then, there exists a unique v ∈ O such that

E[u] = infv∈O

E[v]. (2.6)

It is clear from (2.6) that we can express the optimization problem given by (2.1) as a generalregularization problem as: minimize the cost functional E : H → R with

E[u] =∫

ΩL dx, (2.7)


L : H×Ω → R over a domain Ω ⊂ Rn (n = 2, 3). Next, we present a theorem for the solution of this

problem.

Theorem 2.3. The solution

u∗(x) = minu∈H

E[u](x) (2.8)

of the general variational problem (2.7) is given as the solution of the Euler-Lagrange equations

Lui −n∑

j=1

∂xjL(ui)xj= 0, i = 1 . . . , d (2.9)

under the conditions stated in Theorem 2.2.Lui denotes the (partial) derivative ofL with respect to ui.This solution can be interpreted as the steady-state solution to the following nonlinear elliptic PDE,called gradient descent flow:

Lui −n∑

j=1

∂xjL(ui)xj=∂u

∂t. (2.10)

In both cases, appropriate boundary conditions must be applied.

Now, we solve the optimization problem (2.1) for Dirichlet, anisotropic, and Huberregularization terms. We start with Dirichlet variational integral

Creg[u(x)] =12

∫

Ω‖∇u(x)‖2 dx. (2.11)

For this regularization term, we can write functional (2.5) as

J[R(x),Tu(x);φ(x)] =12

∫

Ω(Tu(x) − R(x))2 dx +

λ

2

∫

Ω‖∇u(x)‖2 dx. (2.12)

Considering the functional (2.12) in the general functional (2.2) and by using Theorem 2.3,the solution of the optimization problem (2.1) is given by

∂u(x)∂t

− λΔu(x) = ∇Tu(x)(Tu(x) − R(x)). (2.13)

Discretizing (2.13) with a finite-difference method, we get

uh+1(x) − uh(x)δ

− λΔuh+1(x) = ∇Tuh(x)(Tuh(x) − R(x)), (2.14)

where Tuh is computed with any interpolation method. Defining

G(uh, x

):= ∇Tuh(x)(Tuh(x) − R(x)), (2.15)


we obtain

(I + δλΔ)uh+1(x) = δG(uh, x

)+ uh(x). (2.16)

We solve this equation with a multigrid method. Now we express the implementation of thisnumerical solution:

(1) set u0;G0(x) = ∇hTu0(x)(Tu0(x) − R(x)),

(2) for i = 0 to max,

(i) compute Gi(x) = ∇hTui(x)(Tui(x) − R(x)),(ii) update δ := εδδ, λ := ελλ,(iii) compute fi(x) = δGi(x) + ui(x),(iv) solve (I − δλΔh)ui+1(x) = fi(x),

(3) end.

We implemented this algorithm in Matlab programming language where we chose the λ =200, δ = 10, ελ = 0.5, εδ = 0.01. Let us notice that because similar implementation procedurescan be written for anisotropic and Huber-type regularization terms, we will omit their detailmatlab programming implementations for the sake of brevity.

Using Theorem 2.3, the minimizer of (2.5) can be interpreted as the steady statesolution to the following nonlinear elliptic PDE, called gradient descent flow:

ut(x) = λ∇ ·(F ′(|∇u(x)|)|∇u(x)| ∇u(x)

)− ∇Tu(x)(Tu(x) − R(x)), (2.17)

assuming homogeneous Neumann boundary conditions.It was proved in [12] that the anisotropic diffusion PDE of Perona and Malik ([13])

given by

ut(x) = ∇ ·(F ′(|∇u(x)|)|∇u(x)| ∇u(x)

)(2.18)

is the gradient descent flow for the variational integral

F =∫

ΩF(|∇u(x)|) dx, (2.19)

where F(z) is given by

F(z) =c2

2log

(1 +

z2

c2

), (2.20)

where z, c ∈ R+. Hence, for these variational integrals we get the same equation with (2.17)

and F(z) defined by (2.20).


Motivated by the robustness of the Huber M-filter in the probabilistic approach ofimage denoising ([6, 7]), we define the Huber variational integral as

Rk(u) =∫

Ωρk(|∇u(x)|) dx, (2.21)

where

ρk(z) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

z2

2if |z| ≤ k,

|z| − k2

2, otherwise.

(2.22)

One may check that the Huber variational integral Rk : H1(Ω) → R+ is well defined,convex, and coercive. It follows from Theorem 2.3 that the minimization problem

u = minu∈H1

12

∫

Ω(T(x + u(x)) − R(x))2 + λ

∫

Ωρk(|∇u|)dx (2.23)

has a unique solution when λ > 0. Using the Euler-Lagrange variational principle, it followsthat the Huber gradient descent flow is given by

ut = ∇ · (gk(|∇u|)∇u), (2.24)

where gk is the Huber M-estimator weight function

gk(z) =ρ′k(z)z

=

⎧⎪⎪⎨

⎪⎪⎩

1 if |z| ≤ k,k

|z| , otherwise(2.25)

with homogeneous boundary conditions. Hence, for the Huber variational integral we canexpress (2.17) as

ut = ∇ · (gk(|∇u|)∇u) − ∇Tu(x)(Tu(x) − R(x)), (2.26)

where gk is defined in (2.25).

3. Computational Results

In this example, we demonstrate the registration of two-dimensional noise-free brain MRimages in the size of 65 × 65. Computational examples indicate that all regularization termsproduce similarly good registration quality but that the cost associated with Huber modelapproach is, on the average, less than that for others. The SSD registers images in an inverse


(a) (b)

(c)

Figure 1: The template image (a), the reference image (b), and registered image (c).

Table 1: SSD versus different models.

Iterations Dirichlet Anisotropic Huber1 320 300 3005 77 60 5575 25 20 18200 10 8 6

way with its rate. In other words, the smaller SSD gives a better registration from the qualityand efficiency point of views. Table 1 shows the changes in SSD for each model used versusnumber of iterations. Duration of the registration with the Huber model is almost 1 minute,and the one with anisotropic and Dirichlet is almost 2 minutes. The template, reference,and registered images with Huber regularization term are shown in Figure 1. Because theregistered image is similar to the other regularization terms, we omit them inhere for the sakeof brevity. We applied the presented method to some other brain MR images and obtainedthe similar results.


4. Concluding Remarks

Deformable image registration is a significant branch of image processing. It has a broadapplication in medical and nonmedical imaging. In this paper, we presented a number ofdeformable image registration techniques. Our method incorporates sum of squareddifferences similarity measure with several different regularization terms such as Dirichlet,anisotropic, and Huber. By variational calculus methods and gradient-descent-based opti-mization techniques, we solve the existing optimization problem and obtain the correspond-ing optimality system in a systematic manner. We were mostly motivated by the efficientperformance of these regularization terms in the statistical approach of image denoisingproblems, (for instance, see [6, 7]).

In a future work, we will investigate the applications of these image registration tech-niques to the registration of noisy and blurred images. Furthermore, we plan to compare thestrength of these registration techniques with some well-known image registration methodsfrom speed, quality, and effectiveness point of view in detail.

References

[1] J. Modersitzki,Numerical Methods for Image Registration, Oxford University Press, New York, NY, USA,2004.

[2] J. Modersitzki, FAIR: Flexible Algorithms for Image Registration, SIAM, Philadelphia, Pa, USA, 2009.[3] M. A. Akinlar,A newmethod for non-rigid registration of 3D images [Ph.D. thesis], The University of Texas

at Arlington, 2009.[4] M. A. Akinlar and R. N. Ibragimov, “Application of an image registration method to noisy images,”

Sarajevo Journal of Mathematics, vol. 7, no. 1, pp. 135–144, 2011.[5] J. Kuangk, “Variational approach to quasi-periodic solution of nonautonomous second-order

hamiltonian systems,” Abstract and Applied Analysis, vol. 2012, Article ID 271616, 14 pages, 2012.[6] A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image

denoising,” in Proceedings of the 6th International Conference: EMMCVPR, pp. 27–29, Ezhou, China,2007.

[7] A. B. Hamza, H. Krim, and G. B. Unal, “Towards a unified estimation theme: probabilistic versusvariational,” IEEE Signal Processing Magazine, pp. 37–47, 2002.

[8] M. A. Akinlar and M. Celenk, “Quality assessment for an image registration method,” InternationalJournal of Contemporary Mathematical Sciences, vol. 6, no. 30, pp. 1483–1489, 2011.

[9] S. Gramsch and E. Schock, “Ill-posed equations with transformed argument,” Abstract and AppliedAnalysis, vol. 2003, no. 13, pp. 785–791, 2003.

[10] H. Kostler,Amultigrid framework for variational approaches in medical image processing and computer vision[Diplom-Informatiker, Diplom-Kaufmann], Universitat Erlangen-Nurnberg, 2008.

[11] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 2000.[12] Y. L. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in

image processing,” IEEE Transactions on Image Processing, vol. 5, no. 11, pp. 1539–1553, 1996.[13] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions

on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990.


Research ArticleThe Difference Problem of Obtainingthe Parameter of a Parabolic Equation

Charyyar Ashyralyyev1, 2 and Oznur Demirdag3

1 Department of Computer Technology, Turkmen Agricultural University, Gerology Street,74400 Asgabat, Turkmenistan

2 Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey3 Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

Correspondence should be addressed to Oznur Demirdag, [email protected]

Received 4 April 2012; Accepted 23 April 2012


Copyright q 2012 C. Ashyralyyev and O. Demirdag. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

The boundary value problem of determining the parameter p of a parabolic equation υ′(t)+Aυ(t) =f(t) + p (0 ≤ t ≤ 1), υ(0) = ϕ, υ(1) = ψ in an arbitrary Banach space E with the stronglypositive operator A is considered. The first order of accuracy stable difference scheme for theapproximate solution of this problem is investigated. The well-posedness of this difference schemeis established. Applying the abstract result, the stability and almost coercive stability estimatesfor the solution of difference schemes for the approximate solution of differential equations withparameter are obtained.

1. Introduction

The differential equations with parameters play a very important role in many branchesof science and engineering. Some examples were given in temperature overspecificationby Dehghan [1], chemistry (chromatography) by Kimura and Suzuki [2], physics (opticaltomography) by Gryazin et al. [3].

The differential equations with parameters have been studied extensively by manyresearchers (see, e.g., [4–20] and the references therein). However, such problems were notwell investigated in general.

As a result, considerable efforts have been expanded in formulating numericalsolution methods that are both accurate and efficient. Methods of numerical solutions ofparabolic problems with parameters have been studied by researchers (see, e.g., [21–29] andthe references therein).


It is known that various boundary value problems for parabolic equations withparameter can be reduced to the boundary value problem for the differential equation withparameter p:

dv(t)dt

+Av(t) = f(t) + p, 0 < t < 1,

v(0) = ϕ, v(1) = ψ(1.1)

in an arbitrary Banach space E with the strongly positive operator A. In the present work,the first order of accuracy difference scheme for the approximate solution of boundaryvalue problem (1.1) is studied. The well-posedness of this difference scheme is established.Applying the abstract result, the stability and almost coercive stability estimates for thesolution of difference schemes for the approximate solution of differential equations withparameter are obtained.

2. The Boundary Value Problem for Parabolic Equations

Throughout this work, E is a Banach space, −A is the generator of the analytic semigroupexp{−tA}(t ≥ 0) with exponentially decreasing norm, when t → +∞, that is, the followingestimates hold:

∥∥exp{−tA}∥∥E→E ≤Me−δt, t∥∥A exp{−tA}∥∥E→E ≤M, t > 0, M > 0, δ > 0. (2.1)

From estimate (2.1), it follows that

‖T‖E→E ≤M(δ). (2.2)

Here, T = (I − exp{−A})−1.Abstract problem (1.1) was investigated in the paper [4] by applying estimates (2.1)

and (2.2). The solvability of problem (1.1) in the space C(E) of the continuous E-valuedfunctions ϕ(t) defined on [0, 1] equipped with the norm

∥∥ϕ∥∥C(E) = max

0≤t≤1

∥∥ϕ(t)∥∥E (2.3)

was studied under the necessary and sufficient conditions for the operator A. The solutiondepends continuously on the initial and boundary data. More pricisely, we have the followingresult.


Theorem 2.1. Assume that −A is the generator of the analytic semigroup exp{−tA}(t ≥ 0) and allpoints 2πik, k ∈ Z, k /= 0 do not belong to the spectrum σ(A). Let v(0) ∈ E, v(1) ∈ D(A), andf(t) ∈ Cβ(E) (0 < β ≤ 1). Then, for the solution (v(t), p) of problem (1.1) in C(E)×E, the estimates

∥∥p∥∥E ≤M

[‖v(0)‖E + ‖v(1)‖E + ‖Av(1)‖E +

1β

∥∥f∥∥Cβ(E)

],

‖v‖C(E) ≤M[‖v(0)‖E + ‖v(1)‖E +

∥∥f∥∥C(E)

] (2.4)

hold, where M does not depend on β, v(0), v(1) and f(t). Here Cβ(E) is the space obtained bycompletion of the space of all smooth E-valued functions ϕ(t) on [0, 1] in the norm

∥∥ϕ∥∥Cβ(E) = max

0≤t≤1

∥∥ϕ(t)∥∥E + sup

0≤t<t+τ≤1

∥∥ϕ(t + τ) − ϕ(t)∥∥Eτβ

. (2.5)

With the help of A, we introduce the fractional space Eα(E,A), 0 < α < 1, consisting of allv ∈ E for which the following norms are finite [6, 30]:

‖v‖α = supλ>0

∥∥∥λ1−αA exp{−λA}v∥∥∥E+ ‖v‖E. (2.6)

We say (v(t), p) is the solution of problem (1.1) in Cβ,γ

0 (E) × E1 if the followingconditions are satisfied:

(i) v′(t), Av(t) ∈ Cβ,γ

0 (E), p ∈ E1 ⊂ E,(ii) (v(t), p) satisfies the equation and boundary conditions (1.1).

Here, Cβ,γ

0 (E), (0 ≤ γ ≤ β, 0 < β < 1) is the Holder space with weight obtained bycompletion of the space of all smooth E-valued functions ϕ(t) on [0, 1] in the norm

∥∥ϕ∥∥Cβ,γ

0 (E) = max0≤t≤1

∥∥ϕ(t)∥∥E + sup

0≤t<t+τ≤1

(t + τ)γ∥∥ϕ(t + τ) − ϕ(t)∥∥E

τβ. (2.7)

In the paper [23], the exact estimates in Cβ,γ

0 (E), (0 ≤ γ ≤ β, 0 < β < 1) and Cβ,γ

0 (Eα−β)(0 ≤ γ ≤ β ≤ α, 0 < α < 1) Holder spaces for the solution of problem (1.1) were proved. Inapplications, exact estimates for the solution of the boundary value problems for parabolicequations were obtained.

Now, we consider the application of Theorem 2.1. First, the boundary-value problemon the range {0 ≤ t ≤ 1, x ∈ Rn} for the 2m-order multidimensional parabolic equation isconsidered:

∂v(t, x)∂t

+∑

|r|=2mar(x)

∂|r|v(t, x)∂xr11 · · · ∂xrnn

+ σv(t, x) = f(t, x) + p(x), 0 < t < 1,

v(0, x) = ϕ(x), v(1, x) = ψ(x), x ∈ Rn, |r| = r1 + · · · + rn,(2.8)


where ar(x) andf(t, x) are given as sufficiently smooth functions. Here, σ is a sufficientlylarge positive constant.

It is assumed that the symbol

Bx(ξ) =∑

|r|=2mar(x)(iξ1)

r1 · · · (iξn)rn , ξ = (ξ1, . . . , ξn) ∈ Rn (2.9)

of the differential operator of the form

Bx =∑

|r|=2mar(x)

∂|r|

∂xr11 · · · ∂xrnn (2.10)

acting on functions defined on the space Rn satisfies the inequalities

0 < M1|ξ|2m ≤ (−1)mBx(ξ) ≤M2|ξ|2m <∞ (2.11)

for ξ /= 0.Problem (2.8) has a unique smooth solution. This allows us to reduce problem (2.8) to

problem (1.1) in a Banach space E = Cμ(Rn) of all continuous bounded functions defined onRn satisfying a Holder condition with the indicator μ ∈ (0, 1).

Theorem 2.2. For the solution of boundary problem (2.8), the following estimates are satisfied:

∥∥p∥∥Cμ(Rn) ≤M

[∥∥ϕ∥∥Cμ(Rn) +

∥∥ψ∥∥C2m+μ(Rn) +

1β

∥∥f∥∥Cβ(Cμ(Rn))

],

‖v‖C(Cμ(Rn)) ≤M[∥∥ϕ∥∥Cμ(Rn) +

∥∥ψ∥∥Cμ(Rn) +

∥∥f∥∥C(Cμ(Rn))

],

(2.12)

whereM is independent of ϕ(x), ψ(x), and f(t, x).

The proof of Theorem 2.2 is based on the abstract Theorem 2.1 and on the stronglypositivity of the operator A = Bx + σI defined by formula (2.10) (see, [31–33]).

Second, letΩ be the unit open cube in the n-dimensional Euclidean space Rn (0 < xk <

1, 1 ≤ k ≤ n)with boundary S,Ω = Ω∪S. In [0, 1]×Ω, we consider the mixed boundary valueproblem for the multidimensional parabolic equation

∂v(t, x)∂t

−n∑

r=1

αr(x)∂2v(t, x)∂x2

r

+ σv(t, x) = f(t, x) + p(x),

x = (x1, . . . , xn) ∈ Ω, 0 < t < 1,

v(0, x) = ϕ(x), v(1, x) = ψ(x), x ∈ Ω,

v(t, x) = 0, x ∈ S,

(2.13)

where αr(x) (x ∈ Ω), ϕ(x), ψ(x)(Ω), and f(t, x)(t ∈ (0, 1), x ∈ Ω) are given smooth functionsand αr(x) ≥ a > 0.Here, σ is a sufficiently large positive constant.


We introduce the Banach spaces Cβ

01(Ω)(β = (β1, . . . , βn), 0 < xk < 1, k = 1, . . . , n)of all continuous functions satisfying a Holder condition with the indicator β = (β1, ..., βn),βk ∈ (0, 1),1 ≤ k ≤ n, and with weight xβk

k(1 − xk − hk)βk , 0 ≤ xk < xk + hk ≤ 1, 1 ≤ k ≤ nwhich

is equipped with the norm

∥∥f∥∥Cβ

01(Ω) =∥∥f∥∥C(Ω) + sup

0≤xk<xk+hk≤1,1≤k≤n

∣∣f(x1, . . . , xn) − f(x1 + h1, . . . , xn + hn)∣∣

×n∏

k=1

h−βkkxβkk (1 − xk − hk)βk ,

(2.14)

where C(Ω) is the space of the all continuous functions defined on Ω, equipped with thenorm

∥∥f∥∥C(Ω) = max

x∈Ω

∣∣f(x)∣∣. (2.15)

It is known that the differential expression [34]

Av(x) = −n∑

r=1

αr(x)∂2v(x)∂x2

r

+ σv(x) (2.16)

defines a positive operator A acting on Cβ

01(Ω) with the domain D(A) = {v(x), ∂2v(x)/∂x2r ∈

Cβ

01(Ω), v(x) = 0 onS}.Therefore, we can replace mixed problem (2.13) by the abstract boundary problem

(1.1). Using the results of Theorem 2.1, we can obtain the following theorem on stability.

Theorem 2.3. For the solution of mixed boundary value problem (2.18), the following estimates arevalid:

∥∥p∥∥Cμ(Rn) ≤M

[∥∥ϕ∥∥Cμ

01(Ω) +∥∥ψ∥∥C

24μ01 (Ω) +

1β

∥∥f∥∥Cβ(Cμ

01(Ω))

],

‖v‖C(Cμ

01(Ω)) ≤M[∥∥ϕ∥∥Cμ

01(Ω) +∥∥ψ∥∥Cμ

01(Ω)) +∥∥f∥∥C(Cμ

01(Ω)))

],

(2.17)

whereM does not depend on ϕ(x), ψ(x), and f(t, x).

Third, we consider the mixed boundary value problem for parabolic equation

∂v(t, x)∂t

− a(x)∂2v(t, x)∂x2

+ σv(t, x) = f(t, x) + p(x), 0 < t < 1, 0 < x < 1,

v(0, x) = ϕ(x), v(1, x) = ψ(x), 0 ≤ x ≤ 1,

v(t, 0) = v(t, 1), vx(t, 0) = vx(t, 1), 0 ≤ t ≤ 1,

(2.18)


where a(x), ϕ(x), ψ(x), and f(t, x) are given sufficiently smooth functions and a(x) ≥ a > 0.Here, σ is a sufficiently large positive constant.

We introduce the Banach spaces Cβ[0, 1] (0 < β < 1) of all continuous functions ϕ(x)satisfying a Holder condition for which the following norms are finite,

∥∥ϕ∥∥Cβ[0,1] =

∥∥ϕ∥∥C[0,1] + sup

0≤x<x+τ≤1

∣∣ϕ(x + τ) − ϕ(x)∣∣τβ

, (2.19)

where C[0, 1] is the space of the all continuous functions ϕ(x) defined on [0, 1]with the usualnorm

∥∥ϕ∥∥C[0,1] = max

0≤x≤1

∣∣ϕ(x)∣∣. (2.20)

It is known that the differential expression [30]

Av = −a(x)v′′(x) + σv(x) (2.21)

defines a positive operator A acting in Cβ[0, 1] with the domain

D(A) ={v(x), v′′(x) ∈ Cβ[0, 1], v(0) = v(1), vx(0) = vx(1)

}. (2.22)

Therefore, we can replace the mixed problem (2.18) by the abstract boundary valueproblem (1.1). Using the result of Theorem 2.1, we can obtain the following theorem onstability.

Theorem 2.4. For the solution of mixed problem (2.18), the following estimates are valid:

∥∥p∥∥Cμ[0,1] ≤M

[∥∥ϕ∥∥Cμ[0,1] +

∥∥ψ∥∥C2+μ[0,1] +

1β

∥∥f∥∥Cβ(Cμ[0,1])

],

‖v‖C(Cμ[0,1]) ≤M[∥∥ϕ∥∥Cμ[0,1] +

∥∥ψ∥∥Cμ[0,1] +

∥∥f∥∥C(Cμ[0,1])

],

(2.23)

whereM is independent of ϕ(x), ψ(x), and f(t, x).

3. Rothe Difference Scheme for Parabolic Equations withan Unknown Parameter

In this section, our focus is the well-posedness of the Rothe difference scheme

τ−1(uk − uk−1) +Auk = ϕk + p, ϕk = f(tk),

tk = kτ, 1 ≤ k ≤N, Nτ = 1,

u0 = ϕ, uN = ψ

(3.1)

for approximately solving problem (1.1).


Let [0, 1]τ = {tk = kτ, k = 0, 1, . . . ,N, Nτ = 1} be the uniform grid space with step sizeτ > 0, whereN is a fixed positive integer.

Throughout the section, C([0, 1]τ , E) denotes the linear space of grid functions ϕτ ={ϕk}N1 with values in the Banach space E.

LetCτ(E) = C([0, 1]τ , E) be the Banach space of bounded grid functions with the norm

∥∥ϕτ∥∥Cτ (E)

= max1≤k≤N

∥∥ϕk∥∥E. (3.2)

For α ∈ [0, 1], let Cα(E) = Cα([0, 1]τ , E) be the Holder space with the following norm:

∥∥ϕτ∥∥Cα(E) =

∥∥ϕτ∥∥Cτ (E)

+ max1≤k<k+r≤N

∥∥ϕk+r − ϕk∥∥E

(rτ)α. (3.3)

Let us start with some lemmas we need in the following.

Lemma 3.1 (see [31]). The following estimates hold:

∥∥∥Rk∥∥∥E→E

≤ 1

(1 + δτ)k, k ≥ 1,

∥∥∥τARk∥∥∥E→E

≤ 1k, k ≥ 1,

(3.4)

for someM,δ > 0, which are independent of τ , where τ is a positive small number andR = (I + τA)−1

is the resolvent of A.

Lemma 3.2. The operator I − RN has an inverse Tτ = (I − RN)−1and the following estimate issatisfied:

‖Tτ‖E→E ≤M(δ). (3.5)

Let us now obtain the formula for the solution of problem (3.1). It is clear that the firstorder of accuracy difference scheme

τ−1(uk − uk−1) +Auk = p + ϕk, ϕk = f(tk),

tk = kτ, 1 ≤ k ≤N, Nτ = 1,

u0 = ϕ,

(3.6)

has a solution and the following formula holds:

uk = Rkϕ +k∑

j=1

Rk−j+1(p + ϕj)τ, 1 ≤ k ≤N. (3.7)


Applying formula (3.7) and the boundary condition

uN = ψ, (3.8)

we can write

ψ = RNϕ +N∑

j=1

RN−j+1ϕjτ +N∑

j=1

RN−j+1τp. (3.9)

Since

N∑

j=1

RN−j+1τ = A−1(I − R)N∑

j=1

RN−j = A−1(I − RN

), (3.10)

we have that

ψ = RNϕ +N∑

j=1

RN−j+1ϕjτ +A−1(I − RN

)p. (3.11)

Using Lemma 3.2, we get

p = Tτ

⎛

⎝Aψ −ARNϕ −N∑

j=1

ARN−j+1ϕjτ

⎞

⎠. (3.12)

Using formulas (3.7) and (3.12), we get

uk = Rkϕ +k∑

j=1

Rk−j+1ϕjτ +k∑

j=1

Rk−j+1τTτ

⎛

⎝Aψ −ARNϕ −N∑

j=1

ARN−j+1ϕjτ

⎞

⎠, 1 ≤ k ≤N.

(3.13)

Since

k∑

j=1

Rk−j+1τ = A−1(I − R)k∑

j=1

Rk−j = A−1(I − Rk

), (3.14)

we have that

uk = Rkϕ +k∑

j=1

Rk−j+1ϕjτ +(I − Rk

)Tτ

⎛

⎝ψ − RNϕ −N∑

j=1

RN−j+1ϕjτ

⎞

⎠, 1 ≤ k ≤N. (3.15)

Hence, difference equation (3.1) is uniquely solvable, and, for the solution, formulas (3.12)and (3.15) are valid.


Theorem 3.3. For the solution ({uk}Nk=1, p) of problem (3.1) in Cτ(E) × E, the stability estimates

∥∥p∥∥E ≤ M

[∥∥ϕ∥∥E +∥∥Aψ

∥∥E +

1β

∥∥∥{ϕk}Nk=1

∥∥∥Cβτ (E)

], (3.16)

∥∥∥{uk}Nk=1∥∥∥Cτ (E)

≤ M

[∥∥ϕ∥∥E +∥∥ψ∥∥E +∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

](3.17)

hold, whereM is independent of τ , ϕ, ψ, and {ϕk}Nk=1.

Proof. From formulas (3.7) and (3.12), it follows that

p = Tτ

⎛

⎝Aψ −ARNϕ −N−1∑

j=1

ARN−j+1(ϕj − ϕN)τ −(I − RN

)ϕN

⎞

⎠. (3.18)

Using this formula, the triangle inequality, and estimates (3.4), we obtain

∥∥p∥∥E ≤ ‖Tτ‖E→E

⎛

⎝∥∥Aψ∥∥H +∥∥∥ARN

∥∥∥E→E

∥∥ϕ∥∥E

+N−1∑

j=1

∥∥∥ARN−j+1∥∥∥E→E

∥∥ϕj − ϕN∥∥Eτ +(1 +∥∥∥RN

∥∥∥E→E

)∥∥ϕN∥∥E

⎞

⎠

≤ M

[∥∥ϕ∥∥E +∥∥Aψ

∥∥E +

1β

∥∥∥{ϕk}Nk=1

∥∥∥Cβτ (E)

].

(3.19)

The estimate (3.16) is proved. Using formula (3.15), the triangle inequality, andestimates (3.4), we obtain

‖uk‖E ≤⎛

⎝∥∥∥Rk∥∥∥E→E

∥∥ϕ∥∥E +

k∑

j=1

∥∥∥Rk−j+1∥∥∥E→E

∥∥ϕj∥∥Eτ +(1 +∥∥∥Rk∥∥∥E→E

)‖Tτ‖E→E

×⎛

⎝‖ψ‖E +∥∥∥RN

∥∥∥E→E

∥∥ϕ∥∥E +

N∑

j=1

∥∥∥RN−j+1∥∥∥E→E

∥∥ϕj∥∥Eτ

⎞

⎠

⎞

⎠

≤M[∥∥ϕ∥∥E +∥∥ψ∥∥E +∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

]

(3.20)

for any k. From that it follows estimate (3.17). Theorem 3.3 is proved.


Theorem 3.4. For the solution ({uk}Nk=1, p) of problem (3.1) inCτ(E)×E, the almost coercive stabilityestimates

∥∥p∥∥E ≤M

[∥∥ϕ∥∥E +∥∥Aψ

∥∥E +min

{ln

1τ, |ln ‖A‖E→E|

}∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

], (3.21)

∥∥∥{τ−1(uk − uk−1)

}Nk=1

∥∥∥Cτ (E)

+∥∥∥{Auk}Nk=1

∥∥∥Cτ (E)

≤M[∥∥Aϕ

∥∥E +∥∥Aψ

∥∥E +min

{ln


}∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

](3.22)

hold, whereM does not depend on τ , ϕ, ψ, and {ϕk}Nk=1.Proof. Using formula (3.12), the triangle inequality and estimates (3.4), we obtain

∥∥p∥∥E ≤ ‖Tτ‖E→E

⎛

⎝∥∥Aψ∥∥H +∥∥∥RN

∥∥∥E→E

∥∥Aϕ∥∥E +

N−1∑

j=1

∥∥∥ARN−j+1∥∥∥E→E

∥∥ϕj∥∥Eτ

⎞

⎠

≤M⎡

⎣∥∥Aϕ∥∥E +∥∥Aψ

∥∥E +

N−1∑

j=1

∥∥∥ARN−j+1∥∥∥E→E

τ∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

⎤

⎦.

(3.23)

Since [31]

N−1∑

j=1

∥∥∥ARN−j+1∥∥∥E→E

τ ≤Mmin{ln


}, (3.24)

we have estimate (3.21). Using formula (3.15), the triangle inequality, and estimates (3.4),(3.24), we obtain

‖Auk‖E ≤⎛

⎝∥∥∥Rk∥∥∥E→E

∥∥Aϕ∥∥E +

k∑

j=1

∥∥∥ARk−j+1∥∥∥E→E

∥∥ϕj∥∥Eτ +(1 +∥∥∥Rk∥∥∥E→E

)‖Tτ‖E→E

×⎛

⎝∥∥Aψ∥∥E +∥∥∥RN

∥∥∥E→E

∥∥Aϕ∥∥E +

N∑

j=1

∥∥∥ARN−j+1∥∥∥E→E

∥∥ϕj∥∥Eτ

⎞

⎠

⎞

⎠

≤M[∥∥Aϕ

∥∥E +∥∥Aψ

∥∥E +min

{ln


}∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

]

(3.25)

for any k. Therefore,

∥∥∥{Auk}Nk=1∥∥∥Cτ (E)

≤M[∥∥Aϕ

∥∥E +∥∥Aψ

∥∥E +min

{ln


}∥∥∥{ϕk}Nk=1

∥∥∥Cτ (E)

].

(3.26)

This estimate, triangle inequality, and (1.1) yield estimate (3.22). Theorem 3.4 is proved.


4. Applications

Now, we consider the applications of Theorems 3.3 and 3.4. The boundary value problem(2.18) for the parabolic differential equation is considered. The discretization of problem(2.18) is carried out in two steps. In the first step, we define the grid space

[0, 1]h = {x = xn : xn = nh, 0 ≤ n ≤M,Mh = 1}. (4.1)

Let us introduce the Banach space Ch = C([0, 1]h) of the grid functions ϕh(x) = {ϕn}M−11

defined on [0, 1]h, equipped with the norm

∥∥∥ϕh∥∥∥Ch

= maxx∈[0,1]h

∣∣∣ϕh(x)∣∣∣. (4.2)

To the differential operator A generated by problem (2.18), we assign the difference operatorAxh by the formula

Axhϕ

h(x) ={−(a(x)ϕx

)x,n + σϕn

}M−1

1(4.3)

acting in the space of grid functions ϕh(x) = {ϕn}M−11 satisfying the conditions ϕ0 = ϕM,

ϕ1 − ϕ0 = ϕM − ϕM−1. It is wellknown that Axh is a strongly positive operator in Ch. With the

help of Axh, we arrive at the boundary value problem

duh(t, x)dt

+Axhu

h(t, x) = ph(x) + fh(t, x), 0 < t < 1, x ∈ [0, 1]h,

uh(0, x) = ϕh(x), uh(1, x) = ψh(x), x ∈ [0, 1]h.

(4.4)

In the second step, we replace (4.4) with the difference scheme (3.1)


+Axhu

hk(x) = p

h(x) + fhk (x),

fhk (x) = fh(tk, x), tk = kτ, 1 ≤ k ≤N, x ∈ [0, 1]h,

uh(0, x) = ϕh(x), uh(1, x) = ψh(x), x ∈ [0, 1]h.

(4.5)

Theorem 4.1. The solution pairs ({uhk(x)}N

0 , ph(x)) of problem (4.5) satisfy the stability estimates

∥∥∥ph∥∥∥Ch

≤M1

[∥∥∥ϕh∥∥∥Ch

+∥∥∥ψh∥∥∥Ch

+∥∥∥Ax

hψh∥∥∥Ch

+1β

∥∥∥∥{fhk

}N1

∥∥∥∥Cβτ (Ch)

],

∥∥∥∥{uhk

}N1

∥∥∥∥Cτ (Ch)

≤M2

[∥∥∥ϕh∥∥∥Ch

+∥∥∥ψh∥∥∥Ch

+∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (Ch)

],

(4.6)

whereM1 andM2 do not depend on β,ϕh, ψh, and fhk, 1 ≤ k ≤N.


Here, Cβτ (Ch) is the grid space of grid functions {fhk }

N

1 defined on [0, 1]τ × [0, 1]h with norm

∥∥∥∥{fhk

}N1

∥∥∥∥Cβτ (Ch)

=∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (Ch)

+ sup1≤k<k+r≤N

∥∥∥fhk+r − fhk∥∥∥L2h

(rτ)β,

∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (Ch)

= max1≤k≤N

∥∥∥fhk∥∥∥Ch

.

(4.7)

The proof of Theorem 4.1 is based on Theorem 3.3 and the positivity property of theoperator Ax

h defined by formula (4.3).

Theorem 4.2. The solution pairs ({uhk(x)}N

0 , ph(x)) of problem (4.5) satisfy the almost coercive

stability estimates

∥∥∥ph∥∥∥Ch

≤M1

[∥∥∥Axhϕ

h∥∥∥Ch

+∥∥∥Ax

hψh∥∥∥Ch

+ ln1

τ + h

∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (Ch)

],

∥∥∥∥∥∥

{uhk − uhk−1τ

}N

1

∥∥∥∥∥∥Cτ (Ch)

+∥∥∥∥{Axhu

hk

}N1

∥∥∥∥Cτ (Ch)

≤M2

[∥∥Ax

hϕh∥∥Ch

+∥∥Ax

hψh∥∥Ch

+ ln1

τ + h

∥∥∥∥{fhk

}N1

∥∥∥∥Cτ (Ch)

],

(4.8)

whereM1 andM2 are independent of ϕh, ψh, and fhk , 1 ≤ k ≤N.

The proof of Theorem 4.2 is based on Theorem 3.4 and the positivity property of theoperator Ax

h defined by formula (4.3) and on the estimate

min{ln

1τ,∣∣∣ln∥∥Ax

h

∥∥Ch →Ch

∣∣∣}

≤M ln1

τ + h. (4.9)

Note that, in a similar manner, we can construct the difference schemes of the firstorder of accuracy with respect to one variable for approximate solutions of boundary valueproblems (2.8) and (2.13). Abstract theorems given from above permit us to obtain thestability, the almost stability estimates for the solutions of these difference schemes.

5. Conclusion

In this work, the first order of accuracy Rothe difference scheme for the approximate solutionof the boundary value problem of determining the parameter p of a parabolic equation

v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(1) = ψ (5.1)

in arbitrary Banach space E with the strongly positive operator A is studied. The well-posedness of the difference scheme is established. Some results in this paper in Hilbert


space H with self adjoint positive definite operator A were obtained in the paper [25]. Theinvestigation of this paper in arbitrary Banach space E with the strongly positive operator Apermits us to obtain the stability and almost stability estimates for the solution of differenceschemes for the approximate solution of differential equations with parameter are obtained.Of course, such type results for the solution of difference scheme for the following boundaryvalue problems

v′(t) +Av(t) = f(t) + p(0 ≤ t ≤ 1), v(0) = ϕ, v(λ) = ψ, 0 < λ ≤ 1,

v′(t) −Av(t) = f(t) + p(0 ≤ t ≤ 1), v(1) = ϕ, v(λ) = ψ, 0 ≤ λ < 1(5.2)

in an arbitrary Banach space with positive operator A and an unknown parameter p hold.

Acknowledgment

The authors are grateful to Professor Allaberen Ashyralyev (Fatih University,Turkey) for hiscomments and suggestions to improve the quality of the paper.

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Research ArticleKamenev-Type Oscillation Criteria for theSecond-Order Nonlinear Dynamic Equations withDamping on Time Scales

M. Tamer Senel

Department of Mathematics, Faculty of Sciences, Erciyes University, 38039 Kayseri, Turkey

Correspondence should be addressed to M. Tamer Senel, [email protected]

Received 6 March 2012; Accepted 22 March 2012


Copyright q 2012 M. Tamer Senel. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The oscillation of solutions of the second-order nonlinear dynamic equation (r(t)(xΔ(t))γ )Δ +p(t)(xΔ(t))γ + f(t, x(g(t))) = 0, with damping on an arbitrary time scale T, is investigated. Thegeneralized Riccati transformation is applied for the study of the Kamenev-type oscillation criteriafor this nonlinear dynamic equation. Several new sufficient conditions for oscillatory solutions ofthis equation are obtained.

1. Introduction

Much recent attention has been given to dynamic equations on time scales, or measure chains,and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatmentof the subject. Since then, several authors have expounded on various aspects of this newtheory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales byBohner and Peterson [3] also summarizes and organizes much of the time scale calculus.

A time scale T is an arbitrary nonempty closed subset of the real numbers R. Theforward and the backward jump operators on any time scale T are defined by σ(t) := inf{s ∈T : s > t}, ρ(t) := sup{s ∈ T : s < t}. A point t ∈ T, t > infT, is said to be left-dense if ρ(t) = t,right dense if t < supT and σ(t) = t, left scattered if ρ(t) < t, and right scattered if σ(t) > t. Thegraininess function μ for a time scale T is defined by μ(t) := σ(t)−t. For a function f : T → R

the (delta) derivative is defined by

fΔ(t) =f(σ(t)) − f(t)

σ(t) − t , (1.1)

if f is continuous at t and t is right scattered. If t is not right scattered, then the derivative is


defined by

fΔ(t) = lims→ t+

f(σ(t)) − f(s)σ(t) − s = lim

s→ t+

f(t) − f(s)t − s , (1.2)

provided this limit exists. A function f : [a, b] → R is said to be right-dense continuous if itis right continuous at each right-dense point and there exists a finite left limit at all left-densepoints, and f is said to be differentiable if its derivative exists. A useful formula dealing withthe time scale is that

fσ = f(σ(t)) = f(t) + μ(t)fΔ(t). (1.3)

Wewill make use of the following product and quotient rules for the derivative of the productfg and the quotient f/g (where ggσ /= 0) of two differentiable functions f and g:

(fg)Δ = fΔg + fσgΔ = fgΔ + fΔgσ,(f

g

)Δ

=fΔg − fgΔ

ggσ.

(1.4)

The integration by parts formula is

∫b

a

fΔ(t)g(t)Δt = f(b)g(b) − f(a)g(a) −∫b

a

fσ(t)gΔ(t)Δ(t). (1.5)

The function f : T → R is called rd-continuous if it is continuous at the right-dense pointsand if the left-sided limits exist in left-dense points. We denote the set of all f : T → R whichare rd-continuous and regressive by �. If p ∈ �, then we can define the exponential functionby

ep(t, s) = exp

(∫ t

s

ξμ(t)(p(τ(t))

)Δτ

)(1.6)

for t ∈ T, s ∈ Tk, where ξh(z) is the cylinder transformation, which is defined by

ξh(z) =

⎧⎨

⎩

log(1 + hz)h

, h /= 0,

z, h = 0.(1.7)

Alternately, for p ∈ � one can define the exponential function ep(·, t0), to be the uniquesolution of the IVP xΔ(t) = p(t)x(t)with x(t0) = 1.

The various-type oscillation and nonoscillation criteria for solutions of ordinary andpartial differential equations have been studied extensively in a large cycle of works (see[4–31]).


In [27], the authors have considered second-order nonlinear neutral dynamic equation

(r(t)((y(t) + p(t)y(t − τ))Δ

)γ)Δ+ f(t, y(t − δ)) = 0 (1.8)

on a time scale T. They have assumed that γ > 0 is a quotient of odd positive integers, τ andδ positive constants such that the delay functions τ(t) = t − τ < t and δ(t) = t − δ < t satisfyτ(t) and δ(t) : T → T for all t ∈ T, r(t) and p(t) real-valued positive functions defined on T

and also they have supposed that

(H1)∫∞t0(1/r(t))1/γΔt = ∞, 0 ≤ p(t) < 1,

(H2) f(t, u) : T × R → R is continuous such that uf(t, u) > 0 for all u/= 0 and there existsa nonnegative function q(t) defined on T such that |f(t, u)| ≥ q(t)|uγ |

and were concerned with oscillation properties of (1.8). In [28], Saker has considered second-order nonlinear neutral delay dynamic equation

(r(t)((y(t) + p(t)y(t − τ))Δ

)γ)Δ+ f(t, y(t − δ)) = 0, (1.9)

when γ ≥ 1 is an odd positive integer with r(t) and p(t) real-valued positive functions definedon T. The author also has improved some well-known oscillation results for second-orderneutral delay difference equations. Agarwal et al. [29] have considered the second-orderperturbed dynamic equation

(r(t)(xΔ)γ)Δ

+ F(t, x(t)) = G(t, x(t), xΔ(t)

), (1.10)

where γ ∈ N is odd and they have interested in asymptotic behavior of solutions of (1.10).Saker et al. [30] have studied the second-order damped dynamic equation with damping

(a(t)xΔ(t)

)Δ+ p(t)xΔσ

(t) + q(t)(foxσ

)= 0, (1.11)

when a(t), p(t), and q(t) are positive real-valued rd-continuous functions and they haveproved that if

∫∞t0(e−p/r(t, t0)/r(t))Δt = ∞ and

∫∞t0(e−p/r(t, t0)/r(t))Δt <∞, then every solution

of (1.11) is oscillatory.In the present paper, we consider the second order nonlinear dynamic equation

(r(t)(xΔ(t)

)γ)Δ+ p(t)

(xΔ(t)

)γ+ f(t, x(g(t)))

= 0, (1.12)

where p, r are real-valued, nonnegative, and right-dense continuous function on a time scaleT ⊂ R, with supT = ∞ and γ is a quotient of odd positive integers. We assume that g : T → T

is a nondecreasing function and such that g(t) ≥ t, for t ∈ T and limt→∞g(t) = ∞. Thefunction f ∈ C(T×R,R) is assumed to satisfy uf(t, u) > 0, for u/= 0 and there exists a positive


rd-continuous function q defined on T such that |f(t, u)/uγ | ≥ q(t) for u/= 0. Throughout thispaper we assume that

∫∞

t0

(e−p/r(t, t0)

r(t)

)1/γ

Δt = ∞. (A∗)

Since we are interested in the oscillatory of solutions near infinity, we assume thatsupT = ∞ and define the time scale interval [t0,∞)

Tby [t0,∞)

T:= [t0,∞)∩T. The oscillation

of solutions of the second-order nonlinear dynamic equation (1.12) with damping on anarbitrary time scale T is investigated. The generalized Riccati transformation is applied forthe study of the Kamenev-type oscillation criteria for this nonlinear dynamic differentialequation. Several new sufficient conditions for oscillatory solutions of this equation areobtained.

A solution x(t) of (1.12) is said to be oscillatory if it is neither eventually positive noreventually negative, otherwise it is nonoscillatory.

2. Preliminary Results

Lemma 2.1. Assume that the condition (A∗) is satisfied and (1.12) has a positive solution x(t) on[t0,∞)

T. Then there exists a sufficiently large t1 ∈ [t0,∞)

Tsuch that

(r(t)(xΔ(t)

)γ)Δ< 0, xΔ(t) > 0 for t ∈ [t1,∞)

T. (2.1)

Proof. Let t1 ∈ [t0,∞) such that x(g(t)) > 0 on [t1,∞). Since x(t) is positive nonoscillatorysolution of (1.12)we can assume that xΔ(t) < 0 for all large t. Then without loss of generalitywe take xΔ(t) < 0 for all t ≥ t2 ≥ t1. From (1.12) it follows that

(r(t)(xΔ(t)

)γ)Δ+ p(t)

(xΔ(t)

)γ= −f(t, x(g(t))) < 0 (2.2)

and so

(r(t)(xΔ(t)

)γ)Δ+ p(t)

(xΔ(t)

)γ< 0. (2.3)

Define y(t) = −r(t)(xΔ(t))γ . Hence

yΔ(t) +p(t)r(t)

y(t) > 0, (2.4)

and it implies that

y(t) > y(t2)e−p/r(·, t2). (2.5)


Then

−r(t)(xΔ(t)

)γ> −r(t2)

(xΔ(t2)

)γe−p/r(·, t2), (2.6)

and therefore

xΔ(t) ≤ r1/γ(t2)(xΔ(t2)

)(e−p/r(·, t2)r(t)

)1/γ

. (2.7)

Next an integration for t > t3 ≥ t2 and by (A∗) gives

x(t) ≤ x(t3) + r1/γ(t2)(xΔ(t2)

)∫ t

t3

(e−p/r(s, t2)

r(s)

)1/γ

Δs −→ −∞ as t −→ ∞ (2.8)

which is a contradiction. Hence xΔ(t) is not negative for all large t and so xΔ(t) > 0 for allt ≥ t1. This completes the proof of Lemma 2.1.

We now define

α1(t) :=(

1r(t)

∫∞

t

q(s)Δs)(1−γ)/γ

α2(t, u) :=

(r1/γ(t)

∫ t

u

Δsr1/γ(s)

)γ−1

α(t) :=

{α1(t), 0 < γ ≤ 1,α2(t, t1), γ ≥ 1.

(2.9)

Lemma 2.2. Assume that (A∗) holds and (1.12) has a positive solution x(t) on [t0,∞)T. Then there

exists a sufficiently large t1 ∈ [t0,∞)Tsuch that if 0 < γ ≤ 1 for t ≥ t1 one has

(xΔ(t)xσ(t)

)1−γ≥ α1(t). (2.10)

Whereas, if γ ≥ 1, one has

(x(t)xΔ(t)

)γ−1≥ α2(t, t1) for t ≥ t1. (2.11)

Proof. As in the proof of Lemma 2.1, there is a sufficiently large t1 ∈ [t0,∞)Tsuch that

x(t) > 0, xΔ(t) > 0,(r(t)(xΔ(t)

)γ)Δ< 0, for t ≥ t1. (2.12)


From (1.12) and (2.12) it follows that

(r(t)(xΔ(t)

)γ)Δ+ p(t)

(xΔ(t)

)γ= −f(t, x(g(t))) < 0, (2.13)

and so

(r(t)(xΔ(t)

)γ)Δ< −f(t, x(g(t))). (2.14)

Then

r(t)(xΔ(t)

)γ ≥∫∞

t

f(s, x(g(s)

))Δs ≥

∫∞

t

q(s)xγ(g(s)

)Δs

≥ xγ(g(t))∫∞

t

q(s)Δs ≥ (xσ(t))γ∫∞

t

q(s)Δs.(2.15)

Next, when 0 < γ ≤ 1, we get

(xΔ(t)xσ(t)

)1−γ≥ α1(t) for t ≥ t1. (2.16)

Finally, since r(t)(xΔ(t))γ is decreasing on [t1,∞)Tfor γ ≥ 1, we get

x(t) ≥ x(t) − x(t1) =∫ t

t1

(r(s)(xΔ(s)

)γ)1/γ

r1/γ(s)Δs

≥(r(t)(xΔ(t)

)γ)1/γ ∫ t

t1

1r1/γ(s)

Δs,

(2.17)

and we obtain

(x(t)xΔ(t)

)γ−1≥ α2(t, t1) for t ≥ t1. (2.18)

3. Main Results

Theorem 3.1. Assume that (A∗) holds and there exist a function φ(t) such that r(t)φ(t) is a Δ-differentiable function and a positive real rd-functions Δ-differentiable function z(t) such that

lim supt→∞

∫ t

t1

[Ψ(s) − 1

4r(s)(ν(s))2

γz(s)α(s)

]Δs = ∞, (3.1)


where

Ψ(t) = −z(t)(q(s) − (r(t)φ(t))Δ +

γα(t)r(t)

(p(t)(r(t)φ(t)

)σ +((r(t)φ(t)

)σ)2)),

ν(t) = zΔ(t) − γz(t)α(t)r(t)

(p(t) − 2

(r(t)φ(t)

)σ).

(3.2)

Then every solution of (1.12) is oscillatory.

Proof. Suppose to the contrary that x(t) is a nonoscillatory solution of (1.12). Without loss ofgenerality, there is a t1 ∈ [t0,∞)

T, sufficiently large, so that x(t) satisfies the conclusions of

Lemmas 2.1 and 2.2 on [t0,∞)T. Define the function w(t) by Riccati substitution

w(t) = z(t)r(t)

((xΔ(t)x(t)

)γ

+ φ(t)

), t ≥ t1. (3.3)

Then w(t) satisfies

wΔ(t) =(z(t)xγ(t)

)(r(t)(xΔ(t)

)γ)Δ+(z(t)xγ(t)

)Δ(r(t)(xΔ(t)

)γ)σ

+ z(t)(r(t)φ(t)

)Δ + zΔ(t)(r(t)φ(t)

)σ,

wΔ(t) =(z(t)xγ(t)

)(r(t)(xΔ(t)

)γ)Δ+

(zΔ(t)xγ(t) − z(t)(xγ(t))Δ

xγ(t)(xγ(t))σ

)(r(t)(xΔ(t)

)γ)σ

+ z(t)(r(t)φ(t)


)σ.

(3.4)

From (1.12) and the definition of w(t) for t ≥ t1 it follows that

wΔ(t) =(z(t)xγ(t)

)(−p(t)

(xΔ(t)

)γ − f(t, x(g(t))))+ zΔ(t)

(r(t)(xΔ(t)

)γ)σ

(xγ(t))σ

− z(t)(xγ(t))Δ

(r(t)(xΔ(t)

)γ)σ

xγ(t)(xγ(t))σ+ z(t)

(r(t)φ(t)


)σ.

(3.5)

Using the fact that f(t, x(g(t))) ≥ q(t)xγ(g(t)) and x(t) is a increasing function, we obtain

wΔ(t) ≤ −z(t)q(t) − z(t)p(t) (xγ(t))Δ

xγ(t)+ zΔ(t)

((r(t)(xΔ(t))γ

xγ(t)

)σ

+(r(t)φ(t)

)σ)

− z(t)(xΔ(t)

)γ

xγ(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ)+ z(t)

(r(t)φ(t)

)Δ.

(3.6)

Now we consider the following two cases: 0 < γ ≤ 1 and γ > 1.


In the first case 0 < γ ≤ 1. Using the Potzsche chain rule (see, [3]), we obtain

(xγ(t))Δ = γ∫1

0

[x(t) + hμ(t)xΔ(t)

]γ−1dhxΔ(t) ≥ γ(xσ(t))γ−1xΔ(t). (3.7)

Using (3.7) in (3.6) for t ≥ t1, we get

wΔ(t) ≤ −z(t)q(t) − γz(t)p(t)xΔ(t)xσ(t)

(xσ(t)x(t)

)γ+ zΔ(t)

wσ(t)zσ(t)

− γz(t)xΔ(t)xσ(t)

(xσ(t)x(t)

)γ(wσ(t)zσ(t)

− (r(t)φ(t))σ)+ z(t)

(r(t)φ(t)

)Δ.

(3.8)

By Lemmas 2.1 and 2.2, for t ≥ t1, we have that

xΔ(t)xσ(t)

=1r(t)

r(t)(xΔ(t)

)γ

(xγ(t))σ

(xΔ(t)xσ(t)

)1−γ≥ α1(t)

r(t)

(r(t)(xΔ(t)

)γ)σ

(xγ(t))σ,

xσ(t)x(t)

≥ 1.

(3.9)

In the view of (3.8), and (3.9) we get

wΔ(t) ≤ −z(t)q(t) + z(t)(r(t)φ(t))Δ − γz(t)p(t)α1(t)r(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ)

+ zΔ(t)wσ(t)zσ(t)

− γz(t)α1(t)r(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ)2

.

(3.10)

In the second case γ > 1. Applying the Potzsche chain rule (see, [3]), we obtain

(xγ(t))Δ = γ∫1

0

[x(t) + hμ(t)xΔ(t)

]γ−1dhxΔ(t) ≥ γ(x(t))γ−1xΔ(t). (3.11)

In the view of (3.11), (3.6) yields

wΔ(t) ≤ −z(t)q(t) + z(t)(r(t)φ(t))Δ − γz(t)p(t) (x(t))γ−1

xγ(t)xΔ(t)


− γz(t) (x(t))γ−1

xγ(t)xΔ(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ).

(3.12)

By Lemmas 2.1 and 2.2, we have that

xΔ(t)x(t)

=1r(t)

r(t)(xΔ(t)

)γ

xγ(t)

(x(t)xΔ(t)

)γ−1≥ α2(t, t1)

r(t)

(r(t)(xΔ(t)

)γ)σ

(xγ(t))σ. (3.13)


By (3.13), (3.12), and then using the definition of w(t), we get

wΔ(t) ≤ −z(t)q(t) + z(t)(r(t)φ(t))Δ − γz(t)p(t)α2(t, t1)r(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ)


− γz(t)α2(t, t1)r(t)

(wσ(t)zσ(t)

− (r(t)φ(t))σ)2

.

(3.14)

Using (3.10), (3.14), and the definitions of Ψ(t), ν(t), and α(t) for γ > 0, we get

wΔ(t) ≤ −Ψ(t) + ν(t)wσ(t)zσ(t)

− γz(t)α(t)r(t)

(wσ(t))2

(zσ(t))2. (3.15)

Then, we can write

wΔ(t) ≤ −Ψ(t) +r(t)(ν(t))2

4γz(t)α(t)−⎡

⎣√γz(t)α(t)r(t)

wσ(t)zσ(t)

− 12

√r(t)

γz(t)α(t)ν(t)

⎤

⎦2

, (3.16)

and so, we get

wΔ(t) ≤ −[Ψ(t) − r(t)(ν(t))2

4γz(t)α(t)

]. (3.17)

Integrating (3.17)with respect to s from t1 to t, we get

w(t) −w(t1) ≤ −∫ t

t1

[Ψ(s) − r(s)(ν(s))2

4γz(s)α(s)

]Δs, (3.18)

and this implies that

∫ t

t1

[Ψ(s) − r(s)(ν(s))2

4γz(s)α(s)

]Δs ≤ |w(t1)| (3.19)

which contradicts to assumption (3.1). This completes the proof of Theorem 3.1.

Corollary 3.2. Assume that (A∗) holds. If

lim supt→∞

∫ t

t1

[q(s) +

γα(s)p2(s)4r(s)

]Δs = ∞, (3.20)

then every solution of (1.12) is oscillatory.


Example 3.3. Consider the nonlinear dynamic equation

(t−γ(xΔ(t)

)γ)Δ+12t−1−γ

(xΔ(t)

)γ+

1t1/γ

xγ(g(t))= 0, t ∈ [t0,∞)

T, T = 2N, (3.21)

where γ ≥ 1 is the quotient of the odd positive integers.We have that p(t) = (1/2)(t−1−γ), q(t) =1/t1/γ and r(t) = t−γ . If T = 2N, then σ(t) = 2t and e−1/σ(t)(t, t0) = t0/t. So we get e−p/r(t, t0) =t0/t. It is clear that (A∗) holds. Indeed,

∫ t

t0

(e−p/r(·, t0)

r(s)

)1/γ

Δs = (t0)1/γ∫ t

t0

1s(1/γ)−1

Δs = ∞,

α2(t, t0) =

((r(t))1/γ

∫ t

t0

Δs

(r(s))1/γ

)γ−1= t1/(γ−1)

(∫ t

t0

Δss−1

)γ−1,

(3.22)

and then

∫ t

t0

Δss−1

= ∞ (3.23)

and so we can find t∗ ≥ t1 such that∫ tt0Δs/r1/γ ≥ 1 for t ≥ t∗. Then we can see from

Corollary 3.2 that it follows that

lim supt→∞

∫ t

t1

[1s1/γ

+γα(s)

(p(s)

)2

4r(s)

]Δs = ∞, (3.24)

and therefore every solution of (3.21) is oscillatory.Now, let us introduce the class of functions �.Let D0 ≡ {(t, s) ∈ T

2 : t > s ≥ t0} and D ≡ {(t, s) ∈ T2 : t ≥ s ≥ t0}. The function

H ∈ Crd(D,R) has the following properties:

H(t, t) = 0, t ≥ t0, H(t, s) > 0, on D0, (3.25)

and H has a continuous Δ-partial derivative HΔs (t, s) on D0 with respect to the second

variable. (H is rd-continuous function ifH is rd-continuous function in t and s.)

Theorem 3.4. Assume that the conditions of Lemma 2.1 are satisfied. Furthermore, suppose thatthere exist functions H,HΔ

s ∈ Crd(D,R) such that (3.25) holds and there exist a function φ(t) withr(t)φ(t) a Δ-differentiable function and a positive Δ-differentiable function z(t) such that

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)Ψ(s) − r(s)

4γH(t, s)z(s)α(s)ϕ2(t, s)

]Δs = ∞, (3.26)

where ϕ(t, s) = [HΔs (t, s) +H(t, s)ν(s)]. Then every solution of (1.12) is oscillatory on [t0,∞)

T.


Proof. Assume that (1.12) has a nonoscillatory solution on [t0,∞)T. Then without loss of

generality, there is a sufficiently large t1 ∈ [t0,∞)Tsuch that x(t) satisfies the conclusions

of Lemmas 2.1 and 2.2 on [t0,∞)T. Consider the generalized Riccati substitution

w(t) = z(t)r(t)

((xΔ(t)x(t)

)γ

+ φ(t)

). (3.27)

We proceed as Theorem 3.1 and from (3.15) it follows that

wΔ(t) ≤ −Ψ(t) + ν(t)wσ(t)zσ(t)

− γz(t)α(t)r(t)

(wσ(t))2

(zσ(t))2. (3.28)

Multiplying both sides of (3.28) by H(t, s) and integrating with respect to s from t1 to t (t ≥t1), we obtain

∫ t

t1

H(t, s)Ψ(s)Δ(s) ≤ −∫ t

t1

H(t, s)wΔ(s) +∫ t

t1

H(t, s)ν(s)wσ(s)zσ(s)

Δs

−∫ t

t1

γH(t, s)z(s)α(s)r(s)

(wσ(s))2

(zσ(s))2Δs.

(3.29)

Integrating by parts, we get

∫ t

t1

H(t, s)Ψ(s)Δ(s) ≤ H(t, t1)w(t1) +∫ t

t1

HΔs (t, s)w

σ(s)Δs +∫ t

t1

H(t, s)ν(s)wσ(s)zσ(s)

Δs

−∫ t

t1


(wσ(s))2

(zσ(s))2Δs,

∫ t

t1

H(t, s)Ψ(s)Δ(s) ≤H(t, t1)w(t1) +∫ t

t1

[HΔ

s (t, s) +H(t, s)ν(s)]wσ(s)zσ(s)

Δs

−∫ t

t1


(wσ(s))2

(zσ(s))2Δs.

(3.30)

It is easy to see that

∫ t

t1

H(t, s)Ψ(s)Δ(s) ≤ H(t, t1)w(t1) +∫ t

t1

ϕ(t, s)wσ(s)zσ(s)

Δs

−∫ t

t1


(wσ(s))2

(zσ(s))2Δs,

(3.31)

where

ϕ(t, s) =[HΔ

s (t, s) +H(t, s)ν(s)]. (3.32)


Then we can write

∫ t

t1

H(t, s)Ψ(s)Δ(s) ≤ H(t, t1)w(t1) +∫ t

t1

r(s)ϕ2(t, s)4γH(t, s)z(s)α(s)

Δs

−∫ t

t1

⎡

⎣√γH(t, s)z(s)α(s)

r(s)wσ(s)zσ(s)

− 12

√r(s)

γH(t, s)z(s)α(s)ϕ(t, s)

⎤

⎦2

Δs.

(3.33)

Hence

∫ t

t1

H(t, s)Ψ(s) − r(s)ϕ2(t, s)4γH(t, s)z(s)α(s)

Δs ≤ H(t, t1)w(t1)

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)Ψ(s) − r(s)ϕ2(t, s)

4γH(t, s)z(s)α(s)

]Δs ≤ w(t1)

(3.34)

which contradicts with assumption (3.26). This completes the proof of Theorem 3.4.

Corollary 3.5. Assume that (A∗) holds. Furthermore, suppose that there exist functionsH,HΔs , and

h ∈ Crd(D,R) such that (3.25) holds and there exist a function φ(t) such that r(t)φ(t) is a Δ-differentiable function and a positive Δ-differentiable function z(t) such that

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)Ψ(s) − h2(s)(zσ(s))2r(s)

4γz(s)α(s)

]Δs = ∞, (3.35)

whereΨ(t) is as defined in Theorem 3.1 andHΔs = −h(t, s)

√H(t, s)−H(t, s)ν(t)/zσ(t). Then every

solution of (1.12) is oscillatory on [t0,∞)T.

Theorem 3.6. Assume that (A∗) holds and there exists a Δ-differentiable positive function z(t) suchthat

lim supt→∞

∫ t

t1

[z(s)q(s) − r(s)ξγ+1(s)

(γ + 1

)γ+1zγ(s)

]Δs = ∞, (3.36)

where

ξ(t) = zΔ(t) − z(t)p(t)r(t)

. (3.37)

Then every solution of (1.12) is oscillatory.


Proof. Suppose that (1.12) has a nonoscillatory solution on [t0,∞)T. Then without loss of



w(t) = z(t)r(t)

(xΔ(t)x(t)

)γ

. (3.38)

From (3.6) it follows that

wΔ(t) ≤ −z(t)q(t) − z(t)p(t)(xΔ(t)

)γ

xγ(t)+ zΔ(t)

wσ(t)zσ(t)

− z(t)(xΔ(t)

)γ

xγ(t)wσ(t)zσ(t)

. (3.39)

In the same manner as in the proof of Theorem 3.1, we get

(xγ(t))Δ ≥{γ(xσ(t))γ−1xΔ, 0 < γ ≤ 1γ(x(t))γ−1xΔ, γ > 1.

(3.40)

If 0 < γ ≤ 1, then we have that

wΔ(t) ≤ −z(t)q(t) +[zΔ(t) − z(t)p(t)

r(t)

]wσ(t)zσ(t)

− γz(t)(xσ(t))γ

xγ(t)xΔ(t)xσ(t)

wσ(t)zσ(t)

, (3.41)

whereas, if γ > 1, we have that

wΔ(t) ≤ −z(t)q(t) +[zΔ(t) − z(t)p(t)

r(t)

]wσ(t)zσ(t)

− γz(t)xσ(t)x(t)

xΔ(t)xσ(t)

wσ(t)zσ(t)

. (3.42)

Using the fact that x(t) is increasing and (r(t)(xΔ(t))γ is decreasing on [t0,∞)T, we get

xσ(t) ≥ x(t), xΔ(t) ≥(rσ(t)r(t)

)1/γ(xΔ(t)

)σ. (3.43)

Using (3.41), (3.42), and (3.43), we obtain

wΔ(t) ≤ −z(t)q(t) + ξ(t)wσ(t)

zσ(t)− z(t) γ

r1/γ(t)

(wσ(t)zσ(t)

)λ, (3.44)

where λ = (γ + 1)/γ . Define A > 0 and B > 0 by

Aλ =γz(t)(wσ(t))λ

(zσ(t))λr1/γ(t), Bλ−1 =

r1/(γ+1)(t)ξ(t)λγ1/λz1/λ(t)

. (3.45)


Then using the inequality (see [32])

λABλ−1 −Aλ ≤ (λ − 1)Bλ, (3.46)

we obtain

ξ(t)wσ(t)zσ(t)

− z(t) γ

r1/γ(t)

(wσ(t)zσ(t)

)λ≤ r(t)ξγ+1(t)(γ + 1

)γ+1zγ(t)

. (3.47)

From this last inequality and (3.44) it follows that

lim supt→∞

∫ t

t1

[z(s)q(s) − r(s)ξγ+1(s)

(γ + 1

)γ+1zγ(s)

]Δs ≤ w(t1) (3.48)

which contradicts with the assumption (3.36). Theorem 3.6 is proved.

Example 3.7. Consider the second-order equation

(tγ(xΔ(t)

)γ)Δ+

1t2

(xΔ(t)

)γ+1txγ(g(t))= 0, (3.49)

where γ = 1/3 ≤ 1, r(t) = t1/3, q(t) = 1/t, t ≥ t0 = 2. Then it follows that

e−p/r(t, 2) ≥ 1 −∫ t

2

p(s)r(s)

Δs = 1 −∫ t

2s−7/3Δs >

12

(3.50)

for t ≥ 2, and so

∫ t

2

(1

r(s)e−p/r(s, 2)

)1/γ

Δs ≥(12

)3 ∫ t

2

1sΔs −→ ∞ as t −→ ∞. (3.51)

Hence (A∗) is satisfied. Now let z(t) = 1 for t ≥ 2. Then

lim supt→∞

∫ t

2

[z(s)q(s) − r(s)ξγ+1(s)

(γ + 1

)γ+1zγ(s)

]Δs = lim sup

t→∞

∫ t

2

[1s− s−25/9

(4/3)4/3

]Δs = ∞, (3.52)

and so (3.36) is satisfied as well. Hence by Theorem 3.6, we have that (3.49) is oscillatory.

Theorem 3.8. Assume that the conditions of Lemma 2.1 hold. Furthermore, suppose that there existfunctions H,HΔ

s ∈ Crd(D,R) such that (3.25) holds and there exists a positive real rd-functionsΔ-differentiable function z(t) such that

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)z(s)q(s) − Cγ+1(t, s)r(s)

(γ + 1

)γ+1zγ(s)(H(t, s))γ

]Δs = ∞, (3.53)


where C(t, s) = HΔs z

σ(s) +H(t, s)ξ(t) and ξ(t) = zΔ(t) − z(t)(p(t)/r(t)). Then every solution of(1.12) is oscillatory on [t0,∞)

T.

Proof . Assume that (1.12) has a nonoscillatory solution on [t0,∞)T. Then without loss of



w(t) = z(t)r(t)

(xΔ(t)x(t)

)γ

. (3.54)

By Theorem 3.6 and inequality (3.44)

wΔ(t) ≤ −z(t)q(t) + ξ(t)wσ(t)

zσ(t)− z(t) γ

r1/γ(t)

(wσ(t)zσ(t)

)λ, (3.55)

where λ = (γ+1)/γ . Multiplying both sides of (3.55)withH(t, s) and integrating with respectto s from t1 to t (t ≥ t1), we get

∫ t

t1

H(t, s)z(s)q(s)Δs ≤ −∫ t

t1

H(t, s)wΔ(s)Δ(s) +∫ t

t1

H(t, s)ξ(s)wσ(s)zσ(s)

−∫ t

t1

H(t, s)z(s)γ

r1/γ(s)

(wσ(s)zσ(s)

)λΔs.

(3.56)

Integrating by parts and using (3.25), we obtain

∫ t

t1

H(t, s)z(s)q(s)Δs ≤ H(t, t1)w(t1)∫ t

t1

C(t, s)wσ(s)zσ(s)

−∫ t

t1

γH(t, s)z(s)r1/γ(s)

(wσ(s)zσ(s)

)λΔs.

(3.57)

Define A > 0 and B > 0 by

Aλ =γH(t, s)z(t)(wσ(t))λ

(zσ(t))λr1/γ(t), Bλ−1 =

r1/(γ+1)(t)C(t, s)

λ(γH(t, s)z(s)

)1/λ . (3.58)

Using the inequality (see [32])

λABλ−1 −Aλ ≤ (λ − 1)Bλ, (3.59)

we get

C(t, s)wσ(t)zσ(t)

− γH(t, s)z(t)r1/γ(t)

(wσ(t)zσ(t)

)λ≤ r(t)Cγ+1(t, s)(γ + 1

)γ+1Hγ(t, s)zγ(t)

. (3.60)


From this last inequality and (3.55) it follows that

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)z(s)q(s) − r(s)Cγ+1(t, s)

(γ + 1

)γ+1Hγ(t, s)zγ(t)

]Δs ≤ w(t1) (3.61)

which contradicts with the assumption (3.53). This completes the proof of Theorem 3.8.

Corollary 3.9. Assume that all conditions of Lemma 2.1 hold. Furthermore, suppose that there existfunctionsH,HΔ

s , and h ∈ Crd(D,R) such that (3.25) holds and there exists a positiveΔ-differentiablefunction z(t) such that

lim supt→∞

1H(t, t1)

∫ t

t1

[H(t, s)z(s)q(s) − (−h(t, s))γ+1r(s)

(γ + 1

)γ+1zγ(s)

]Δs = ∞, (3.62)

where HΔs + H(t, s)ξ(t)/zσ(s) = −h(t, s)(H(t, s))γ/(γ+1)/zσ(t). Then every solution of (1.12) is

oscillatory on [t0,∞)T.

Example 3.10. Consider the second-order dynamic equation

(tγ(xΔ(t)

)γ)Δ+

1t2

(xΔ(t)

)γ+1txγ(g(t))= 0, (3.63)

where t ∈ [t0,∞)T, t1 ≥ t0 = 2, γ = 5/3 ≥ 1, q(t) = 1/t. It is easy to check that (A∗) holds. For

z(t) = 1 andH(t, s) = (t − s)2, it immediately follows that

h(t, s) ={(t − s) − (t − s)2 + (t − σ(s))

}(t − s)2γ/(γ+1) (3.64)

and so −h(t, s) = 0. Hence,

lim supt→∞

1H(t, 2)

∫ t

2

[H(t, s)z(s)q(s) − (−h(t, s))γ+1r(s)

(γ + 1

)γ+1zγ(s)

]Δs =lim sup

t→∞

1t2

∫ t

2

1s(t − s)2Δs = ∞.

(3.65)

Therefore by Corollary 3.9, every solution of (3.63) is oscillatory.

Acknowledgments

The author would like to thank Professor A. Ashyralyev and anonymous referee for theirhelpful suggestions to the improvement of this paper. This work was supported by ResearchFund of the Erciyes University Project no. FBA-11-3391.


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Research ArticleGeneralizations of Wendroff Integral Inequalitiesand Their Discrete Analogues

Maksat Ashyraliyev

Department of Mathematics and Computer Sciences, Bahcesehir University, Besiktas,34353 Istanbul, Turkey

Correspondence should be addressed to Maksat Ashyraliyev, [email protected]



Copyright q 2012 Maksat Ashyraliyev. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Generalizations ofWendroff type integral inequalities with four dependent limits and their discreteanalogues are obtained. In applications, these results are used to establish the stability estimatesfor the solution of the Goursat problem.

1. Introduction

Integral inequalities play a significant role in the theory of ordinary and partial differentialequations. They are useful to investigate some properties of the solutions of equations, suchas existence, uniqueness, and stability, see for instance [1–6].

Most scientific and technical problems can be solved by using mathematical modellingand new numerical methods. This is based on the mathematical description of real processesand the subsequent solving of the appropriate mathematical problems on the computer.The mathematical models of many scientific and technical problems lead to already knownor new problems of partial differential equations. In most of the cases it is difficult tofind the exact solutions of the problems for partial differential equations. For this reasondiscrete methods play a significant role, especially due to the increasing role of mathematicalmethods of solving problems in various areas of science and engineering. A well-known andwidely applied method of approximate solutions for problems of differential equations is themethod of difference schemes. Modern computers allow us to implement highly accuratedifference schemes. Hence, the task is to construct and investigate highly accurate differenceschemes for various types of partial differential equations. The investigation of stabilityand convergence of these difference schemes is based on the discrete analogues of integralinequalities, see for instance [1, 7–9].


Due to various motivations, several generalizations and applications of Wendroff-typeintegral inequality have been obtained and used extensively, see for instance [10–12]. In[12], the following generalizations of Wendroff-type integral inequality in two independentvariables are obtained.

Theorem 1.1. Assume that u(x, y) ≥ 0 and a(x, y) ≥ 0 are continuous functions on 0 ≤ x ≤ L,0 ≤ y ≤M and the inequalities

u(x, y) ≤ C +

∫x

0

∫y

0a(s, t)u(s, t)dt ds, 0 ≤ x ≤ L, 0 ≤ y ≤M (1.1)

hold, where C = const ≥ 0. Then, for u(x, y) the inequalities:

u(x, y) ≤ C exp

[∫x

0

∫y

0a(s, t)dt ds

], 0 ≤ x ≤ L, 0 ≤ y ≤M (1.2)

are satisfied.

Theorem 1.2. Assume that u(x, y) ≥ 0 is a continuous function on 0 ≤ x ≤ L, 0 ≤ y ≤ M and theinequalities:

u(x, y) ≤ f(x, y) +

∫x

0a(s)u

(s, y)ds +

∫y

0b(t)u(x, t)dt, 0 ≤ x ≤ L, 0 ≤ y ≤M (1.3)

hold, where f(x, y) > 0 is a continuous function on 0 ≤ x ≤ L, 0 ≤ y ≤ M and increasing withrespect to each variable, a(x) ≥ 0 and b(y) ≥ 0 are integrable functions on 0 ≤ x ≤ L and 0 ≤ y ≤M,respectively. Then, for u(x, y) the inequalities:

u(x, y) ≤ f(x, y) exp

[∫x

0a(s)ds +

∫y

0b(t)dt +

∫x

0

∫y

0a(s)b(t)dt ds

], 0 ≤ x ≤ L, 0 ≤ y ≤M

(1.4)

are satisfied.

In this paper, generalizations of Wendroff-type integral inequalities in two indepen-dent variables with four dependent limits and their discrete analogues are obtained. Inapplications, these results are used to obtain the stability estimates of solutions for theGoursat problem.

2. Wendroff-type Integral Inequalities with Four DependentLimits and Their Discrete Analogues

First of all, let us give the discrete analogue of the Gronwall-type integral inequality with twodependent limits. We will need this result in the remaining part of the paper.


Theorem 2.1. Assume that vi ≥ 0, ai ≥ 0, δi ≥ 0, i = −N, . . . ,N are the sequences of real numbersand the inequalities:

vi ≤ δi + h|i|−1∑

j=−|i|+1j /= 0

ajvj , i = −N, . . . ,N (2.1)

hold. Then, for vi the inequalities:

vi ≤ δi + exp

⎡⎢⎢⎣h

|i|−1∑

j=−|i|+1j /= 0

aj

⎤⎥⎥⎦h

|i|−1∑

j=−|i|+1j /= 0

ajδj exp

⎡⎢⎢⎣−h

|j|∑

n=−|j|n/= 0

an

⎤⎥⎥⎦, i = −N, . . . ,N (2.2)

are satisfied.

Proof. The proof of (2.2) for i = −1, 0, 1 follows directly from (2.1). Let us prove (2.2) fori = −N, . . . ,−2, 2, . . . ,N. We denote

yi = h|i|−1∑

j=−|i|+1j /= 0

ajvj , i = −N, . . . ,N. (2.3)

Then, (2.1) gets the form

vi ≤ δi + yi, i = −N, . . . ,N. (2.4)

Moreover, we have

y−i = yi, i = −N, . . . ,N. (2.5)

Then, using (2.3)–(2.5) for i = 1, . . . ,N − 1, we obtain

yi+1 − yi = h(aivi + a−iv−i) ≤ hai(yi + δi

)+ ha−i

(y−i + δ−i

)

= h(ai + a−i)yi + h(aiδi + a−iδ−i).(2.6)

So,

yi+1 ≤ [1 + h(ai + a−i)]yi + h(aiδi + a−iδ−i), i = 1, . . . ,N − 1. (2.7)


Then by induction, we can prove that

yi ≤i−1∏

j=1

[1 + h

(aj + a−j

)]y1 +

i−2∑

j=1

h(ajδj + a−jδ−j

) i−1∏

n=j+1

[1 + h(an + a−n)]

+ h(ai−1δi−1 + a−i+1δ−i+1),

(2.8)

for i = 2, . . . ,N. Since y1 = 0, using (2.4) and the inequality 1 + x < exp[x], x > 0, we obtain

vi ≤ δi + hi−2∑

j=1

(ajδj + a−jδ−j

)exp

⎡

⎣hi−1∑

n=j+1

(an + a−n)

⎤

⎦ + h(ai−1δi−1 + a−i+1δ−i+1)

= δi + exp

[hi−1∑

n=1

(an + a−n)

]hi−2∑

j=1


)exp

[−h

j∑

n=1

(an + a−n)

]

+ h(ai−1δi−1 + a−i+1δ−i+1)

= δi + exp

⎡

⎣hi−1∑

j=1

(aj + a−j

)⎤

⎦hi−1∑

j=1


)exp

[−h

j∑

n=1

(an + a−n)

]

= δi + exp

⎡⎢⎢⎣h

i−1∑

j=−i+1j /= 0

aj

⎤⎥⎥⎦h

i−1∑

j=1


)exp

⎡⎢⎢⎣−h

j∑

n=−jn /= 0

an

⎤⎥⎥⎦

= δi + exp

⎡⎢⎢⎣h

i−1∑

j=−i+1j /= 0

aj

⎤⎥⎥⎦h

i−1∑

j=−i+1j /= 0

ajδj exp

⎡⎢⎢⎢⎣−h

|j|∑

n=−|j|n/= 0

an

⎤⎥⎥⎥⎦.

(2.9)

So, we proved (2.2) for i = 2, . . . ,N.Let us prove (2.2) for i = −N, . . . ,−2. Using (2.3)–(2.5) for i = −N + 1, . . . ,−1, we have

yi−1 − yi = h(aivi + a−iv−i) ≤ hai(yi + δi

)+ ha−i

(y−i + δ−i

)

= h(ai + a−i)yi + h(aiδi + a−iδ−i).(2.10)

So,

yi−1 ≤ [1 + h(ai + a−i)]yi + h(aiδi + a−iδ−i), i = −N + 1, . . . ,−1. (2.11)



yi ≤−i−1∏

n=1

[1 + h(an + a−n)]y−1 +−i−2∑

j=1

h(ajδj + a−jδ−j

) −i−1∏

n=j+1

[1 + h(an + a−n)]

+ h(a−i−1δ−i−1 + ai+1δi+1),

(2.12)

for i = −N, . . . ,−2. Since y−1 = 0, using (2.4) and the inequality 1 + x < exp[x], x > 0, weobtain

vi ≤ δi + h−i−2∑

j=1


)exp

⎡

⎣h−i−1∑

n=j+1

(an + a−n)

⎤

⎦ + h(a−i−1δ−i−1 + ai+1δi+1)

= δi + exp

[h−i−1∑

n=1

(an + a−n)

]h−i−2∑

j=1


)exp

[−h

j∑

n=1

(an + a−n)

]

+ h(a−i−1δ−i−1 + ai+1δi+1)

= δi + exp

⎡

⎣h−i−1∑

j=1

(aj + a−j

)⎤

⎦h−i−1∑

j=1


)exp

[−h

j∑

n=1

(an + a−n)

]

= δi + exp

⎡⎢⎢⎣h

−i−1∑

j=i+1j /= 0

aj

⎤⎥⎥⎦h

−i−1∑

j=1


)exp

⎡⎢⎢⎣−h

j∑

n=−jn /= 0

an

⎤⎥⎥⎦

= δi + exp

⎡⎢⎢⎣h

−i−1∑

j=i+1j /= 0

aj

⎤⎥⎥⎦h

−i−1∑

j=i+1j /= 0

ajδj exp

⎡⎢⎢⎢⎣−h

|j|∑

n=−|j|n/= 0

an

⎤⎥⎥⎥⎦.

(2.13)

So, we proved (2.2) for i = −N, . . . ,−2.

By putting Nh = 1 and passing to limit h → 0 in the Theorem 2.1, we obtain thefollowing generalization of Gronwall’s integral inequality with two dependent limits.

Theorem 2.2. Assume that v(t) ≥ 0, δ(t) ≥ 0 are continuous functions on [−1, 1], a(t) ≥ 0 is anintegrable function on [−1, 1], and the inequalities

v(t) ≤ δ(t) + sgn(t)∫ t

−ta(s)v(s)ds, −1 ≤ t ≤ 1 (2.14)


hold. Then for v(t), the inequalities:

v(t) ≤ δ(t) + exp

[sgn(t)

∫ t

−ta(s)ds

]sgn(t)

∫ t

−ta(s)δ(s) exp

[− sgn(s)

∫ s

−sa(τ)dτ

]ds,

−1 ≤ t ≤ 1

(2.15)

are satisfied.

Now, we consider the generalizations of Wendroff inequality for integrals in twoindependent variables with four dependent limits and their discrete analogues.

Theorem 2.3. Assume that vkn ≥ 0, akn ≥ 0, n = −N, . . . ,N, k = −K, . . . , K are sequences of realnumbers, and the inequalities:

vkn ≤ C + h1h2|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

aτsvτs , n = −N, . . . ,N, k = −K, . . . , K (2.16)

hold, where C = const > 0, h1 = const > 0, h2 = const > 0. Then for vkn, the following inequalities:

vkn ≤ C exp

⎡⎢⎢⎣h1h2

|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

aτs

⎤⎥⎥⎦, n = −N, . . . ,N, k = −K, . . . , K. (2.17)

are satisfied.

Proof. The proof of (2.17) for n = −1, 0, 1, k = −K, . . . , K and k = −1, 0, 1, n = −N, . . . ,N followsdirectly from (2.16). Let us prove (2.17) for n = −N, . . . ,−2, 2, . . . ,N, k = −K, . . . ,−2, 2, . . . , K.We denote

ωkn = C + h1h2

|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

aτsvτs , n = −N, . . . ,N, k = −K, . . . , K. (2.18)

Then, (2.16) gets the form:

vkn ≤ ωkn, n = −N, . . . ,N, k = −K, . . . , K. (2.19)

Furthermore, we have

ωkn = ωk

−n = ω−kn = ω−k

−n, n = −N, . . . ,N, k = −K, . . . , K. (2.20)


From (2.18) for n = −N, . . . ,N, k = 1, . . . , K − 1, we have

ωk+1n −ωk

n = h1h2|n|−1∑

s=−|n|+1s /= 0

(aksv

ks + a

−ks v

−ks

)≥ 0. (2.21)

So,

ωkn ≤ ωk+1

n , n = −N, . . . ,N, k = 1, . . . , K − 1. (2.22)

Then using (2.18)–(2.22) for n = 1, . . . ,N − 1, k = −K, . . . ,−2, 2, . . . , K, we obtain

ωkn+1 −ωk

n = h1h2|k|−1∑

τ=−|k|+1τ /= 0

(aτnvτn + a

τ−nv

τ−n) ≤ h1h2

|k|−1∑

τ=−|k|+1τ /= 0

(aτnωτn + a

τ−nω

τ−n)

= h1h2|k|−1∑

τ=−|k|+1τ /= 0

ωτn(a

τn + a

τ−n) = h1h2

|k|−1∑

τ=1

ωτn(a

τn + a

τ−n) + h1h2

|k|−1∑

τ=1

ω−τn

(a−τn + a−τ−n

)

= h1h2|k|−1∑

τ=1

ωτn

(aτn + a

τ−n + a

−τn + a−τ−n

) ≤ h1h2ω|k|n

|k|−1∑

τ=1

(aτn + a

τ−n + a

−τn + a−τ−n

)

= h1h2ωkn

|k|−1∑

τ=−|k|+1τ /= 0

(aτn + aτ−n).

(2.23)

So,

ωkn+1 ≤ ωk

n

⎡⎢⎢⎣1 + h1h2

|k|−1∑

τ=−|k|+1τ /= 0

(aτn + aτ−n)

⎤⎥⎥⎦, n = 1, . . . ,N − 1, k = −K, . . . ,−2, 2, . . . , K. (2.24)


ωkn ≤ ωk

1

n−1∏

s=1

⎡⎢⎢⎣1 + h1h2

|k|−1∑

τ=−|k|+1τ /= 0

(aτs + aτ−s)

⎤⎥⎥⎦, n = 2, . . . ,N, k = −K, . . . ,−2, 2, . . . , K. (2.25)


Since ωk1 = C, k = −K, . . . , K, using (2.19) and the inequality 1 + x < exp[x], x > 0, we obtain

vkn ≤ ωkn ≤ C exp

⎡⎢⎢⎣h1h2

n−1∑

s=1

|k|−1∑

τ=−|k|+1τ /= 0

(aτs + aτ−s)

⎤⎥⎥⎦ = C exp

⎡⎢⎢⎣h1h2

n−1∑

s=−n+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

aτs

⎤⎥⎥⎦. (2.26)

So, we proved (2.17) for n = 2, . . . ,N, k = −K, . . . ,−2, 2, . . . , K.Using (2.18)–(2.22) for n = −N + 1, . . . ,−1, k = −K, . . . ,−2, 2, . . . , K, we obtain

ωkn−1 −ωk

n = h1h2|k|−1∑

τ=−|k|+1τ /= 0

(aτnvτn + a

τ−nv

τ−n) ≤ h1h2

|k|−1∑

τ=−|k|+1τ /= 0

(aτnωτn + a

τ−nω

τ−n)

= h1h2|k|−1∑

τ=−|k|+1τ /= 0

ωτn(a

τn + a

τ−n) = h1h2

|k|−1∑

τ=1

ωτn(a

τn + a

τ−n) + h1h2

|k|−1∑

τ=1

ω−τn

(a−τn + a−τ−n

)

= h1h2|k|−1∑

τ=1

ωτn

(aτn + a

τ−n + a

−τn + a−τ−n

) ≤ h1h2ω|k|n

|k|−1∑

τ=1

(aτn + a

τ−n + a

−τn + a−τ−n

)

= h1h2ωkn

|k|−1∑

τ=−|k|+1τ /= 0

(aτn + aτ−n).

(2.27)

So,

ωkn−1 ≤ ωk

n

⎡⎢⎣1 + h1h2

|k|−1∑τ=−|k|+1τ /= 0

(aτn + aτ−n)

⎤⎥⎦, n = −N + 1, . . . ,−1, k = −K, . . . ,−2, 2, . . . , K.

(2.28)


ωkn ≤ ωk

−1−n−1∏

s=1

⎡⎢⎢⎣1 + h1h2

|k|−1∑

τ=−|k|+1τ /= 0

(aτs + aτ−s)

⎤⎥⎥⎦, n = −N, . . . ,−2, k = −K, . . . ,−2, 2, . . . , K. (2.29)


Since ωk−1 = C, k = −K, . . . , K using (2.19) and the inequality 1 + x < exp[x], x > 0, we obtain

vkn ≤ ωkn ≤ C exp

⎡⎢⎢⎣h1h2

−n−1∑

s=1

|k|−1∑

τ=−|k|+1τ /= 0

(aτs + aτ−s)

⎤⎥⎥⎦ = C exp

⎡⎢⎢⎣h1h2

−n−1∑

s=n+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

aτs

⎤⎥⎥⎦. (2.30)

So, we proved (2.17) for n = −N, . . . ,−2, k = −K, . . . ,−2, 2, . . . , K.

Theorem 2.4. Assume that vkn ≥ 0, δkn > 0, an ≥ 0, bk ≥ 0, n = −N, . . . ,N, k = −K, . . . , K aresequences of real numbers, and the inequalities:

vkn ≤ δkn + h1|n|−1∑

s=−|n|+1s /= 0

asvks + h2

|k|−1∑

τ=−|k|+1τ /= 0

bτvτn, n = −N, . . . ,N, k = −K, . . . , K, (2.31)

δkn ≤ δkn+1, n = −N, . . . ,N − 1, k = −K, . . . , K,

δkn ≤ δk+1n , n = −N, . . . ,N, k = −K, . . . , K − 1(2.32)

hold, where h1 = const > 0, h2 = const > 0. Then for vkn, n = −N, . . . ,N, k = −K, . . . , K, theinequalities:

vkn ≤ δkn exp

⎡⎢⎢⎣h1

|n|−1∑

s=−|n|+1s /= 0

as + h2|k|−1∑

τ=−|k|+1τ /= 0

bτ + h1h2|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

asbτ

⎤⎥⎥⎦ (2.33)

are satisfied.

Proof. We denote ukn = vkn/δkn, n = −N, . . . ,N, k = −K, . . . , K. Then, (2.31) takes the form

ukn ≤ 1 + h1|n|−1∑

s=−|n|+1s /= 0

asuks + h2

|k|−1∑

τ=−|k|+1τ /= 0

bτuτn, n = −N, . . . ,N, k = −K, . . . , K. (2.34)

Next, by denoting

Tkn = 1 + h1|n|−1∑

s=−|n|+1s /= 0

asuks , n = −N, . . . ,N, k = −K, . . . , K, (2.35)

we have

ukn ≤ Tkn + h2|k|−1∑

τ=−|k|+1τ /= 0

bτuτn, n = −N, . . . ,N, k = −K, . . . , K. (2.36)


By using Theorem 2.1, we obtain

ukn ≤ Tkn + exp

⎡⎢⎢⎣h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦h2

|k|−1∑

τ=−|k|+1τ /= 0

bτTτn exp

⎡⎢⎢⎣−h2

|τ |∑

j=−|τ |j /= 0

bj

⎤⎥⎥⎦. (2.37)

Inserting (2.35) yields

ukn ≤ 1 + h1|n|−1∑

s=−|n|+1s /= 0

asuks + exp

⎡⎢⎢⎣h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ exp

⎡⎢⎢⎣−h2

|τ |∑

j=−|τ |j /= 0

bj

⎤⎥⎥⎦

+ exp

⎡⎢⎢⎣h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦h1h2

|k|−1∑

τ=−|k|+1τ /= 0

|n|−1∑

s=−|n|+1s /= 0

asbτuτs exp

⎡⎢⎢⎣−h2

|τ |∑

j=−|τ |j /= 0

bj

⎤⎥⎥⎦

≤ exp

⎡⎢⎢⎣h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦ + h1

|n|−1∑

s=−|n|+1s /= 0

asuks

+ exp

⎡⎢⎢⎣h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦h1h2

|k|−1∑

τ=−|k|+1τ /= 0

|n|−1∑

s=−|n|+1s /= 0

asbτuτs exp

⎡⎢⎢⎣−h2

|τ |∑

j=−|τ |j /= 0

bj

⎤⎥⎥⎦.

(2.38)

Now, by denoting

wkn = ukn exp

⎡⎢⎢⎣−h2

|k|−1∑

τ=−|k|+1τ /= 0

bτ

⎤⎥⎥⎦, n = −N, . . . ,N, k = −K, . . . , K, (2.39)

we have

wkn ≤ 1 + h1

|n|−1∑

s=−|n|+1s /= 0

aswks + h1h2

|k|−1∑

τ=−|k|+1τ /= 0

|n|−1∑

s=−|n|+1s /= 0

asbτwτs . (2.40)

Let us denote the right-hand side of (2.40) by Rkn. Then, (2.40) gets the form

wkn ≤ Rk

n, n = −N, . . . ,N, k = −K, . . . , K. (2.41)


Then by using induction, we can prove that

Rkn exp

⎡⎢⎢⎣−h1

|n|−1∑

s=−|n|+1s /= 0

as

⎤⎥⎥⎦ ≤ 1 + h1h2

|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

asbτRτs exp

⎡⎢⎢⎣−h1

|s|−1∑

j=−|s|+1j /= 0

aj

⎤⎥⎥⎦. (2.42)

By using Theorem 2.3, we obtain

Rkn exp

⎡⎢⎢⎣−h1

|n|−1∑

s=−|n|+1s /= 0

as

⎤⎥⎥⎦ ≤ exp

⎡⎢⎢⎣h1h2

|n|−1∑

s=−|n|+1s /= 0

|k|−1∑

τ=−|k|+1τ /= 0

asbτ

⎤⎥⎥⎦, n = −N, . . . ,N, k = −K, . . . , K.

(2.43)

Finally, by combining (2.39), (2.41), and (2.43), we finish the proof of (2.33).

By puttingNh1 = L,Kh2 =M and passing to limit as h1 → 0, h2 → 0 in Theorems 2.3and 2.4, we obtain the following two theorems about the generalizations of Wendroff integralinequality with four dependent limits.

Theorem 2.5. Assume that v(x, y) ≥ 0 and b(x, y) ≥ 0 are continuous functions on −L ≤ x ≤ L,−M ≤ y ≤M and the inequalities

v(x, y) ≤ C + sgn

(xy) ∫x

−x

∫y

−yv(s, t)b(s, t)dt ds, −L ≤ x ≤ L, −M ≤ y ≤M (2.44)

hold, where C = const ≥ 0. Then for v(x, y) the inequalities

v(x, y) ≤ C exp

[sgn(xy) ∫x

−x

∫y

−yb(s, t)dt ds

], −L ≤ x ≤ L, −M ≤ y ≤M (2.45)

are satisfied.

Theorem 2.6. Assume that v(x, y) ≥ 0 is a continuous function on −L ≤ x ≤ L, −M ≤ y ≤M, andthe inequalities:

v(x, y) ≤ f

(x, y)+ sgn(x)

∫x

−xv(s, y)a(s)ds

+ sgn(y) ∫y

−yv(x, t)b(t)dt, −L ≤ x ≤ L, −M ≤ y ≤M

(2.46)


hold, where f(x, y) > 0 is a continuous function on −L ≤ x ≤ L, −M ≤ y ≤ M and increasingwith respect to each variable, a(x) ≥ 0 and b(y) ≥ 0 are integrable functions on −L ≤ x ≤ L and−M ≤ y ≤M, respectively. Then for v(x, y), the inequalities

v(x, y) ≤ f(x, y) exp

[sgn(x)

∫x−x a(s)ds+sgn

(y) ∫y

−y b(t)dt+sgn(xy) ∫x

−x∫y−y a(s)b(t)dsdt

]

(2.47)

are satisfied.

Finally, we formulate (without proofs) the generalizations of Wendroff-type inequali-ties for the integrals in three independent variables with six dependent limits.

Theorem 2.7. Assume that v(x, y, z) ≥ 0, b(x, y, z) ≥ 0 (−l1 ≤ x ≤ l1, −l2 ≤ y ≤ l2, −l3 ≤ z ≤ l3)are continuous functions, and the inequalities:

v(x, y, z

) ≤ C + sgn(xyz) ∫x

−x

∫y

−y

∫z

−zv(s, t, τ)b(s, t, τ)dτ dt ds (2.48)

hold, where C = const ≥ 0. Then for v(x, y, z), the inequalities

v(x, y, z

) ≤ C exp

[sgn(xyz) ∫x

−x

∫y

−y

∫z

−zb(s, t, τ)dτ dt ds

],

−l1 ≤ x ≤ l1, −l2 ≤ y ≤ l2, −l3 ≤ z ≤ l3(2.49)

are satisfied.

Theorem 2.8. Assume that v(x, y, z) ≥ 0 (−l1 ≤ x ≤ l1, −l2 ≤ y ≤ l2, −l3 ≤ z ≤ l3) is a continuousfunction, and the inequalities:

v(x, y, z

) ≤ δ(x, y, z

)+ sgn(x)

∫x

−xv(s, y, z

)a(s)ds + sgn

(y) ∫y

−yv(x, t, z)b(t)dt

+ sgn(z)∫z

−zv(s, t, τ)c(τ)dτ

(2.50)

hold, where a(x) ≥ 0 (−l1 ≤ x ≤ l1), b(y) ≥ 0 (−l2 ≤ y ≤ l2), and c(z) ≥ 0 (−l3 ≤ z ≤ l3) areintegrable functions, δ(x, y, z) > 0 is continuous and increasing with respect to each variable. Thenfor v(x, y, z), the inequalities:

v(x, y, z

)≤ δ(x, y, z) exp[T(x, y, z

)+ 3

√sgn(xyz) ∫x

−x

∫y

−y

∫z

−za(s)b(t)c(τ)dτ dt ds eT(x,y,z)

]

−l1 ≤ x ≤ l1, −l2 ≤ y ≤ l2, −l3≤z≤ l3(2.51)


are satisfied, where

T(x, y, z

)= sgn(x)

∫x

−xa(s)ds + sgn

(y) ∫y

−yb(t)dt + sgn(z)

∫z

−zc(τ)dτ

+ sgn(xy) ∫x

−x

∫y

−ya(s)b(t)dt ds + sgn(xz)

∫x

−x

∫z

−za(s)c(τ)dτ ds

+ sgn(yz) ∫y

−y

∫z

−zb(t)c(τ)dτ dt.

(2.52)

3. Applications

In applications, we consider the Goursat problem for hyperbolic equations:

uxy = f(x, y)+

∂

∂y

(a(x, y)u(x, y)+ a(−x, y)u(−x, y))

+∂

∂x

(b(x, y)u(x, y)+ b(x,−y)u(x,−y)), −l1 < x < l1, −l2 < y < l2,

u(x, 0) = φ(x), −l1 ≤ x ≤ l1,u(0, y)= ψ(y), −l2 ≤ y ≤ l2.

(3.1)

A function u(x, y) is called a solution of the Goursat problem (3.1) if the followingconditions are satisfied:u(x, y) is twice continuously differentiable on the region [−l1, l1] ×[−l2, l2], and the derivatives at the endpoints are understood as the appropriate unilateralderivatives;

Theorem 3.1. Assume that the functions φ(x) and ψ(y) are continuously differentiable and φ(0) =ψ(0). Let a(x, y), b(x, y), and f(x, y) be continuously differentiable functions. Then, there is a uniquesolution of the problem (3.1) and the stability inequalities:

∣∣u(x, y)∣∣ ≤

(l1l2f +

32(φ + ψ

)+ 2al1φ + 2bl2ψ

)e2a|x|+2b|y|+4ab|xy|,

∣∣ux(x, y)∣∣,∣∣uy(x, y)∣∣ ≤M1,

∣∣uxy(x, y)∣∣ ≤M2

(3.2)

hold, whereM1 andM2 do not depend on x and y and

f = max|x|≤l1|y|≤l2

∣∣f(x, y)∣∣, a = max

|x|≤l1|y|≤l2

∣∣a(x, y)∣∣, b = max

|x|≤l1|y|≤l2

∣∣b(x, y)∣∣,

φ = max|x|≤l1

∣∣φ(x)∣∣, ψ = max

|y|≤l2∣∣ψ(y)∣∣.

(3.3)


The proof of this theorem is based on the formula:

u(x, y)= φ(x) + ψ

(y) − φ(0) −

∫x

−xa(s, 0)φ(s)ds −

∫y

−yb(0, t)ψ(t)dt +

∫x

0

∫y

0f(s, t)dt ds

+∫x

−xa(s, y)u(s, y)ds +

∫y

−yb(x, t)u(x, t)dt, −l1 ≤ x ≤ l1, −l2 ≤ y ≤ l2

(3.4)

and on the Theorem 2.6.Now, we consider the difference schemes for approximate solutions of problem (3.1):

uky,n

− uk−1y,n

h1=aknu

kn − ak−1n uk−1n

h2+ak−nu

k−n − ak−1−n u

k−1−n

h2+bknu

kn − bkn−1ukn−1

h1

+b−kn u−kn − b−kn−1u−kn−1

h1+ fkn , −N + 1 ≤ n ≤N, −K + 1 ≤ k ≤ K,

akn = a(xn, yk

), bkn = b

(xn, yk

), fkn = f

(xn, yk

), −N ≤ n ≤N, −K ≤ k ≤ K,

xn = nh1, −N ≤ n ≤N, Nh1 = l1, yk = kh2, −K ≤ k ≤ K, Kh2 = l2,

u0n = φ(xn), −N ≤ n ≤N, uk0 = ψ(yk), −K ≤ k ≤ K,

(3.5)

where uky,n

= (ukn − ukn−1)/h2.

Theorem 3.2. For the solution of difference schemes (3.5), the following estimates are satisfied:

max−N≤n≤N−K≤k≤K

∣∣∣ukn∣∣∣ ≤M

⎡

⎣ max−N+1≤n≤N−K+1≤k≤K

∣∣∣fkn∣∣∣ + max

−N≤n≤N

∣∣φn∣∣ + max

−K≤k≤K

∣∣ψk∣∣⎤

⎦, (3.6)

whereM does not depend on h1, h2, fkn , φn, ψk(−N ≤ n ≤N, −K ≤ k ≤ K).

The proof of this theorem is based on the following formula:

ukn = φn + ψk − φ0 − h1n∑

s=−ns/= 0

a0sφs − h2k∑

τ=−kτ /= 0

bτ0ψτ + h1h2n∑

s=1

k∑

τ=1

fτs

+ h1n∑

s=−ns/= 0

aksuks + h2

k∑

τ=−kτ /= 0

bτnuτn, −N ≤ n ≤N, −K ≤ k ≤ K

(3.7)

and on Theorem 2.4.


4. Conclusion

In this paper, generalizations of Wendroff-type inequalities for the integrals in two in-dependent variables with four dependent limits and their discrete analogues are studied. Thegeneralizations of Wendroff-type integral inequalities are used to establish stability estimatesfor the solution of the Goursat problem. A difference scheme approximately solving theGoursat problem is presented. Bu using the discrete analogues of the generalizations ofWendroff-type integral inequalities, stability estimates for the solution of this differencescheme are established.

References

[1] M. Ashyraliyev, “A note on the stability of the integral-differential equation of the hyperbolic type ina Hilbert space,” Numerical Functional Analysis and Optimization, vol. 29, no. 7-8, pp. 750–769, 2008.

[2] S. G. Krein, Linear Differential Equations in a Banach Space, Nauka, Moscow, Russia, 1966.[3] R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in

Banach-valued weighted space,” Boundary Value Problems, no. 1, pp. 9–42, 2005.[4] S. Ashirov and N. Kurbanmamedov, “Investigation of the solution of a class of integral equations of

Volterra type,” Izvestiya Vysshikh Uchebnykh ZavedeniıMatematika, vol. 9, pp. 3–8, 1987.[5] S. Ashirov and Y. D. Mamedov, “A Volterra-type integral equation,” Ukrainian Mathematical Journal,

vol. 40, no. 4, pp. 438–442, 1988.[6] A. Ashyralyev, E. Misirli, and O. Mogol, “A note on the integral inequalities with two dependent

limits,” Journal of Inequalities and Applications, vol. 2010, Article ID 430512, 2010.[7] A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations,

Birkhauser, Berlin, Germany, 2004.[8] A. Ashyralyev and P. E. Sobolevskiı,Well-Posedness of Parabolic Difference Equations, Birkhauser, Berlin,

Germany, 1994.[9] P. Henrici,Discrete Variable Methods in Ordinary Differential Equations, JohnWiley & Sons, London, UK,

1962.[10] E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin, Germany, 1961.[11] A. Corduneanu, “A note on the Gronwall inequality in two independent variables,” Journal of Integral

Equations, vol. 4, no. 3, pp. 271–276, 1982.[12] T. Nurimov and D. Filatov, Integral Inequalities, FAN, Tashkent, Uzbekistan, 1991.


Research ArticleA Note on Nonlocal Boundary Value Problems forHyperbolic Schrodinger Equations

Yildirim Ozdemir and Mehmet Kucukunal

Department of Mathematics, Duzce University, Konuralp, 81620 Duzce, Turkey

Correspondence should be addressed to Yildirim Ozdemir, [email protected]



Copyright q 2012 Y. Ozdemir and M. Kucukunal. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The nonlocal boundary value problem d2u(t)/dt2 +Au(t) = f(t) (0 ≤ t ≤ 1), i(du(t)/dt) +Au(t) =g(t) (−1 ≤ t ≤ 0), u(0+) = u(0−), ut(0+) = ut(0−), Au(−1) = αu(μ) + ϕ, 0 < μ ≤ 1, for hyperbolicSchrodinger equations in a Hilbert space H with the self-adjoint positive definite operator A isconsidered. The stability estimates for the solution of this problem are established. In applications,the stability estimates for solutions of the mixed-type boundary value problems for hyperbolicSchrodinger equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for partial differential equationsand partial differential equations of mixed type have been studied extensively by manyresearches (see, e.g., [1–12] and the references given therein).

In the present paper, the nonlocal boundary value problem

d2u(t)dt2

+Au(t) = f(t) (0 ≤ t ≤ 1),

idu(t)dt

+Au(t) = g(t) (−1 ≤ t ≤ 0),

u(0+) = u(0−), ut(0+) = ut(0−),

Au(−1) = αu(μ) + ϕ, 0 < μ ≤ 1

(1.1)

for differential equations of hyperbolic Schrodinger type in a Hilbert space H with self-adjoint positive definite operator A is considered.


It is known that various nonlocal boundary value problems for the hyperbolicSchrodinger equations can be reduced to problem (1.1).

A function u(t) is called a solution of the problem (1.1) if the following conditions aresatisfied.

(i) u(t) is twice continuously differentiable on the interval (0,1] and continuouslydifferentiable on the segment [−1, 1]. The derivatives at the endpoints of thesegment are understood as the appropriate unilateral derivatives.

(ii) The element u(t) belongs to D(A) for all t ∈ [−1, 1], and the function Au(t) iscontinuous on the segment [−1, 1].

(iii) u(t) satisfies the equations and nonlocal boundary condition (1.1).

In the present paper, the stability estimates for the solution of the problem (1.1) forthe hyperbolic Schrodinger equation are established. In applications, the stability estimatesfor the solutions of the mixed-type boundary value problems for hyperbolic Schrodingerequations are obtained.

Finally note that hyperbolic Schrodinger equations play important role in physics andengineering (see, e.g., [13–16] and the references given therein).

Furthermore, the investigation of the numerical solution of initial value problems andSchrodinger equations is the subject of extensive research activity during the last decade(indicatively [17–25] and the references given therein).

2. The Main Theorem

Let H be a Hilbert space, and let A be a positive definite self-adjoint operator with A ≥ δI,where δ > δ0 > 0. Throughout this paper, {c(t), t ≥ 0} is a strongly continuous cosine operatorfunction defined by

c(t) =eitA

1/2+ e−itA

1/2

2. (2.1)

Then, from the definition of the sine operator function s(t)

s(t)u =∫ t

0c(s)uds, (2.2)

it follows that

s(t) = A−1/2 eitA1/2 − e−itA1/2

2i. (2.3)

For the theory of cosine operator function, we refer to Fattorini [26] and Piskarev and Shaw[27].

We begin with two lemmas that will be needed as follows.



‖c(t)‖H→H ≤ 1,∥∥∥A1/2s(t)

∥∥∥H→H

≤ 1, t ≥ 0,∥∥∥e±itA

∥∥∥H→H

≤ 1, t ≥ 0.(2.4)

Lemma 2.2. Let

|α| < δ√1 + δ

. (2.5)

Then, the operator

I − α[A−1c

(μ)+ is(μ)]eiA (2.6)

has an inverse

T =(I − α[A−1c

(μ)+ is(μ)]eiA)−1 , (2.7)

and the estimate

‖T‖H→H ≤M (2.8)

holds, whereM does not depend on α and μ.

Proof. Actually, the proof of estimate (2.8) is based on the following estimate:

∥∥∥−α[A−1c

(μ)+ is(μ)]eiA∥∥∥H→H

< 1. (2.9)

Using the definitions of cosine and sine operator functions, A ≥ δI, δ > 0 (positivity), andA = A∗ (self-adjointness property), we obtain

∥∥−α[A−1c(μ)+ is(μ)]eiA∥∥H→H ≤ sup

δ≤ρ<∞

∣∣∣∣∣−α[1ρcos(√

ρμ)+

i√ρsin(√

ρμ)]eiρ∣∣∣∣∣

≤ supδ≤ρ<∞

|α|∣∣∣∣∣1ρcos(√

ρμ)+

i√ρsin(√

ρμ)∣∣∣∣∣

∣∣∣eiρ∣∣∣

≤ |α| supδ≤ρ<∞

√1ρ2

cos2(√

ρμ)+1ρsin2(√ρμ

)

≤ |α|√1 + ρρ

.

(2.10)


Since

√1 + ρρ

≤√1 + δδ

, (2.11)

we have that

∥∥∥−α[A−1c

(μ)+ is(μ)]eiA∥∥∥H→H

<δ√1 + δ

·√1 + δδ

= 1. (2.12)

Hence, Lemma 2.2 is proved.

Now, we will obtain the formula for solution of problem (1.1). It is known that forsmooth data of initial value problems

d2u(t)dt2

+Au(t) = f(t) (0 ≤ t ≤ 1),

u(0) = u0, u′(0) = u′0,

idu(t)dt

+Au(t) = g(t) (−1 ≤ t ≤ 0),

u(−1) = u−1,

(2.13)

there are unique solutions of problems (2.13), and following formulas hold:

u(t) = c(t)u(0) + s(t)u′(0) +∫ t

0s(t − y)f(y)dy, 0 ≤ t ≤ 1, (2.14)

u(t) = ei(t+1)Au−1 − i∫ t

−1ei(t−y)Ag

(y)dy, −1 ≤ t ≤ 0. (2.15)

Using (2.14), (2.15), and (1.1), we can write

u(t) = [c(t) + iAs(t)]

{eiAu−1 − i

∫0

−1e−iAyg

(y)dy

}

− is(t)g(0) +∫ t

0s(t − y)f(y)dy.

(2.16)

Now, using the nonlocal boundary condition

Au(−1) = αu(μ) + ϕ, (2.17)


we obtain the operator equation:{I − α

[A−1c

(μ)+ is(μ)]eiA}u−1

= α

{−iA−1c

(μ) ∫0

−1e−iAyg

(y)dy

−s(μ)[iA−1g(0) −

∫0

−1e−iAyg

(y)dy

]+A−1

∫μ

0s(μ − y)f(y)dy

}+A−1ϕ.

(2.18)

Since the operatorI − α

[A−1c

(μ)+ is(μ)]eiA (2.19)

has an inverse

T =(I − α[A−1c

(μ)+ is(μ)]eiA)−1

, (2.20)

for the solution of the operator equation (2.18), we have the formula

u−1 = T

(α

{−iA−1c

(μ) ∫0

−1e−iAyg

(y)dy

−s(μ)[iA−1g(0) −

∫0

−1e−iAyg

(y)dy

]+A−1

∫μ

0s(μ − y)f(y)dy

}+A−1ϕ

)

(2.21)

Thus, for the solution of the nonlocal boundary value problem (1.1) we obtain (2.15),(2.16), and (2.21).

Theorem 2.3. Suppose that ϕ ∈ D(A1/2), f(0) ∈ D(A1/2), and g(0) ∈ D(A1/2). Let f(t)be continuously differentiable on [0, 1] and let g(t) be twice continuously differentiable on [−1, 0]functions. Then, there is a unique solution of the problem (1.1) and the following stability inequalities

max−1≤t≤1

‖u(t)‖H

≤M[∥∥∥A−1/2ϕ

∥∥∥H+∥∥∥A−1/2g(0)

∥∥∥H+ max

−1≤t≤0

∥∥∥A−1g ′(t)∥∥∥H+max

0≤t≤1

∥∥∥A−1/2f(t)∥∥∥H

],

(2.22)

max−1≤t≤1

∥∥∥∥du(t)dt

∥∥∥∥H

+ max−1≤t≤1

∥∥∥A1/2u(t)∥∥∥H

≤M[∥∥ϕ∥∥H

+∥∥g(0)

∥∥H + max

−1≤t≤0

∥∥∥A−1/2g ′(t)∥∥∥H+max

0≤t≤1

∥∥f(t)∥∥H

],

(2.23)

max−1≤t≤0

∥∥∥∥du(t)dt

∥∥∥∥H

+max0≤t≤1

∥∥∥∥∥d2u(t)dt2

∥∥∥∥∥H

+ max−1≤t≤1

‖Au(t)‖H

≤M[∥∥∥A1/2f ′(t)

∥∥∥∥∥∥A1/2ϕ

∥∥∥H+∥∥∥A1/2g(0)

∥∥∥H+∥∥g ′(0)

∥∥H

+max−1≤t≤0

∥∥g ′′(t)∥∥H +∥∥∥A1/2f(0)

∥∥∥H+max

0≤t≤1

∥∥∥A1/2f ′(t)∥∥∥]

H

(2.24)

hold, whereM is independent of f(t), t ∈ [0, 1], g(t), t ∈ [−1, 0], and ϕ.


Note that there are three inequalities in Theorem 2.3 on the stability of solution, stabil-ity of first derivative of solution and stability of second derivative of solution. That means thesolution of problem (1.1) u(t) and its first and second derivatives are continuously dependenton f(t), g(t) and ϕ.

Proof. First, estimate (2.22) will be obtained. Using formula (2.21) and integration by parts,we obtain

u−1 = T

(α

{−A−2c

(μ)[g(0) − eiAg(−1) −

∫0

−1e−iAyg ′(y

)dy

]

+ iA−1s(μ)(eiAg(−1) +

∫0

−1e−iAyg ′(y

)dy

)

+A−1∫μ

0s(μ − y)f(y)dy

}+A−1ϕ

).

(2.25)

Using estimates (2.4), and (2.8), we get

‖u−1‖H ≤M[∥∥∥A−1/2ϕ

∥∥∥H+∥∥∥A−1g(0)

∥∥∥H+ max

−1≤t≤0

∥∥∥A−1g ′(t)∥∥∥H+max

0≤t≤1

∥∥∥A−1f(t)∥∥∥H

]. (2.26)

Applying A1/2 to the formula (2.25) and using estimates (2.4) and (2.8), we can write

∥∥∥A1/2u−1∥∥∥H

≤M[∥∥ϕ∥∥H +∥∥∥A−1/2g(0)

∥∥∥H+∥∥∥A−1/2g ′(t)

∥∥∥H+max

0≤t≤1

∥∥∥A−1/2f(t)∥∥∥H

]. (2.27)

Using formulas (2.15) and (2.16) and integration by parts, we obtain

u(t) = ei(t+1)Au−1 +A−1[g(t) − ei(t+1)Ag(−1) −

∫ t

−1ei(t−y)Ag ′(y

)dy

], −1 ≤ t ≤ 0,

u(t) = [c(t) + iAs(t)]

{eiAu−1 +A−1

(g(0) − eiAg(−1) −

∫0

−1e−iAyg ′(y

)dy

)}

− is(t)g(0) +∫ t

0s(t − y)f(y)dy, 0 ≤ t ≤ 1.

(2.28)

Using estimates (2.4) we get

‖u(t)‖H ≤M[‖u−1‖H +

∥∥∥A−1g(0)∥∥∥H+ max

−1≤t≤0

∥∥∥A−1g ′(t)∥∥∥H

], −1 ≤ t ≤ 0,

‖u(t)‖H ≤M[∥∥∥A1/2u−1

∥∥∥H+∥∥∥A−1/2g(0)

∥∥∥H+ max

−1≤t≤0

∥∥∥A−1g ′(t)∥∥∥H+ max

0≤t≤1

∥∥∥A−1/2f(t)∥∥∥H

],

1 ≤ t ≤ 1.(2.29)

Then, from estimates (2.26), (2.27), and (2.29) it follows (2.22).


Second, (2.23) will be obtained. Applying A1/2 to the formula (2.25) and using esti-mates (2.4),and (2.8), we obtain

∥∥∥A1/2u−1∥∥∥H

≤M[∥∥ϕ∥∥H +∥∥∥A−1/2g(0)

∥∥∥H+ max

−1≤t≤0

∥∥∥A−1/2g ′(t)∥∥∥H+max

0≤t≤1

∥∥∥A−1/2f(t)∥∥∥H

].

(2.30)

Applying A to the formula (2.25) and using estimates (2.4), (2.8), we get

‖Au−1‖H ≤M[∥∥∥A1/2ϕ

∥∥∥H+∥∥g(0)

∥∥H + max

−1≤t≤0

∥∥g ′(t)∥∥H +max

0≤t≤1

∥∥f(t)∥∥H

]. (2.31)

Applying A1/2 to the formulas (2.28), and using estimates (2.4)we can write

∥∥∥A1/2u(t)∥∥∥H

≤M[∥∥∥A1/2u−1

∥∥∥H+∥∥∥A−1/2g(0)

∥∥∥H+ max

−1≤t≤0

∥∥∥A−1/2g ′(t)∥∥∥H

], −1 ≤ t ≤ 0,

∥∥∥A1/2u(t)∥∥∥H

≤M[‖Au−1‖H +

∥∥g(0)∥∥H + max

−1≤t≤0

∥∥∥A−1/2g ′(t)∥∥∥H+max

0≤t≤1

∥∥f(t)∥∥H

], 0 ≤ t ≤ 1.

(2.32)

Combining estimates (2.30), (2.31),and (2.32), we get estimate (2.23).Third, estimate (2.24) will be obtained. Using formula (2.25) and integration by parts,

we obtain

u−1 =

⎛⎜⎝α

{−A−2c

(μ)(g(0) − e−iAg(−1) − iA−1

×[g ′(0) − eiAg ′(−1) −

∫0

−1e−iAyg ′′(y

)dy

])+ iA−1s

(μ)

×[eiAg(−1) + iA−1

(g ′(0) − eiAg ′(−1) −

∫0


)dy

)]

−A−2[f(μ)+ c(μ)f(0) −

∫μ

0c(μ − y)f ′(y

)dy

]}+A−1ϕ

⎞⎟⎠

(2.33)

Applying A to formula (2.33) and using estimates (2.4) and (2.8), we get

‖Au−1‖H ≤M[∥∥∥A1/2ϕ

∥∥∥H

+∥∥g(0)

∥∥H +∥∥∥A−1/2g ′(0)

∥∥∥H

+max−1≤t≤0

∥∥∥A−1/2g ′′(t)∥∥∥H+∥∥f(0)

∥∥H +max

0≤t≤1

∥∥∥A−1/2f ′(t)∥∥∥H

].

(2.34)


Applying A3/2 to formula (2.33) and using estimates (2.4), and (2.8) we can write

∥∥∥A3/2u−1∥∥∥H

≤M[∥∥Aϕ

∥∥H +∥∥∥A1/2g(0)

∥∥∥H+∥∥g ′(0)

∥∥H

+max−1≤t≤0

∥∥g ′′(t)∥∥H +∥∥∥A1/2f(0)

∥∥∥H+max

0≤t≤1

∥∥f ′(t)∥∥H

].

(2.35)

Using formulas (2.28), and integration by parts, we obtain

u(t) = ei(t+1)Au−1 +A−1[g(t) − ei(t+1)Ag(−1)

−iA−1(g ′(t) − ei(t+1)Ag ′(−1) −

∫ t

−1ei(t−y)Ag ′′(y

)dy

)], −1 ≤ t ≤ 0,

u(t) =[c(t) + iAs(t)

]{eiAu−1 +A−1

(g(0) − eiAg(−1)

−iA−1[g ′(0) − eiAg ′(−1) −

∫0


)dy

])}

− is(t)g(0) −A−1[f(t) − c(t)f(0) −

∫ t

0c(t − y)f ′(y

)dy

], 0 ≤ t ≤ 1.

(2.36)

Applying A to the formulas (2.36), and using estimates (2.4), we get

‖Au(t)‖H ≤M[‖Au−1‖H +

∥∥∥A1/2g(0)∥∥∥H+∥∥g ′(0)

∥∥H + max

−1≤t≤0

∥∥g ′′(t)∥∥H

], −1 ≤ t ≤ 0,

‖Au(t)‖H ≤M[∥∥∥A3/2u−1

∥∥∥H

+∥∥∥A1/2g(0)

∥∥∥H+∥∥g ′(0)

∥∥H

+max−1≤t≤0

∥∥g ′′(t)∥∥H +∥∥∥A1/2f(0)

∥∥∥H+max

0≤t≤1

∥∥∥A1/2f ′(t)∥∥∥H

], 0 ≤ t ≤ 1.

(2.37)

From (2.34) and (2.35) and estimates (2.37) it follows (2.24). This completes the proof ofTheorem 2.3.

Remark 2.4. We can obtain the same stability results for the solution of the following multi-point nonlocal boundary value problem:

d2u(t)dt2

+Au(t) = f(t) (0 ≤ t ≤ 1),

id(t)dt

+Au(t) = g(t) (−1 ≤ t ≤ 0),

Au(−1) =N∑

j=1

αju(μj)+ ϕ,

0 < μj ≤ 1, 1 ≤ j ≤N,

(2.38)


for differential equations of mixed type in a Hilbert spaceH with self-adjoint positive definiteoperator A.

3. Applications

Initially, the mixed problem for the hyperbolic Schrodinger equation

vyy − (a(x)vx)x + δv = f(y, x), 0 < y < 1, 0 < x < 1,

ivy − (a(x)vx)x + δv = g(y, x), −1 < y < 0, 0 < x < 1,

−(a(x)vx(−1, x))x + δv(−1, x) = αv(1, x) + ϕ(x), 0 ≤ x ≤ 1,

v(y, 0)= v(y, 1), vx

(y, 0)= vx

(y, 1), −1 ≤ y ≤ 1,

v(0+, x) = v(0−, x

), vy(0+, x) = vy

(0−, x

), 0 ≤ x ≤ 1,

|α| < δ√1 + δ

(3.1)

is considered, where δ = const > 0. The problem (3.1) has a unique smooth solution v(y, x)for smooth a(x) ≥ a > 0 (x ∈ (0, 1)), ϕ(x) (x ∈ [0, 1]), f(y, x) (y ∈ [0, 1], x ∈ [0, 1]), andg(y, x) (y ∈ [−1, 0], x ∈ [0, 1]) functions.

We introduce the Hilbert space L2[0, 1] of all the square integrable functions definedon [0, 1] and Hilbert spacesW1

2 [0, 1] andW22 [0, 1] equipped with norms

∥∥ϕ∥∥W1

2 [0,1]=

(∫1

0

∣∣ϕ(x)∣∣2dx

)1/2

+

(∫1

0

∣∣ϕx(x)∣∣2dx

)1/2

,

∥∥ϕ∥∥W2

2 [0,1]=

(∫1

0

∣∣ϕ(x)∣∣2dx

)1/2

+

(∫1

0

∣∣ϕx(x)∣∣2dx

)1/2

+

(∫1

0

∣∣ϕxx(x)∣∣2dx

)1/2

,

(3.2)

respectively. This allows us to reduce the mixed problem (3.1) to the nonlocal boundary valueproblem (1.1) in Hilbert space H with a self-adjoint positive definite operator A defined byproblem (3.1).

Theorem 3.1. The solutions of the nonlocal boundary value problem (3.1) satisfy the following sta-bility estimates:

max−1≤y≤1

∥∥vy(y, ·)∥∥

L2[0,1]+ max

−1≤y≤1

∥∥v(y, ·)∥∥W1

2 [0,1]

≤M[∥∥ϕ∥∥L2[0,1]

+∥∥g(0, ·)∥∥L2[0,1]

+ max−1≤y≤0

∥∥gy(y, ·)∥∥

L2[0,1]+ max

0≤y≤1

∥∥f(y, ·)∥∥L2[0,1]

],

max−1≤y≤1

∥∥v(y, ·)∥∥W2

2 [0,1]+ max

−1≤y≤0

∥∥vy(y, ·)∥∥

L2[0,1]+max

0≤y≤1

∥∥vyy(y, ·)∥∥

L2[0,1]

≤M[∥∥ϕ∥∥W1

2 [0,1]+∥∥g(0, ·)∥∥L2[0,1]

+∥∥gy(0, ·)

∥∥L2[0,1]

+ max−1≤y≤0

∥∥gyy(y, ·)∥∥

L2[0,1]+∥∥f(0, ·)∥∥W1

2 [0,1]+ max

0≤y≤1

∥∥fy(y, ·)∥∥

W12 [0,1]

],

(3.3)

where M does not depend on not only f(y, x) (y ∈ [0, 1], x ∈ [0, 1]) and g(y, x)(y ∈ [−1, 0], x ∈[0, 1]) but also ϕ(x)(x ∈ [0, 1]).


The proof of Theorem 3.1 is based on the abstract Theorem 2.3 and symmetry proper-ties of the space operator defined by problem (3.1).

Next, we consider the mixed nonlocal boundary value problem for the multidimen-sional hyperbolic Schrodinger equation:

vyy−m∑

r=1

(ar(x)vxr )xr = f(y, x), 0 ≤ y ≤ 1,

x = (x1, . . . , xm) ∈ Ω,

ivy−m∑

r=1

(ar(x)vxr )xr = g(y, x), −1 ≤ y ≤ 0,

x = (x1, . . . , xm) ∈ Ω,

−n∑

r=1

(ar(x)vxr (−1, x))xr = v(1, x) + ϕ(x), x ∈ Ω,

u(y, x)= 0, x ∈ S, −1 ≤ y ≤ 1,

(3.4)

where Ω is the unit open cube in them-dimensional Euclidean space Rm:

(x : x = (x1, . . . , xm), 0 < xk < 1, 1 ≤ k ≤ m) (3.5)

with boundary S and Ω = Ω ∪ S. Here, ar(x) (x ∈ Ω), ϕ(x) (x ∈ Ω), and f(y, x) (y ∈(0, 1), x ∈ Ω), g(y, x) (y ∈ (−1, 0), x ∈ Ω) are given smooth functions in [0, 1]×Ω and ar(x) ≥a > 0.

We introduce the Hilbert space L2(Ω) of all square integrable functions defined on Ω,equipped with the norm

∥∥f∥∥L2(Ω) =

{∫· · ·∫

x∈Ω

∣∣f(x)∣∣2dx1 · · ·dxn

}1/2(3.6)

and Hilbert spacesW12 (Ω) andW2

2 (Ω) defined on Ω, equipped with norms

∥∥ϕ∥∥W1

2 (Ω) =∥∥ϕ∥∥L2(Ω) +

{∫· · ·∫

x∈Ω

n∑

r=1

|ϕxr |2dx1 · · ·dxn}1/2

,

∥∥ϕ∥∥W2

2 (Ω) =∥∥ϕ∥∥L2(Ω) +

{∫· · ·∫

x∈Ω

n∑

r=1

|ϕxr |2dx1 · · ·dxn}1/2

+

{∫· · ·∫

x∈Ω

n∑

r=1

|ϕxrxr |2dx1 · · ·dxn}1/2

,

(3.7)

respectively. The problem (3.4) has a unique smooth solution v(y, x) for smoothar(x), f(y, x), and g(y, x) functions. This allows us to reduce the mixed problem (3.4) tothe nonlocal boundary value problem (1.1) in Hilbert space H with a self-adjoint positivedefinite operator A defined by problem (3.4).


Theorem 3.2. The following stability inequalities for solutions of the nonlocal boundary valueproblem (3.4)

max−1≤y≤1

∥∥vy(y, ·)∥∥

L2(Ω) + max−1≤y≤1

∥∥v(y, ·)∥∥W1

2 (Ω)

≤M[∥∥g(0, ·)∥∥L2(Ω) + max

−1≤y≤0

∥∥gy(y, ·)∥∥

L2(Ω) + max0≤y≤1

∥∥f(y, ·)∥∥L2(Ω) +∥∥ϕ∥∥L2(Ω)

],

max−1≤y≤1

∥∥v(y, ·)∥∥W22 (Ω) + max

−1≤y≤0

∥∥vy(y, ·)∥∥

L2(Ω) + max0≤y≤1

∥∥vyy(y, ·)∥∥L2(Ω)

≤M[∥∥ϕ∥∥W1

2 (Ω) +∥∥g(0, ·)∥∥L2(Ω) +

∥∥gy(0, ·)∥∥L2(Ω)

+ max−1≤y≤0

∥∥gyy(y, ·)∥∥

L2(Ω) +∥∥f(0, ·)∥∥W1

2 (Ω) + max0≤y≤1

∥∥fy(y, ·)∥∥

W12 (Ω)

]

(3.8)

hold. Here,M is independent of f(y, x) (y ∈ [0, 1], x ∈ [0, 1]), g(y, x) (y ∈ [−1, 0], x ∈ [0, 1]),and ϕ(x) (x ∈ [0, 1]).

The proof of Theorem 3.2 is based on the abstract Theorem 2.3, symmetry propertiesof the space operator defined by problem (3.4), and the following theorem on the coercivityinequality for the solution of the elliptic differential problem in L2(Ω) in Sobolevskii [28].

Theorem 3.3. For the solutions of the elliptic differential problem

−m∑

r=1

(ar(x)uxr )xr = ω(x), x ∈ Ω,

u(x) = 0, x ∈ S,(3.9)


m∑

r=1

‖uxrxr‖L2(Ω) ≤M‖ω‖L2(Ω). (3.10)

Acknowledgment

The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey)for his helpful suggestions to the improvement of this paper.

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Research ArticleA New Approach for Linear Eigenvalue Problemsand Nonlinear Euler Buckling Problem

Meltem Evrenosoglu Adiyaman and Sennur Somali

Department of Mathematics, Faculty of Science, Dokuz Eylul University, 35160 Tinaztepe,Buca, Izmir, Turkey

Correspondence should be addressed to Meltem Evrenosoglu Adiyaman,[email protected]



Copyright q 2012 M. E. Adiyaman and S. Somali. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We propose a numerical Taylor’s Decomposition method to compute approximate eigenvaluesand eigenfunctions for regular Sturm-Liouville eigenvalue problem and nonlinear Euler bucklingproblem very accurately for relatively large step sizes. For regular Sturm-Liouville problem, thetechnique is illustrated with three examples and the numerical results show that the approximateeigenvalues are obtained with high-order accuracy without using any correction, and they arecompared with the results of other methods. The numerical results of Euler Buckling problem arecompared with theoretical aspects, and it is seen that they agree with each other.

1. Introduction

We investigate the computation of eigenvalues of regular Sturm-Liouville eigenvalueproblems:

−y′′(x) + r(x)y(x) = λy(x), 0 ≤ x0 < x < xn,y(x0) = y(xn) = 0,

(1.1)

where r(x) ∈ Cp+q[x0, xn] and p, q ∈ N and Euler Buckling problem:

y′′ + λ siny = 0,

y′(0) = 0, y′(1) = 0.(1.2)

Regular Sturm-Liouville problems arise in many applications, and many methods areavailable for their numerical solution Pryce [1].


We also examine an elementary, classical problem buckling of an end-loaded rodwhich possesses a completely soluble continuous model in the form of a nonlinear, second-order boundary value problem as described in elsewhere [2–5]. An essential completeanalysis of this problem was provided by Euler [6]. For the nonlinear eigenvalue problem(1.2), one may find that for small λ the only solution is zero solution as in the linear case.But as the eigenvalue λ increases, it reaches a critical value λ1 at which a nonzero solutionappears, corresponding to buckling of the rod. For λ > λ1, the nonlinear problem behavesquite differently from the linear problem: for a range of values λ1 < λ < λ2, there is exactlyone nonzero solution of (1.2) for each λ, and when λ exceeds λ2, a second nonzero solutionappears; similarly, there is a value λ3 beyond which there are three nonzero solutions, and soon. Namely, one may give inductively

0 ≤ λ ≤ π2, only the trivial solution,

π2 < λ ≤ 4π2, one nontrivial solution,

n2π2 < λ ≤ (n + 1)2π2, n nontrivial solutions,

(1.3)

as given by Stakgold [2]. This behavior is a simple example of the phenomenon of bifurcationor branching; it occurs in many different areas of applied mathematics.

The method considered here is a Taylor decomposition which was used by Adiyamanand Somali [7] for the solution of certain nonlinear problems. Like classical finite-differenceand finite-element methods, this high order method is best suited to the fundamentaleigenvalue and small eigenvalues.

In Section 2, the behavior of eigenvalues and corresponding eigenfunctions for regularSturm-Liouville problem is obtained by Taylor’s decomposition method, and convergenceof the method for regular Sturm-Liouville problem with constant function r(x) is given. Weestablish a Lemma and a Theorem, and thenwe give an application of Taylor’s decompositionmethod to the Euler Buckling problem in Section 3. The technique is illustrated withthree examples, and the numerical results of regular Sturm-Liouville problem are givenby comparing the results of other methods in Section 4. The numerical results of EulerBuckling problem accompanying the theoretical results and the behavior of solution arealso discussed in Section 4. In the conclusion, we summarize the study and present oursuggestions regarding future work.

2. Application and Error Analysis of Taylor’s Decomposition Methodfor Regular Sturm-Liouville Eigenvalue Problems

2.1. Application of Taylor’s Decomposition on Two Points for RegularSturm-Liouville Eigenvalue Problems

We consider the regular Sturm-Liouville eigenvalue problem (1.1) by introducing the newdepending variable y′(x) = z(x), (1.1) can be written as

Y ′(x) = A(x)Y (x),

C0Y (x0) + C1Y (xn) = 0,(2.1)


where

Y (x) =

[y(x)

z(x)

], A(x) =

[0 1

r(x) − λ 0

], C0 =

[1 0

0 0

], C1 =

[0 0

1 0

]. (2.2)

From Ashyralyev and Sobolevskii [8], we will consider the application of Taylor’s decompo-sition of function Y (x) on two points. We need to find Y (j)(x) for any 1 ≤ j ≤ p and q. Usingthe equation Y ′(x) = A(x)Y (x), we get

Y (j)(x) = Aj(x)Y (x), (2.3)

with

A0(x) = I,

A1(x) = A(x),

Aj(x) = A′j−1(x) +Aj−1(x)A(x), 2 ≤ j ≤ p,

(2.4)

where I is the 2 × 2 identity matrix. By using the structure of the matrix A(x), we obtain theentries of the matrix of

Aj(x) =

[aj(1,1)(λ;x) aj(1,2)(λ;x)

aj(2,1)(λ;x) aj(2,2)(λ;x)

](2.5)

as in the following formulas:

aj(1,1)(λ;x) =∂aj−1(1,1)(λ;x)

∂x+ (r(x) − λ)aj−2(2,2)(λ;x) = aj−1(2,1)(λ;x),

aj(2,2)(λ;x) =∂aj−1(2,2)(λ;x)

∂x+ aj(1,1)(λ;x),

aj(1,2)(λ;x) = aj−1(2,2)(λ;x),

aj(2,1)(λ;x) = −∂aj(2,2)(λ;x)∂x

+ aj+1(2,2)(λ;x),

(2.6)

for 2 ≤ j ≤ p, where

a0(1,1)(λ;x) = 1, a1(1,1)(λ;x) = 0,

a0(1,2)(λ;x) = 0, a1(1,2)(λ;x) = 1,

a0(2,1)(λ;x) = 0, a1(2,1)(λ;x) = r(x) − λ,a0(2,2)(λ;x) = 1, a1(2,2)(λ;x) = 0.

(2.7)


From the theorem given in Ashyralyev and Sobolevskii [8], we have the followingrelation:

Y (xk) − Y (xk−1) +p∑

j=1

αjY(j)(xk)hj −

q∑

j=1

βjY(j)(xk−1)hj

=(−1)p(p + q

)!

∫xk

xk−1(xk − s)q(s − xk−1)pY (p+q+1)(s)ds

(2.8)

on the uniform grid

[x0, xn]h = {xk = x0 + kh, k = 0, 1, . . . , n, nh = xn − x0, n ∈N}, (2.9)

where

αj =

(p + q − j)!p!(−1)j(p + q

)!j!(p − j)! , 1 ≤ j ≤ p,

βj =

(p + q − j)!q!

(p + q

)!j!(q − j)! , 1 ≤ j ≤ q.

(2.10)

Rewriting (2.8) by neglecting the last term, we obtain the single-step difference scheme of(p + q)-order of accuracy for the approximate solution of problem (2.1):

Yk − Yk−1 +p∑

j=1

αjAj(xk)Ykhj −q∑

j=1

βjAj(xk−1)Yk−1hj = 0, (2.11)

where

Yk =

[yk

zk

](2.12)

is the approximate value of Y (xk). For the simple computation, let p = q, then we have

⎛

⎝I +p∑

j=1

αjAj(xk)hj⎞

⎠Yk =

⎛

⎝I +p∑

j=1

(−1)jαjAj(xk−1)hj⎞

⎠Yk−1, (2.13)

where βj = (−1)jαj . Letting M(xk) = (I +∑p

j=1 αjAj(xk)hj) and N(xk−1) = (I +∑p

j=1(−1)jαjAj(xk−1)hj), we write

Yk =M−1(xk)N(xk−1)Yk−1. (2.14)


Since the accuracy and convergence of the method not only depend on h, they also dependon p, we can increase the order of accuracy by increasing p for fixed h. So h is chosen as lengthof the whole interval as follows. Now, taking h = xn − x0 gives

Y1 =M−1(xn)N(x0)Y0, (2.15)

and substituting into the boundary condition of (2.1), we get

(C1M

−1(xn)N(x0) + C0

)Y0 = 0. (2.16)

To obtain a nontrivial solution Y0, we must have the following equation:

det(C1M

−1(xn)N(x0) + C0

)= 0. (2.17)

Defining

M(xn) =

[m11 m12

m21 m22

], N(x0) =

[n11 n12

n21 n22

], (2.18)

we have the following statement

m22n12 −m12n22 = 0. (2.19)

Since

M(xn) =

⎡⎢⎢⎢⎢⎣

1 +p∑

j=1

αjaj(1,1)(λ;xn)hjp∑

j=1

αjaj(1,2)(λ;xn)hj

p∑

j=1

αjaj(2,1)(λ;xn)hj 1 +p∑

j=1

αjaj(2,2)(λ;xn)hj

⎤⎥⎥⎥⎥⎦,

N(x0) =

⎡⎢⎢⎢⎢⎣

1 +p∑

j=1

(−1)jαjaj(1,1)(λ;x0)hjp∑

j=1

(−1)jαjaj(1,2)(λ;x0)hj

p∑

j=1

(−1)jαjaj(2,1)(λ;x0)hj 1 +p∑

j=1

(−1)jαjaj(2,2)(λ;x0)hj

⎤⎥⎥⎥⎥⎦,

(2.20)


using the entriesm12, m22, n12, and n22 of the above matrices and the properties of the entriesof Aj(x), we obtain (2.19) in terms of λ:

F(λ) = m22n12 −m12n22

=

⎛

⎝1 +p∑

j=1

αjaj(2,2)(λ;xn)hj⎞

⎠

⎛

⎝p∑

j=1

(−1)jαjaj−1(2,2)(λ;x0)hj⎞

⎠

−⎛

⎝p∑

j=1

αjaj−1(2,2)(λ;xn)hj⎞

⎠

⎛

⎝1 +p∑

j=1

(−1)jαjaj(2,2)(λ;x0)hj⎞

⎠.

(2.21)

Solving nonlinear equation F(λ) = 0 by Newton’s method, we find the approximateeigenvalues. This method appears to require a separate calculation for the eigenfunctions.

To find the corresponding eigenfunctions of the regular Sturm-Liouville eigenvalueproblem (2.1), we substitute the eigenvalue to (2.1) and we solve the obtained boundaryvalue problem by Taylor’s decompositionmethod on two points xk−1 and xk with the uniformgrid [0, 1]h for p = q. Then, we get a homogeneous linear equation system of 2n equationswith 2n unknown z0, y1, z1, y2, z2 . . . , yn−1, zn−1, zn which are the approximated values ofy′(x0), y(x1), y′(x1), y(x2), y′(x2), . . . , y(xn−1), y′(xn−1), y′(xn), respectively. Solving the 2n×2nhomogeneous system, we obtain approximate values of the eigenfunction and its derivativeof (1.1) at the point x = xk.

2.2. Error Analysis for Regular Sturm-Liouville Problem When r(x) = c

In this section, we will show the convergence of the method for eigenfunctions with theconstant function r(x) = c by obtaining approximate value of eigenfunction at the pointx ∈ [x0, xn] of the problem (1.1). Without loss of generality, we may choose r(x) = 0, thenAj(x) = Aj , that is, aj(2,2)(λ;xn) = aj(2,2)(λ; 0) = aj(2,2)(λ). Using (2.6), we can find explicitvalues of aj(1,1), aj(2,2) as follows:

a2j(1,1) = (−1)jλj ,

a2j(2,2) = (−1)jλj ,a2j+1(1,1) = 0,

a2j+1(2,2) = 0, j ≥ 0.

(2.22)


This yields

m22 = 1 +�p/2�∑

j=1

α2j(−1)jλjh2j ,

n22 = m22,

m12 =�(p−1)/2�∑

j=0

α2j+1(−1)jλjh2j+1,

n12 = −m12,

m11 = m22,

m21 = −λm12.

(2.23)

Using (2.14) for k = 1, we have

Y1 =M−1(x)N(x0)Y0, (2.24)

where Y0 and Y1 are the approximated values of Y (x0) and Y (x), respectively, with thestepsize h = x − x0:

Y1 =z0

det(M)

[ −2m22m12

−λ(m12)2 + (m22)2

]. (2.25)

The first component of the above vector (2.25) gives the approximate eigenfunction y1, andthe second component of the above vector (2.25) gives the derivative of the approximateeigenfunction z1 of the regular Sturm-Liouville problem (1.1) at x. Now, we will show thaty1 and z1 converge to exact functions y(x) and y′(x), respectively, as p → ∞.

Using the Stirling’s Formula n! ≈ √2πn(n+1)/2e−n for αj in (2.10), we obtain

αj =

(2p − j)!p!(−1)j(2p)!j!(p − j)! ≈ (−1)j 1

j!12j

(p − j/2p − j

)(p−j+1)/2(p − j/2p

)p/2. (2.26)

This gives

limp→∞

αj = (−1)j 1j!

12j. (2.27)


Thus,

limp→∞

m22 = limp→∞

⎛

⎝1 +�p/2�∑

j=1

α2j(−1)jλjh2j⎞

⎠ = 1 +∞∑

j=1

1(2j)!122j

(−1)jλjh2j

=∞∑

j=0(−1)j

(√λh

2

)2j1(2j)!= cos

(√λ)h2.

(2.28)

By using the same idea, we obtain

limp→∞

m12 = limp→∞

⎛

⎝�(p−1)/2�∑

j=0(−1)jα2j+1λjh2j+1

⎞

⎠ =1√λsin(√

λ)h2. (2.29)

It follows from (2.28) and (2.29) that

limp→∞

det(M) = m222 + λm

212 = cos2

(√λ)h2+ λ(

1√λsin(√λ)h

2

)2

= 1. (2.30)

Hence, for r(x) = 0, the approximate eigenfunction of (1.1) to the corresponding eigenvalueλ converges to exact eigenfunction:

limp→∞

y1 = 2z0

det(M)1√λ

(cos(√

λh

2

))(sin(√

λh

2

))=z0√λsin(√

λ(x − x0)). (2.31)

Since we have z(x) = y′(x), the derivative of approximate eigenfunction of (1.1) to thecorresponding eigenvalue λ converges to derivative of the exact solution:

limp→∞

z1 =z0

det(M)

((−λ) 1

λsin2(√

λh

2

)+ cos2

(√λh

2

))= z0 cos

(√λ(x − x0)

), (2.32)

where λ = k2π2, k = 1, 2, . . ..This demonstration shows that approximate eigenfunction and eigenvalue converges

to exact one as p → ∞ for fixed step-size “h.”


2.3. Taylor’s Decomposition Method to the Euler Buckling Problem

For convenience, we introduce the following notations as in (2.1) and Adiyaman and Somali[7]:

Y (x) =

[y(x)

z(x)

], F(Y (x)) =

⎡

⎣f(0)1

(y, z)

f(0)2

(y, z)

⎤

⎦,

C0 =

[0 1

0 0

], C1 =

[0 0

0 1

], f

(0)1

(y, z)= z, f

(0)2

(y, z)= −λ siny.

(2.33)

Thus, the Euler Buckling Problem (1.2) can be written in the form:

Y ′(x) = F(Y (x)), 0 < x < 1,

C0Y (0) + C1Y (1) = 0,(2.34)

Defining the following recurrence relations for j = 1, . . . , 2p:

f(j)i

(y, z)= z

∂f(j−1)i

(y, z)

∂y− λ siny ∂f

(j−1)i

(y, z)

∂z, i = 1, 2, (2.35)

we obtain

Y (j)(x) =

⎡

⎣f(j−1)1

(y, z)

f(j−1)2

(y, z)

⎤

⎦ =

⎡

⎣f(j−2)2

(y, z)

f(j−1)2

(y, z)

⎤

⎦ for j = 2, . . . , 2p + 1. (2.36)

We first give the following lemma which defines f (j−1)2 (y, z) explicitly.

Lemma 2.1. For j = 0, . . . , 2p, let f (j)2 (y, z) satisfy the recurrence relation (2.35) with f (0)

2 (y, z) =−λ siny. Then

f(2m)2

(y, z)=

m∑

i=0

λi+1z2m−2i�i/2�∑

k=0

(−1)m+1−ka2m+1,i,k(cosy

)i−2k(siny)2k+1

, (2.37)

where

a2m+1,i,k =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

1, i = 0, k = 0,

(2k + 2)a2m,i,k+1 + (i + 1 − 2k)a2m,i,k

+(2m + 1 − 2i)a2m,i−1,k, 1 ≤ i ≤ m, 0 ≤ k ≤ i

2,

0, else,

(2.38)


form = 0, . . . , p, and

f(2m+1)2

(y, z)=

m∑

i=0

λi+1z2m+1−2i�(i+1)/2�∑

k=0


)i+1−2k(siny)2k

, (2.39)

where

a2m+2,i,k =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

1, i = 0, k = 0,

(2k + 1)a2m+1,i,k + (i − 2k)a2m+1,i,k−1

+(2m + 2 − 2i)a2m+1,i−1,k−1, 1 ≤ i ≤ m, 0 ≤ k ≤ i

20, else,

(2.40)

form = 0, . . . , p − 1.

Proof. The proof follows induction argument based on (2.35).

Theorem 2.2. If f (j)1 (y, z) and f (j)

2 (y, z), are sufficiently smooth and satisfy (2.36), (2.37), and(2.39) then the following relations hold: (a) it holds that

y1 − y0 +p∑j=1βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]= −2

(y0 +

p∑j=1βjf

(j−1)1

(y0, 0

)),

p∑j=1βj[(−1)jf (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)]= 0,

(2.41)

for y1 = −y0, (b) it holds that

p∑

j=1

βj[(−1)jf (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)]= −2

p∑

j=1

βjf(j−1)2

(y0, 0

),

y1 − y0 +p∑

j=1

βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]= 0,

(2.42)

for y1 = y0.

Proof. Let j = 2m + 1 form = 0, . . . , p, then f (2m+1)1 (y, z) becomes

f(2m)1

(y, z)= f (2m−1)

2

(y, z)=

(m−1)∑

i=0

λi+1z2m−2i−1�(i+1)/2�∑

k=0

(−1)m−ka2m,i,k(cosy

)i+1−2k(siny)2k

,

(2.43)


by Lemma 2.1. Since all terms of previous sum contain z, f2m1 (y, 0) = f2m−1

2 (y, 0) = 0 form = 1, . . . , p, hence, we get the following equations:

y1 − y0 +p∑

j=1

βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]

= y1 − y0 +p∑

j=1

βj[f(j−1)1

(y1, 0

) − f (j−1)1

(y0, 0

)],

p∑

j=1

βj[(−1)jf (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)]=

p∑

j=1

βj[−f (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)].

(2.44)

Letting j = 2m + 2 form = 0, . . . , p − 1 and using (2.37), we get

f(2m+2−1)1

(y, z)= f (2m)

2

(y, z)=

m∑

i=0

λi+1z2m−2i�i/2�∑

k=0


)i−2k(siny)2k+1

.

(2.45)

Substituting the value z = 0 into (2.45), we obtain

f(2m+1)1

(−y0, 0)= f2m

2

(−y0, 0)= λm+1

�m/2�∑

k=0

(−1)m+1−ka2m+1,m,k(cosy0

)i−2k[−(siny0)2k+1]

= −f (2m)2

(y0, 0

),

(2.46)

which gives the following relations:

f(2m+1)1

(−y0, 0)= −f (2m+1)

1

(y0, 0

),

f(2m)2

(−y0, 0)= −f (2m)

2

(y0, 0

).

(2.47)


Table 1: Corresponding to the initial values y1,0, y2,0, y3,0, and y4,0 for various λ obtained from (2.54) and(2.55).

λ y1,0 y2,0 y3,0 y4,0

15 > π2 1.7471 — — —45 > 4π2 2.8578 1.0092 — —90 > 9π2 3.0718 2.3413 0.3236 —160 > 16π2 3.1272 2.7999 2.0239 0.3771

Using (2.47) for y1 = −y0, we obtain the following relations:

y1 − y0 +p∑

j=1

βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]

= −y0 − y0 +p∑

j=1

βj[f(j−1)1

(−y0, 0) − f (j−1)

1

(y0, 0

)]

= −2⎛

⎝y0 +p∑

j=1

βjf(j−1)1

(y0, 0

)⎞

⎠

p∑

j=1

βj[−f (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)]= 0.

(2.48)

Similarly for y1 = y0 using (2.47), we observe that

y1 − y0 +p∑

j=1

βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]= 0,

p∑

j=1

βj[(−1)jf (j−1)

2

(y1, 0

)f(j−1)2

(y0, 0

)]= −2

p∑

j=1

βjf(j−1)2

(y0, 0

).

(2.49)

So, our assertions (a) and (b) are proved.Again, we consider the application of Taylor’s decomposition method to (2.34) on two

points xk and xk−1:

Yk − Yk−1 +p∑

j=1

αjY(j)khj −

q∑

j=1

βjY(j)k−1h

j = 0, (2.50)

where Y (j)k is the approximate value of Y (j)

k (xk). For the computation of the eigenvalues of(1.2), putting h = 1 and p = q, the approximation (2.50) gives

Y1 − Y0 +p∑

j=1

(−1)jβjY (j)1 −

p∑

j=1

βjY(j)0 = 0, (2.51)


Table 2: Comparison of the first eigenvalue and solutions of Example 3.1 using Taylor’s decompositionmethod, exact values, and Table 4 from [9], when n = 0.

xHWSM FDM TDM

Exact

Errors ofTDM

Errors ofTDM

Errors ofTDM

(h = 1/16) (h = 1/16) (p = 3)(h = 1/16)

(p = 3)(h = 1/16)

(p = 4)(h = 1/8)

(p = 5)(h = 1/4)

0 0 0 0 0 0 0 00.0625 0.27521 0.278599 0.275999 0.275899 1.54E − 10 — —0.125 0.54181 0.541196 0.541196 0.541196 2.91E − 10 1.13E − 11 —0.1875 0.78549 0.785695 0.785695 0.785695 3.93E − 10 — —0.25 1.00482 1 1 1 4.45E − 10 1.74E − 11 6.87E − 120.3125 1.17851 1.17588 1.17588 1.17588 4.37E − 10 — —0.375 1.31285 1.30656 1.30656 1.30656 3.61E − 10 1.41E − 11 —0.4375 1.38376 1.38704 1.38704 1.38704 2.15E − 10 — —0.5 1.41103 1.41421 1.41421 1.41421 2.22E − 16 2.22E − 16 00.5625 1.38376 1.38704 1.38704 1.38704 2.76E − 10 — —0.625 1.31285 1.30656 1.30656 1.30656 6.03E − 10 2.35E − 11 —0.6875 1.17851 1.17588 1.17588 1.17588 9.63E − 10 — —0.75 1.00482 1 1 1 1.33E − 9 5.22E − 11 2.06E-110.8125 0.78549 0.785695 0.785695 0.785695 1.70E − 9 — —0.875 0.54181 0.541196 0.541196 0.541196 2.03E − 9 7.96E − 11 —0.9375 0.27521 0.275899 0.275999 0.275899 2.31E − 9 — —1 0 0 0 0 0 0 0λ1 = 10.9334 (HWSM), 10.8379 (FDM), 10.8696 (TDM), 10.8696 (Exact).

Table 3: Comparison of the first eigenvalue and solutions of Example 3.1 using TDM and Table 4 from [9],when p = 16, n = 2, and h = 0.0625.

x HWSM FDM TDM0 0 0 00.0625 0.27521 0.27756 0.2775630.125 0.54181 0.54434 0.5443370.1875 0.78949 0.78996 0.7899530.25 1.00485 1.00488 1.004870.3125 1.18153 1.18075 1.180740.375 1.31286 1.31082 1.310760.4375 1.39372 1.38996 1.389940.5 1.42102 1.41527 1.415290.5625 1.39371 1.38591 1.385980.625 1.31285 1.30323 1.303340.6875 1.18154 1.18066 1.170810.75 1.0048 0.99361 0.9937920.8125 0.77949 0.77917 0.7793570.875 0.53481 0.53577 0.5359340.9375 0.27726 0.27277 0.2728781 0 0 0λ1 = 10.3452 (HWSM), 9.95067 (FDM), 9.98317 (TDM).


Table 4: The errors between exact and approximate fundamental eigenvalue for various p and for h = 1and n = 0 in Example 3.1.

p 2 3 4 6 7 9 10 11Errors 2.13E0 1.30E − 1 5.49E − 3 2.60E − 6 3.38E − 8 2.60E − 12 1.59E − 14 1.77E − 15

Table 5: Observed orders of Example 3.1 for n = 2 at x = 1/2 using Taylor’s decomposition method.

p = 2 p = 3 p = 4ord(1/8) 4.23247 7.90607 10.6574ord(1/16) 3.86153 5.22141 8.13719ord(1/32) 3.98674 5.93074 7.63932

where αj = (−1)jβj . Writing (2.51) with respect to the components and imposing theboundary conditions z0 = z(0) = y′(0) = 0 and z1 = z(1) = y′(1) = 0, we have the followingequations

y1 − y0 +p∑

j=1

βj[(−1)jf (j−1)

1

(y1, 0

) − f (j−1)1

(y0, 0

)]= 0, (2.52)

p∑

j=1

βj[(−1)jf (j−1)

2

(y1, 0

) − f (j−1)2

(y0, 0

)]= 0. (2.53)

Using Theorem 2.2(a) for y1 = −y0, (2.52) becomes

G1(y0, λ

)= −2

⎛

⎝y0 +p∑

j=1

βjf(j−1)1

(y0, 0

)⎞

⎠ = 0, (2.54)

and (2.53) is satisfied. For y1 = y0, (2.52) is satisfied by Theorem 2.2(b) and (2.53) becomes

G2(y0, λ

)= −2

p∑

j=1

βjf(j−1)2

(y0, 0

)= 0. (2.55)

From Table 1, we observe that there is only trivial initial condition for 0 ≤ λ ≤ π2, thereis one nontrivial initial condition from (2.54) for π2 < λ ≤ 4π2, there are n nontrivial initialconditions for n2π2 < λ ≤ (n + 1)2π2. These results show that the numerical results obtainedusing Taylor’s decomposition method agree with the theoretical results of Euler bucklingproblem given in [2].

Now, we find an approximate solution to the problem

Y ′(x) = F(Y (x)),

Y (0) = Y0,(2.56)

Which corresponds to Euler buckling problem (1.2) for an eigenvalue λ and the initialvalue y0. Using Taylor’s Decomposition on two points xk−1, xk for p = q then y0 � y(0),


Table 6: Comparison of higher eigenvalues for Mathieu’s equation obtained from FDM, HWSM, and TDMcorresponding to θ = 5.

n n2 λn (FDM) λn (HWSM) λn (TDM)1 1 −5.7311 −5.4665 −5.790082 4 2.0992 2.6161 2.099463 9 9.2365 9.4227 9.236334 16 16.648 16.3707 16.64825 25 25.511 24.1471 25.51086 36 36.359 36.6577 36.3589λ1 = −5.46653 (HWSM), −5.73115 (FDM), −5.79008 (TDM).

Table 7: Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov’smethod (Λk) with correction and TDM (μk) for Example 3.3 corresponding to β = 10.

k λk λk −Λ(40)k

λk −Λ(80)k

λk − μ70k

λk − μ80k

1 0.0000000 2.32E − 3 1.43E − 4 3.63E − 3 7.60E − 52 37.7596285 7.52E − 3 4.42E − 4 9.15E − 4 3.81E − 63 37.8059002 1.61E − 3 9.93E − 5 5.66E − 7 1.12E − 74 37.8525995 6.88E − 3 4.43E − 4 8.97E − 4 3.85E − 55 70.5475097 3.22E − 2 1.97E − 3 1.57E − 2 3.74E − 46 92.6538177 2.25E − 2 1.37E − 3 2.25E − 3 2.20E − 57 96.2058159 1.20E − 2 7.28E − 4 2.18E − 3 1.78E − 48 102.254347 3.45E − 2 2.09E − 3 4.38E − 3 2.44E − 4h = π for Taylor’s decomposition method.

z0 = z(0). Solving the obtained nonlinear system by Newton’s method, we obtain the approx-imate value yk of the eigenfunction y(x) at x = xk with O(h2p).

It is clear that f (0)2 (y, z) = siny is Lipschitz in y in 2-dimensional box D. Using the

results (Adiyaman and Somali [7, Lemma 2 and Theorem 3]), the global error for (2.50) isbounded by

‖Y (xk) − Yk‖ ≤ C0‖Y (0) − Y0‖ + C1ξh2pMp+1(2p)!

, (2.57)

where C0 = ex((2LB(h))/(1−LB(h))), C1 = const (C0/L)(1/(1 + (β2/β1)h + · · · + βp/β1hp−1)), M =max(y,z)∈D{|f (0)

1 (y, z)|, |f (0)2 (y, z)|}, D is 2-dimensional box in R2, ξ = max{∑�(i+1)/2�

k=0 aj,i,k},j = 1, . . . , 2p, i = 0, . . . , p, const is a constant independent of h, p, ‖ · ‖ denotes ‖ · ‖∞,L = max1≤j≤p{l1,j , l2,j} with l1,j = max1≤j≤p{d1,j , s1,j}, l2,j = max1≤j≤p{d2,j , s2,j}, dk,j =

max(y,z)∈D|(∂f (j)k(y, z)/∂y)|, sk,j = max(y,z)∈D|(∂f (j)

k(y, z)/∂z)|, k = 1, 2, and B(h) =

L∑p

j=1 βjhj−1 for some x > 0.


Table 8: Comparison of the exact eigenvalues with approximate eigenvalues obtained from Numerov’smethod (Λk) with correction and TDM (μk) for Example 3.3 corresponding to β = 20.

k λk λk −Λ(40)k

λk −Λ(80)k

λk − μ100k

λk − μ110k

1 0.0000000 1.93E − 2 1.17E − 3 1.04E − 1 7.64E − 32 77.9161943 1.22E − 1 7.38E − 3 4.37E − 2 6.98E − 23 77.9161957 1.63E − 2 9.95E − 4 9.89E − 4 1.67E − 44 77.9161972 1.62E − 2 9.89E − 4 1.13E − 1 7.48E − 25 151.463224 3.90E − 1 2.33E − 2 7.04E − 1 1.56E − 1h = π for Taylor’s decomposition method.


3.1. Numerical Results for Regular Sturm-Liouville Eigenvalue Problems

We consider three regular Sturm-Liouville eigenvalue problems, one of them has polynomialcoefficients and the others have periodic coefficients taken from Bujurke et al. [9] andAndrew[10].

Example 3.1. Consider the Titchmarch equation:

y′′ +(λ − x2n

)y(x) = 0,

y(0) = y(1) = 0,(3.1)

where n is a nonnegative integer. We obtain the numerical solutions taking n = 0, 2. Theaccuracy of the method is tested by comparing with the exact solution which exists whenn = 0 and finite-difference method (FDM) solution andHaar wavelet series method (HWSM)solution when n = 2.

Tables 2 and 3 give computed eigenvalues and solution y(x) of Titchmarch problemusing Taylor’s decomposition method (TDM) with different values of p, HWSM and FDMfor n = 0, 2, the integer parameter in Titchmarch problem. In Table 2, it is easily seen that theerror between approximate eigenfunction and exact eigenfunction decreases as p increases orthe step-size decreases or both happen. So, we can find good approximation to eigenfunctionsfor relatively large step-sizes by increasing p. In Table 3 m is the number of intervals. Table 4gives the errors between exact and approximate eigenvalues for fixed step-size h = 1 forn = 0. Notice that, as p increases, the accuracy of approximation almost doubles in digitswhich demonstrates a fast convergence.

Example 3.2. Consider the Mathieu’s equation:

y′′ + (λ − 2θ cos(2x))y = 0,

y(0) = y(π) = 0.(3.2)

We will solve these two problems approximately using Taylor’s decomposition method(TDM), and we will compare our results with the results in Bujurke et al. [9]. Bujurke etal. [9] solved Examples 3.1 and 3.2 approximately using Haar wavelets. They transform


2

1

0

−1

−2

0 0.2 0.4 0.6 0.8 1

n = 1n = 2

n = 3n = 4

Eigen

func

tion

y(x)

x

Figure 1: Higher eigenfunctions of Mathieu’s equation for a fixed parameter θ = 5.

θ = 1θ = 2θ = 5

2

1

0 0.2 0.4 0.6 0.8 1

2.5

0

1.5

0.5

Eigen

func

tion

y(x)

x

Figure 2: Solutions of Mathieu’s equation for different parameters of θ.

the interval [0, π] to [0, 1] because of the properties of Haar wavelets. So to compare theresults we normalize the interval [0, π] by using x = πt, Mathieu’s equation in Example 3.2transformed into

y′′ +(π2λ − 2π2θ cos(2πt)

)y = 0,

y(0) = y(1) = 0.(3.3)

The eigenvalues for a fixed value for θ = 5 are obtained in Table 6 which gives theasymptotic behavior of higher eigenvalues of Mathieu’s equation, and these eigenvaluesare λn = n2 + O(1). This result agrees with the classical theorem on asymptoticity of the


y0 = 1.7471

0

1

−1

0 0.2

1.5

0.5

0.60.4 0.8 1

−0.5

−1.5

Figure 3: Solution of (1.2) corresponding to the initial values y0 for π2 ≤ λ = 15 < 4π2.

eigenvalues limn→∞λ1/2n /n = 1 from van Brunt [11]. Figure 1 demonstrates that the nth

eigenfunction has n − 1 zeros in (0,1) which is consistent with the relevant graph in Bujurkeet al. [9]. The selected values of parameter θ shifts the symmetry of the solutions and thisproperty is given in Figure 2.

Example 3.3. Consider the equation

−y′′ +(2β cos 2x + β2sin22x

)y = λy,

y′(0) = y′(π) = 0.(3.4)

We give the comparison of approximate eigenvalues obtained using Taylor’s Decompositionmethod with the approximate eigenvalues obtained using Numerov’s method Andrew [10]for β = 10 and β = 20 in Tables 7 and 8. The values shown as the “exact” λk and the correctedapproximate eigenvalues Λk obtained using Numerov’s method for step-sizes h = 40 andh = 80 in Tables 7 and 8 are taken from Andrew [10]. The values shown as μk are evaluatedusing Taylor’s Decomposition method for p = 70 and 80 in Table 7 and for p = 100 and 110in Table 8. From the tables, it can be seen that Taylor’s Decomposition method approximatessmall eigenvalues with high-order accuracy without using any correction.

In comparison to Example 3.1, the estimation of eigenvalues for Examples 3.2 and 3.3is more complicated. But Example 3.1 is important to show the high accuracy of the methodwhile calculating the eigenfunctions for relatively large step-sizes. Other two examples showthe accuracy of the method while calculating the eigenvalues for large step-sizes which equalto whole interval.


y0 = 2.8578y0 = 1.0092

2

1

0

−1

−2

0 0.2 0.4 0.6 0.8 1

Figure 4: Solution of (1.2) corresponding to the initial values y0 for 4π2 ≤ λ = 45 < 9π2.

In Table 5, the observed orders ord(h) are computed using the following formula

ord(h) =log((y4h − y2h

)/(y2h − yh

))

log 2, (3.5)

where y4h, y2h, and yh are the approximated value of eigenfunctions at xk to thecorresponding eigenvalue λ when the problems are solved with step sizes 4h, 2h, and hrespectively. The observed orders given in Table 5 well confirm the theoretical results. Thatis, the order of TDM is order of 2p.

The numerical calculations and all figures in this work are performed usingMathematica.

3.2. Numerical Results for Euler Buckling Problem

The approximate solutions of Euler Buckling problem for λ = 15, λ = 45, λ = 90, and λ = 160generated using Taylor’s Decomposition method for step size h = 1/20 are illustrated inFigures 3, 4, 5, and 6, respectively.

4. Conclusion

In this paper, we have described Taylor’s Decomposition method for regular Sturm-Liouville eigenvalue problems with Dirichlet and Neumann boundary conditions to obtainapproximate eigenvalues and eigenfunctions and for Euler Buckling Problem to obtain


y0 = 3.0718y0 = 2.3413y0 = 0.3236

2

3

1

0

−1

−3

−2

0 0.2 0.4 0.6 0.8 1


y0 = 3.1272y0 = 2.7999

y0 = 2.0239y0 = 0.3771

0 0.2 0.4 0.6 0.8 1

2

3

1

0

−1

−3

−2



approximate initial values and eigenfunctions. The obtained results for Euler Bucklingproblem give the behavior of eigenvalues and corresponding eigenfunctions with high-order accuracy without using small stepsize. We have seen that these results agree with thetheoretical aspects. This method can be extended to solve regular Sturm-Liouville eigenvalueproblems with Robin (mixed) boundary conditions and to some nonlinear eigenvalueproblems to investigate the behavior of the eigenvalues and eigenfunction. However, thismethod is best suited to find small eigenvalues for the other nonlinear problems in literature.One possible method of improving its efficiency for higher eigenvalues may be to follow theideas of [10, 12, 13] and for eigenvalue problems for partial differential equations given inelsewhere [14–18].

References

[1] J. D. Pryce, Numerical Solution of Sturm-Liouville Problems, Monographs on Numerical Analysis, TheClarendon Press Oxford University Press, New York, NY, USA, 1993.

[2] I. Stakgold, “Branching of solutions of nonlinear equations,” SIAM Review, vol. 13, pp. 289–332, 1971.[3] R. M. Jones, Buckling of Bars, Plates, and Shells, Bull Ridge, Virginia, Va, USA, 2006.[4] G. Domokos and P. Holmes, “Euler’s problem, Euler’s method, and the standardmap; or, The discrete

charm of buckling,” Journal of Nonlinear Science, vol. 3, no. 1, pp. 109–151, 1993.[5] D. H. Griffel, Applied Functional Analysis, Ellis Horwood Series in Mathematics and Its Applications,

Ellis Horwood, Chichester, UK, 1981.[6] L. Euler, “Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes ostwald’s

klassiker der exakten wiss,” Laussane and Geneva, vol. 75, 1774.[7] M. E. Adiyaman and S. Somali, “Taylor’s decomposition on two points for one-dimensional Bratu

problem,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 412–425, 2010.[8] A. Ashyralyev and P. E. Sobolevskii, New difference schemes for partial differential equations, vol. 148 of

Operator Theory: Advances and Applications, Birkhauser, Basel, Switzerland, 2004.[9] N. M. Bujurke, C. S. Salimath, and S. C. Shiralashetti, “Computation of eigenvalues and solutions

of regular Sturm-Liouville problems using Haar wavelets,” Journal of Computational and AppliedMathematics, vol. 219, no. 1, pp. 90–101, 2008.

[10] A. L. Andrew, “Asymptotic correction of Numerov’s eigenvalue estimates with natural boundaryconditions,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 359–366, 2000.

[11] B. van Brunt, The Calculus of Variations, Universitext, Springer, New York, NY, USA, 2004.[12] R. S. Anderssen and F. R. de Hoog, “On the correction of finite difference eigenvalue approximations

for Sturm-Liouville problems with general boundary conditions,” BIT Numerical Mathematics, vol. 24,no. 4, pp. 401–412, 1984.

[13] S. Somali and V. Oger, “Improvement of eigenvalues of Sturm-Liouville problem with t-periodicboundary conditions,” Journal of Computational and Applied Mathematics, vol. 180, no. 2, pp. 433–441,2005.

[14] V. Mehrmann and A. Miedlar, “Adaptive computation of smallest eigenvalues of self-adjoint ellipticpartial differential equations,” Numerical Linear Algebra with Applications, vol. 18, no. 3, pp. 387–409,2011.

[15] Q.-M. Cheng, T. Ichikawa, and S. Mametsuka, “Estimates for eigenvalues of the poly-Laplacian withany order in a unit sphere,” Calculus of Variations and Partial Differential Equations, vol. 36, no. 4, pp.507–523, 2009.

[16] C. Lovadina, M. Lyly, and R. Stenberg, “A posteriori estimates for the Stokes eigenvalue problem,”Numerical Methods for Partial Differential Equations, vol. 25, no. 1, pp. 244–257, 2009.

[17] S. Jia, H. Xie, X. Yin, and S. Gao, “Approximation and eigenvalue extrapolation of biharmoniceigenvalue problem by nonconforming finite element methods,” Numerical Methods for PartialDifferential Equations, vol. 24, no. 2, pp. 435–448, 2008.

[18] C. V. Verhoosel, M. A. Gutierrez, and S. J. Hulshoff, “Iterative solution of the random eigenvalueproblem with application to spectral stochastic finite element systems,” International Journal forNumerical Methods in Engineering, vol. 68, no. 4, pp. 401–424, 2006.


Research ArticleAn Approximation of Ultra-Parabolic Equations

Allaberen Ashyralyev1, 2 and Serhat Yılmaz1

1 Department of Mathematics, Fatih University, Istanbul, Turkey2 Department of Mathematics, ITTU, Ashgabat, Turkmenistan, Turkey

Correspondence should be addressed to Serhat Yılmaz, [email protected]


Academic Editor: Hasan Ali Yurtsever

Copyright q 2012 A. Ashyralyev and S. Yılmaz. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The first and second order of accuracy difference schemes for the approximate solution of the initialboundary value problem for ultra-parabolic equations are presented. Stability of these differenceschemes is established. Theoretical results are supported by the result of numerical examples.

1. Introduction

Mathematical models that are formulated in terms of ultraparabolic equations are of greatimportance in various problems for instance in age-dependent population model, in themathematical model of Brownian motion, in the theory of boundary layers, and so forth, see[1–5]. We refer also to [6–9] and the references therein for existence and uniqueness resultsand other properties of these models. On the other hand, Akrivis et al. [10] and Ashyralyevand Yılmaz [11, 12] developed numerical methods for ultraparabolic equations. In this paper,our interest is studying the stability of first- and second-order difference schemes for theapproximate solution of the initial boundary value problem for ultraparabolic equations

∂u(t, s)∂t

+∂u(t, s)∂s

+Au(t, s) = f(t, s), 0 < t, s < T,

u(0, s) = ψ(s), 0 ≤ s ≤ T,u(t, 0) = ϕ(t), 0 ≤ t ≤ T,

(1.1)


in an arbitrary Banach space Ewith a strongly positive operatorA. For approximately solvingproblem (1.1), the first-order of accuracy difference scheme

uk,m − uk−1,mτ

+uk−1,m − uk−1,m−1

τ+Auk,m = fk,m,

fk,m = f(tk, sm), tk = kτ, sm = mτ, 1 ≤ k, m ≤N, Nτ = 1,

u0,m = ψm, 0 ≤ m ≤N, uk,0 = ϕk, 0 ≤ k ≤N

(1.2)

and second-order of accuracy difference scheme

uk,m − uk−1,mτ

+uk−1,m − uk−1,m−1

τ+12A(uk,m + uk−1,m−1) = fk,m,

fk,m = f(tk − τ

2, sm − τ

2

), tk = kτ, sm = mτ, 1 ≤ k, m ≤N, Nτ = 1,

u0,m = ψm, 0 ≤ m ≤N, uk,0 = ϕk, 0 ≤ k ≤N

(1.3)

are presented. The stability estimates for the solution of difference schemes (1.2) and (1.3)are established. In applications, the stability in maximum norm of difference shemes formultidimensional ultraparabolic equations with Dirichlet condition is established. Applyingthe difference schemes, the numerical methods are proposed for solving one-dimensionalultraparabolic equations.

Theorem 1.1. For the solution of (1.2), we have the following stability inequality:

max1≤k≤N

max1≤m≤N

‖uk,m‖E ≤ C(

max0≤m≤N

∥∥ψm∥∥E + max

0≤k≤N

∥∥ϕk∥∥E + max

1≤k≤Nmax1≤m≤N

∥∥fk,m∥∥E

), (1.4)

where C is independent of τ , ψm, ϕk, and fk,m.

Proof. Using (1.2), we get

uk,m − uk−1,m−1τ

+Auk,m = fk,m. (1.5)

From that it follows

uk,m = Ruk−1,m−1 + τRfk,m, (1.6)

where R = (I + τA)−1. By the mathematical induction, we will prove that

uk,m = Rnuk−n,m−n +n∑

j=1

τRn−j+1fk−n+j,m−n+j (1.7)


is true for all positive integers n. It is obvious that for n = 1, 2 formula (1.7) is true. Assumethat for n = r

uk,m = Rruk−r,m−r +r∑

j=1

τRr−j+1fk−r+j,m−r+j (1.8)

is true. In formula (1.6), replacing k andm with k − r andm − r, respectively, we have

uk−r,m−r = Ruk−r−1,m−r−1 + τRfk−r,m−r . (1.9)

Then, using (1.8) and (1.9), we get

uk,m = Rr+1uk−r−1,m−r−1 + τRr+1fk−r,m−r +r∑

j=1

τRr−j+1fk−r+j,m−r+j . (1.10)

From that it follows

uk,m = Rr+1uk−r−1,m−r−1 +r+1∑

j=1

τRr−j+2fk−r−1+j,m−r−1+j (1.11)

is true for n = r + 1. So, formula (1.7) is proved. For m > k, replacing n with k in formula(1.7), we obtain that

uk,m = Rkψm−k +k∑

j=1

τRk−j+1fj,m−k+j . (1.12)

Using estimate (see [13])

∥∥∥Rk∥∥∥E→E

≤M (1.13)

and triangle inequality, we get

‖uk,m‖E ≤∥∥∥Rk∥∥∥E→E

∥∥ψm−k∥∥E +

k∑

j=1

τ∥∥∥Rk−j+1

∥∥∥E→E

∥∥fj,m−k+j∥∥E

≤M[

max0≤k,m≤N

∥∥ψm−k∥∥ + max

1≤j≤Nmax1≤m≤N

∥∥fj,m∥∥] (1.14)

for any k andm. For k > m, replacing nwithm in formula (1.7), we get

uk,m = Rmϕk−m +m∑

j=1

τRm−j+1fk−m+j,j . (1.15)


From estimate (1.13) and triangle inequality, it follows that

‖uk,m‖E ≤ ‖Rm‖E→E

∥∥ϕk−m∥∥E +

m∑

j=1

τ∥∥∥Rm−j+1

∥∥∥E→E

∥∥fk−m+j,j∥∥E

≤M[

max0≤k,m≤N

∥∥ϕk−m∥∥ + max

1≤j≤Nmax1≤m≤N

∥∥fj,m∥∥] (1.16)

for any k andm. Thus, Theorem 1.1 is proved.

Theorem 1.2. For the solution of (1.3), we have the following stability inequality:

max1≤k≤N

max1≤m≤N

∥∥∥∥uk,m + uk−1,m

2

∥∥∥∥E

+ max1≤k≤N

max1≤m≤N

∥∥∥∥uk,m + uk,m−1

2

∥∥∥∥E

≤ C(

max0≤m≤N

∥∥ψm∥∥E + max

0≤k≤N

∥∥ϕk∥∥E + max

1≤k≤Nmax1≤m≤N

∥∥fk,m∥∥E

),

(1.17)

where C is independent of τ , ψm, ϕk, and fk,m.

The proof of Theorem 1.2 is based on the following formulas:

uk,m = Buk−1,m−1 + τBCfk,m,

uk,m = Bkψm−k +k∑

j=1

τBk−jCfj,m−k+j , m > k,

uk,m = Bmϕk−m +m∑

j=1

τBm−jCfk−m+j,j , k > m

(1.18)

for the solution of difference scheme (1.3) and the following estimate [14]:∥∥∥BkC

∥∥∥E→E

≤M, (1.19)

where B = (I − τA/2)(I + τA/2)−1 and C=(I + τA/2)−1.

2. Application

Let Ω be the unit open cube in the n-dimensional Euclidean space Rn (0 < xk < 1, 1 ≤ k ≤ n)

with boundary S, Ω = Ω ∪ S. In [0, 1] × [0, 1] ×Ω, we consider the boundary-value problemfor the multidimensional parabolic equation

∂u(t, s, x)∂t

+∂u(t, s, x)

∂s−

n∑

r=1

αr(x)∂2u(t, s, x)

∂x2r

+ δu(t, s, x) = f(t, s, x),

x = (x1, . . . , xn) ∈ Ω, 0 < t, s < 1,

u(0, s, x) = ψ(s, x), s ∈ [0, 1], u(t, 0, x) = ϕ(t, x), t ∈ [0, 1], x ∈ Ω,

u(t, s, x) = 0, t, s ∈ [0, 1], x ∈ S,

(2.1)


where αr(x) > a > 0 (x ∈ Ω) and f(t, s, x) (t, s ∈ (0, 1), x ∈ Ω) are given smooth functionsand δ > 0 is a sufficiently large number.

We introduce the Banach spaces Cβ

01(Ω) (β = (β1, . . . , βn), 0 < xk < 1, k = 1, . . . , n)of all continuous functions satisfying a Holder condition with the indicator β = (β1, . . . , βn),βk ∈ (0, 1), 1 ≤ k ≤ n, and with weight xβkk (1 − xk − hk)βk , 0 ≤ xk < xk + hk ≤ 1, 1 ≤ k ≤ n,which is equipped with the norm

∥∥f∥∥Cβ

01(Ω) =∥∥f∥∥C(Ω) + sup

0≤xk<xk+hk≤11≤k≤n

∣∣f(x1, . . . , xn) − f(x1 + h1, . . . , xn + hn)∣∣

×n∏

k=1

h−βkkxβkk (1 − xk − hk)βk ,

(2.2)

where C(Ω) stands for the Banach space of all continuous functions defined on Ω, equippedwith the norm

∥∥f∥∥C(Ω) = max

x∈Ω

∣∣f(x)∣∣. (2.3)

It is known that the differential expression

Av = −n∑

r=1

αr(x)∂2v(t, s, x)

∂x2+ δv(t, s, x) (2.4)

defines a positive operator A acting on Cβ

01(Ω) with domain D(Ax) ⊂ C2+β01 (Ω) and satisfying

the condition v = 0 on S.The discretization of problem (2.1) is carried out in two steps. In the first step, let us

define the grid sets

Ωh = {x = xm = (h1m1, . . . , hnmn), m = (m1, . . . , mn),

0 ≤ mr ≤Nr, hrNr = L, r = 1, . . . , n},


(2.5)

We introduce the Banach spaces Ch = Ch(Ωh), Cβ

h = Cβ

01(Ωh) of grid functions ϕh(x) ={ϕ(h1m1, . . . , hnmn)} defined on Ωh, equipped with the norms

∥∥∥ϕh∥∥∥C(Ωh)

= maxx∈Ωh

∣∣∣ϕh(x)∣∣∣,

∥∥∥ϕh∥∥∥Cβ

01(Ωh)=∥∥∥ϕh∥∥∥C(Ωh)

+ sup0≤xk<xk+hk≤1

1≤k≤n

∣∣∣ϕh(x1, . . . , xn) − ϕh(x1 + h1, . . . , xn + hn)∣∣∣

×n∏

k=1

h−βkkxβkk (1 − xk − hk)βk .

(2.6)



hby the formula

Axhu

hx = −

n∑

r=1

ar(x)(uh−xr

)

xr ,jr(2.7)

acting in the space of grid functions uh(x), satisfying the condition uh(x) = 0 for all x ∈ Sh.With the help of Ax

h, we arrive at the initial boundary-value problem

∂uh(t, s, x)∂t

+∂uh(t, s, x)

∂s+Ax

huh(t, s, x) = fh(t, s, x), 0 < t, s < 1, x ∈ Ωh,

uh(0, s, x) = ψh(s, x), 0 ≤ s ≤ 1, uh(t, 0, x) = ϕh(t, x), 0 ≤ t ≤ 1, x ∈ Ωh

(2.8)

for an infinite system of ordinary differential equations.

In the second step, we replace problem (2.8) by difference scheme(1.2)

uhk,m

− uhk−1,m

τ+uhk−1,m − uh

k−1,m−1τ

+Axhu

hk,m = fhk,m(x), x ∈ Ωh,

fhk,m(x) = fh(tk, sm,x), tk = kτ, sm = mτ, 1 ≤ k,m ≤N, x ∈ Ωh,

uh0,m = ψhm, 0 ≤ m ≤N, uhk,0 = ϕhk, 0 ≤ k ≤N

(2.9)

and by difference scheme(1.3)

uhk,m

− uhk−1,m

τ+uhk−1,m − uh

k−1,m−1τ

+12Axh(uk,m + uk−1,m−1) = fhk,m(x), x ∈ Ωh,

fhk,m(x) = fh(tk − τ

2, sm − τ

2, x), tk = kτ, sm = mτ, 1 ≤ k, m ≤N, x ∈ Ωh,

uh0,m = ψhm, 0 ≤ m ≤N, uhk,0 = ϕhk, 0 ≤ k ≤N.

(2.10)

It is known that Axhis a positive operator in C(Ωh) and C

β

01(Ωh). Let us give a numberof corollaries of Theorems 1.1 and 1.2.

Theorem 2.1. For the solution of difference scheme (2.9), we have the following stability inequality:

max1≤k≤N

max1≤m≤N

∥∥∥uhk,m∥∥∥C(Ωh)

≤ C1

(max0≤m≤N

∥∥∥ψhm∥∥∥C(Ωh)

+ max0≤k≤N

∥∥∥ϕhk∥∥∥C(Ωh)

+ max1≤k≤N

max1≤m≤N

∥∥∥fhk,m∥∥∥C(Ωh)

),

(2.11)

where C1 is independent of τ , ψhm, ϕhk, and fh

k,m.


Theorem 2.2. For the solution of difference scheme (2.10), we have the following stability inequality:

max1≤k≤N

max1≤m≤N

∥∥∥∥∥uhk,m + uhk−1,m−1

2

∥∥∥∥∥C(Ωh)

≤ C1

(max0≤m≤N

∥∥∥ψhm∥∥∥C(Ωh)

+ max0≤k≤N

∥∥∥ϕhk∥∥∥C(Ωh)

+ max1≤k≤N

max1≤m≤N

∥∥∥fhk,m∥∥∥C(Ωh)

),

(2.12)

where C1 does not depend on τ , ψhm, ϕhk, and fh

k,m.


In this section, the initial boundary value problem

∂u(s, t, x)∂t

+∂u(s, t, x)

∂s− ∂2u(s, t, x)

∂x2+ 2u(t, s, x) = f(t, s, x),

f(t, s, x) = e−(t+s) sinπx, 0 < s, t < 1, 0 < x < 1,

u(0, t, x) = e−t sinπx, 0 < t < 1, 0 < x < 1,

u(s, 0, x) = e−s sinπx, 0 < s < 1, 0 < x < 1,

u(s, t, 0) = u(s, t, π) = 0, 0 < s, t < 1

(3.1)

for one-dimensional ultraparabolic equations is considered.The exact solution of problem (3.1) is

u(t, s, x) = e−(t+s) sinπx. (3.2)

Using the first order of accuracy in t and s implicit difference scheme (2.9), we obtainthe difference scheme first order of accuracy in t and s and second-order of accuracy in x

uk,mn − uk−1,mn

τ− uk−1,mn − uk−1,m−1

n

τ− uk,mn+1 − 2uk,mn + uk,mn−1

h2+ 2uk,mn = fhk,m,

fhk,m = f(tk, sm, xn) = e−(tk+sm) sinxn, 1 ≤ k,m ≤N, 1 ≤ n ≤M − 1,

u0,mn = e−sm sinxn, 0 ≤ m ≤N, 0 ≤ n ≤M,

uk,0n = e−tk sinxn, 0 ≤ k ≤N, 0 ≤ n ≤M,

uk,m0 = uk,mM = 0, 0 ≤ k,m ≤N,

tk = kτ, sm = mτ, 1 ≤ k,m ≤N, Nτ = 1,

xn = nh, 1 ≤ n ≤M, Mh = π

(3.3)


for approximate solutions of initial boundary value problem (3.3). It can be written in thematrix form

Aun+1 + Bun + Cun−1 = ϕn, 1 ≤ n ≤M − 1,

u0 = 0, uM = 0.(3.4)

Here

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · · · · · · · · · · 0 0

0 0 · · · · · · · · · · · · 0

.... . . . . .

......

......

0 · · · 0 a 0 · · · 0

0 0 · · · 0 a 0 · · ·...

. . . . . . . . . . . . . . ....

0 · · · · · · · · · · · · · · · a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 · · · · · · · · · · · · 0 0 0

0 1 0 · · · · · · · · · · · · 0 0

0 0 1 0 · · · · · · · · · · · · 0

......

. . . . . . . . . . . . . . . . . ....

0 b 0 · · · c d 0 · · · 0

0 0 b 0 · · · c d 0 · · ·...

. . . . . . . . . . . . . . . . . . . . ....

0 0 · · · · · · b 0 · · · c d

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2

,

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · · · · · · · · · · 0 0

0 0 · · · · · · · · · · · · 0

.... . . . . .

......

......

0 · · · 0 a 0 · · · 0

0 0 · · · 0 a 0 · · ·...

. . . . . . . . . . . . . . ....

0 · · · · · · · · · · · · · · · a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2

,

(3.5)

where

a =−1h2, b =

−1τ, c =

−1τ, d =

1τ+1τ+

2h2

+ 2,


ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0,0n

ϕ1,0n

...

ϕN,0n

ϕ0,1n

ϕ1,1n

...

ϕN,1n

...

ϕ0,Nn

ϕ1,Nn

...

ϕN,Nn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×1

, ϕk,mn = f(tk, sm, xn) = e−(tk+sm) sinxn,

un =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

u0,0n

u1,0n

...

uN,0n

u0,1n

u1,1n

...

uN,1n

...

u0,Nn

u1,Nn

...

uN,Nn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×1

.

(3.6)


This type system was used by Samarskii and Nikolaev [15] for difference equations. For thesolution of matrix equation (3.4), we will use the modified Gauss elimination method. Weseek a solution of the matrix equation by the following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 2, 1, (3.7)

where uM = 0, αj(j = 1, . . . ,M−1) are (N + 1)2 × (N + 1)2 square matrices, βj(j = 1, . . . ,M−1)are (N + 1)2 × 1 coloumn matrices, α1, β1 are zero matrices, and

αn+1 = −(B + Cαn)−1A,

βn+1 = (B + Cαn)−1(ϕn − Cβn

), n = 1, 2, 3, . . . , (M − 1).

(3.8)

Using the second-order of accuracy in t and s implicit difference scheme (2.10), weobtain the difference scheme second-order of accuracy in t and s and second-order of accuracyin x

uk,mn − uk−1,mn

τ− uk−1,mn − uk−1,m−1

n

τ

− 12

[uk,mn+1 − 2uk,mn + uk,mn−1

h2+ 2uk,mn +

uk−1,m−1n+1 − 2uk−1,m−1

n + uk−1,m−1n−1

h2+ 2uk−1,m−1

n

]= fhk,m,

fhk,m = f(tk, sm, xn) = e−(tk+sm−τ) sinxn, 1 ≤ k,m ≤N, 1 ≤ n ≤M − 1,

u0,mn = e−sm sinxn, 0 ≤ m ≤N, 0 ≤ n ≤M,

uk,0n = e−tk sinxn, 0 ≤ k ≤N, 0 ≤ n ≤M,

uk,m0 = uk,mM = 0, 0 ≤ k,m ≤N,

tk = kτ, sm = mτ, 1 ≤ k,m ≤N, Nτ = 1,

xn = nh, 1 ≤ n ≤M, Mh = π(3.9)


for approximate solutions of initial boundary value problem (3.9). The matrix form (3.4) canbe written. Here,

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · · · · · · · · · · 0 0

0 0 · · · · · · · · · · · · 0

.... . . . . .

......

......

a 0 · · · a 0 · · · 0

0 a 0 · · · a 0 · · ·...

. . . . . . . . . . . . . . ....

0 · · · · · · a 0 · · · a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2(N+1)2×(N+1)2

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 · · · · · · · · · · · · 0 0 0

0 1 0 · · · · · · · · · · · · 0 0

0 0 1 0 · · · · · · · · · · · · 0

......

. . . . . . . . . . . . . . . . . ....

b 0 0 · · · 1 c 0 · · · 0

0 b 0 0 · · · 1 c 0 · · ·...

. . . . . . . . . . . . . . . . . . . . ....

0 0 · · · b 0 0 · · · 1 c

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2

,

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 · · · · · · · · · · · · 0 0

0 0 · · · · · · · · · · · · 0

.... . . . . .

......

......

a 0 · · · a 0 · · · 0

0 a 0 · · · a 0 · · ·...

. . . . . . . . . . . . . . ....

0 · · · · · · a 0 · · · a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×(N+1)2

,

(3.10)


where

a = − 12h2

, b = − 1τ+

1h2

+ 1, c =1τ+

1h2

+ 1,

ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0,0n

ϕ1,0n

...

ϕN,0n

ϕ0,1n

ϕ1,1n

...

ϕN,1n

...

ϕ0,Nn

ϕ1,Nn

...

ϕN,Nn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×1

, ϕk,mn = f(tk − τ

2, sm − τ

2, xn)= e−(tk+sm) sinxn,

un =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

u0,0n

u1,0n

...

uN,0n

u0,1n

u1,1n

...

uN,1n

...

u0,Nn

u1,Nn

...

uN,Nn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)2×1

.

(3.11)


Table 1

Difference schemes N =M = 10 N =M = 20 N =M = 40 N =M = 80

Difference scheme (3.3) 0.028100 0.014400 0.006811 0.003225

Difference scheme (3.9) 0.000511 0.000121 0.000028 0.000006

We seek a solution of the matrix equation by the same algorithm (3.7) and (3.8).

4. Error Analysis

The errors are computed by

EK,MN = max1≤k,m≤N,1≤n≤M−1

∣∣∣u(tk, sm, xn) − uk,mn∣∣∣ (4.1)

of the numerical solutions, where u(tk, sm, xn) represents the exact solution and uk,mnrepresents the numerical solution at (tk, sm, xn), and the results are given in Table 1.

It may be noted from Table 1 that asN,M increase, the value of the errors associatedwith difference scheme (3.3) decreases by a factor of approximately 1/2 and the errorsassociated with difference scheme (3.9) decrease by a factor of approximately 1/4. Thisconfirms that difference scheme (3.3) is first order and difference scheme (3.9) is second-orderas stated in Section 1. Moreover, the results show that the second-order of accuracy differencescheme (3.9) are more accurate comparing with the first order of accuracy difference scheme(3.3).

References

[1] J. Dyson, E. Sanchez, R. Villella-Bressan, and G. F. Webb, “An age and spatially structured model oftumor invasion with haptotaxis,”Discrete and Continuous Dynamical Systems B, vol. 8, no. 1, pp. 45–60,2007.

[2] K. Kunisch, W. Schappacher, and G. F. Webb, “Nonlinear age-dependent population dynamics withrandom diffusion,” Computers & Mathematics with Applications, vol. 11, no. 1–3, pp. 155–173, 1985,Hyperbolic partial differential equations, II.

[3] A. N. Kolmogorov, “Zur Theorie der stetigen zufalligen Prozesse,” Mathematische Annalen, vol. 108,pp. 149–160, 1933.

[4] A. N. Kolmogorov, “Zufallige Bewegungen,” Annals of Mathematics, vol. 35, pp. 116–117, 1934.[5] T. G. Gencev, “On ultraparabolic equations,”Doklady Akademii Nauk SSSR, vol. 151, pp. 265–268, 1963.[6] Q. Deng and T. G. Hallam, “An age structured population model in a spatially heterogeneous

environment: existence and uniqueness theory,” Nonlinear Analysis, vol. 65, no. 2, pp. 379–394, 2006.[7] G. di Blasio and L. Lamberti, “An initial-boundary value problem for age-dependent population

diffusion,” SIAM Journal on Applied Mathematics, vol. 35, no. 3, pp. 593–615, 1978.[8] G. di Blasio, “Nonlinear age-dependent population diffusion,” Journal of Mathematical Biology, vol. 8,

no. 3, pp. 265–284, 1979.[9] S. A. Tersenov, “Boundary value problems for a class of ultraparabolic equations and their

applications,”Matematicheskiı Sbornik, vol. 133(175), no. 4, pp. 529–544, 1987.[10] G. Akrivis, M. Crouzeix, and V. Thomee, “Numerical methods for ultraparabolic equations,” Calcolo,

vol. 31, no. 3-4, pp. 179–190, 1994.[11] A. Ashyralyev and S. Yılmaz, “On the approximate solution of ultra parabolic equations,” in

Proceedings of the 2nd International Symposium on Computing in Science & Engineering, M. Gunes, A.K. Cınar, and I. Gurler, Eds., pp. 533–535, Gediz University, Izmir, Turkey, 2011.


[12] A. Ashyralyev and S. Yılmaz, “Second order of accuracy difference schemes for ultra parabolicequations,” AIP Conference Proceedings, vol. 1389, pp. 601–604, 2011.

[13] A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations Operator TheoryAdvance and Applications, vol. 69, Birkh auser, Basel, Switzerland, 1994.

[14] P. E. Sobolevskii, “On the Crank-Nicolson difference scheme for parabolic equations,” NonlinearOscillations and Control Theory, pp. 98–106, 1978.

[15] A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. 2: Iterative Methods,Birkhauser, Basel, Switzerland, 1989.


Research ArticleApproximate Solutions of Delay ParabolicEquations with the Dirichlet Condition

Deniz Agirseven

Department of Mathematics, Trakya University, 22030 Edirne, Turkey

Correspondence should be addressed to Deniz Agirseven, [email protected]

Received 21 February 2012; Accepted 28 March 2012


Copyright q 2012 Deniz Agirseven. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Finite difference and homotopy analysis methods are used for the approximate solution of theinitial-boundary value problem for the delay parabolic partial differential equation with theDirichlet condition. The convergence estimates for the solution of first and second orders ofdifference schemes in Holder norms are obtained. A procedure of modified Gauss eliminationmethod is used for the solution of these difference schemes. Homotopy analysis method is applied.Comparison of finite difference and homotopy analysis methods is given on the problem.

1. Introduction

Increase in interest in the theoretical aspects of numerical methods for delay differentialequations points out that delay differential equations are capable of generating extensiveand conceivable models for phenomena in many branches of sciences. Numerical solutionsof the delay ordinary differential equations have been studiedmostly for ordinary differentialequations (cf., e.g., [1–14] and the references therein). Nevertheless, delay partial differentialequations are less in demand than delay ordinary differential equations. Different kinds ofproblems for delay partial differential equations are solved by using operator approach (see,e.g., [15–17]).

In recent years, Ashyralyev and Sobolevskii considered the initial-value problem forlinear delay partial differential equations of parabolic type in the spaces C(Eα) of functionsdefined on the segment [0,∞)with values in a Banach space Eα and the stability inequalitieswere established under stronger assumption than the necessary condition of the stabilityof the differential problem. The stability estimates for the solutions of difference schemesof the first- and second-order accuracy difference schemes for approximately solving thisinitial-value problem for delay differential equations of parabolic type were presented. They


obtained the stability estimates in Holder norms for solutions of the initial-value problemof the delay differential and difference equations of the parabolic type [15, 16]. Gabriellaused extrapolation spaces to solve Banach spaces valued delay differential equations withunbounded delay operators. The author proved regularity properties of various types of solu-tions and investigated the existence of strong and weak solutions for a class of abstract semi-linear delay equations [17].

In this paper, finite difference (see, e.g., [18–28]) and homotopy analysis methods(HAM) (see, e.g., [29–37]) for the approximate solutions of the delay differential equationof the parabolic type

ut(t, x) + (−a(x)uxx(t, x) + b(x)ux(t, x) + c(x)u(t, x))= d(t)(−a(x)uxx(t −ω, x) + b(x)ux(t −ω, x)+c(x)u(t −ω, x)), 0 < t <∞, x ∈ (0, l),

u(t, x) = g(t, x), −ω ≤ t ≤ 0, x ∈ [0, l],

u(t, 0) = u(t, l) = 0, t ≥ 0,

(1.1)

are studied. Here g(t, x) (t ∈ (−∞, 0), x ∈ [0, l]), a(x), b(x), c(x) (x ∈ (0,∞)) are givensmooth bounded functions and a(x) ≥ a > 0.

Difference schemes which are accurate to first and second orders for the approximatesolution of problem (1.1) are presented. The convergence estimates for the solution of thesedifference schemes are obtained. For the numerical study, procedure of modified Gausselimination method is used to solve these difference schemes. Homotopy analysis method isapplied to find the solution of problem (1.1). The numerical results are obtained at the samepoints for each method. Comparison of finite difference and homotopy analysis methods isgiven on the problem.

2. The Finite Difference Method

In this section, the first and second orders of accuracy in t for the approximate solution ofproblem (1.1) are considered. The convergence estimates for the solution of these differenceschemes are established. A procedure of modified Gauss elimination method is used to solvethese difference schemes.

2.1. The Difference Scheme, Convergence Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, we define thegrid space

[0, L]h = {x = xn : xn = nh, 0 ≤ n ≤M, Mh = L}. (2.1)


To formulate our results, we introduce the Banach space◦Cα

h =◦Cα

[0, L]h, α ∈ [0, 1), of all gridfunctions ϕh = {ϕn}M−1

n=1 defined on [0, L]h with ϕ0 = ϕM = 0 equipped with the norm

∥∥∥ϕh∥∥∥ ◦Cα

h

=∥∥∥ϕh

∥∥∥Ch

+ sup1≤n<n+r≤M−1

∣∣ϕn+r − ϕn∣∣(rh)−α,

∥∥∥ϕh∥∥∥Ch

= max1≤n≤M−1

∣∣ϕn∣∣.

(2.2)

Moreover,Cτ(E) = C([0,∞)τ , E) is the Banach space of all grid functions fτ = {fk}∞k=1 definedon

[0,∞)τ = {tk = kτ, k = 0, 1, . . .} (2.3)

with values in E equipped with the norm

∥∥fτ∥∥Cτ (E)

= sup1≤k<∞

∥∥fk∥∥E. (2.4)

To the differential operator A generated by problem (1.1), we assign the difference operatorsAxh, Bx

hby the formulas

Axhϕ

h(x) ={−a(xn)

ϕn+1 − 2ϕn + ϕn−1h2

+ b(xn)ϕn+1 − ϕn−1

2h+ c(xn)ϕn

}M−1

1,

Bxh(t)ϕh(x) = d(t)Ax

hϕh,

(2.5)

acting in the space of grid functions ϕh(x) = {ϕn}M−11 satisfying the conditions ϕ0 = ϕM = 0.

It is well known thatAxhis a strongly positive operator in Ch. With the help ofAx

hand d(t)Ax

h,

we arrive at the initial value problem

duh(t, x)dt

+Axhu

h(t, x) = d(t)Axhu

h(t −w,x), 0 < t <∞, 0 < x < L,

uh(t, x) = gh(t, x), −ω ≤ t ≤ 0, 0 ≤ x ≤ L.(2.6)

In the second step, we consider difference schemes of first and second orders of accuracy

1τ

(uhk(x) − uhk−1(x)

)+Ax

huhk(x) = d(tk)A

xhu

hk−N(x), tk = kτ, 1 ≤ k, Nτ = w,

uhk(x) = gh(tk, x), tk = kτ, −N ≤ k ≤ 0,

(2.7)

1τ

(uhk(x) − uhk−1(x)

)+(Axh +

12τ(Axh

)2)uhk(x)

=12

(I +

τ

2Axh

)d(tk − τ

2

)Axh

(uhk−N(x) + uhk−N−1(x)

), tk = kτ, 1 ≤ k,

uhk = gh(tk, x), tk = kτ, −N ≤ k ≤ 0.

(2.8)



sup0≤t<∞

|d(t)| ≤ 1 − αM22−α

. (2.9)

Suppose that problem (1.1) has a smooth solution u(t, x) and

∫∞

0

[max0≤x≤L

|uss(s, x)| + sup0<x<x+y<L

∣∣uss(s, x + y

) − uss(s, x)∣∣

y2α

]ds <∞,

∫∞

0

[max0≤x≤L

|uxxxx(s, x)| + sup0<x<x+y<L

∣∣uxxxx(s, x + y

) − uxxxx(s, x)∣∣

y2α

]ds <∞.

(2.10)

Then, for the solution of difference scheme (2.7), the following convergence estimate holds:

supk

∥∥∥uhk − uh(tk, ·)∥∥∥ ◦C

2α

h

≤M1

(τ + h2

)(2.11)

withM1 being a real number independent of τ , α, and h.

Proof. Using notations of Axh and B

xh , we can obtain the following formula for the solution:

uhk(x) = Rkgh(0, x) +

k∑

j=1

Rk−j+1Bxj gh(tj−N, x

)τ, 1 ≤ k ≤N,

uhk(x) = Rk−nNuhnN(x) +

k∑

j=nN+1

Rk−j+1Bxj uhj−N(x)τ,

nN ≤ k ≤ (n + 1)N,

(2.12)

where R = (I + τAxh)

−1. The proof of Theorem 2.1 is based on the formulas (2.12), on theconvergence theorem, on the difference schemes in Cτ(Ehα) (see [38]), on the estimate

‖ exp{−tkAxh

}‖Ch →Ch ≤M, k ≥ 0, (2.13)

and on the fact that inEhα = Eα(Axh, Ch) the norms are equivalent to the norms in

◦C

2α

h uniformlyin h for 0 < α < 1/2 (see, [18]).


Theorem 2.2. Assume that assumption (2.9) of Theorem 2.1 and the following conditions hold:

∫∞

0

[max0≤x≤L

|usss(s, x)| + sup0<x<x+y<L

∣∣usss(s, x + y

) − usss(s, x)∣∣

y2α

]ds <∞,

∫∞

0

[max0≤x≤L

|uxxss(s, x)| + sup0<x<x+y<L

∣∣uxxss(s, x + y

) − uxxss(s, x)∣∣

y2α

]ds <∞,

∫∞

0

[max0≤x≤L

|uxxxxs(s, x)| + sup0<x<x+y<L

∣∣uxxxxs(s, x + y

) − uxxxxs(s, x)∣∣

y2α

]ds <∞.

(2.14)

Then for the solution of difference scheme (2.8), the following convergence estimate is satisfied:

supk

∥∥∥uhk − uh(tk, ·)∥∥∥ ◦C

2α

h

≤M2

(τ2 + h2

)(2.15)

withM2 being a real number independent of τ , α, and h.

Proof. Using notations of Axhand Bx

hagain, we can obtain the following formula for the solu-

tion:

uhk(x) = Rkgh(0, x) +

k∑

j=1

Rk−j+1(I +

τAxh

2

)(gh(tj−N, x

) − gh(tj−N−1, x))τ, 1 ≤ k ≤N,

uhk(x) = Rk−nNuhnN(x) +

k∑

j=nN+1

Rk−j+1(I +

τAxh

2

)Bxj

12

(uhj−N(x) + uhj−N−1(x)

)τ,

nN ≤ k ≤ (n + 1)N,(2.16)

where R = (I + τAxh + (τAx

h)2/2)

−1. The proof of Theorem 2.2 is based on the formulas (2.16),

on the convergence theorem, on the difference schemes in Cτ(Ehα) (see, [38]), on the estimate(2.13), and on the equivalence of the norms as in Theorem 2.1.

Finally, the numerical methods are given in the following section for the solution ofdelay parabolic differential equation with the Dirichlet condition. The method is illustratedby numerical examples.

2.2. Numerical Results

We consider the initial-boundary-value problem

∂u(t, x)∂t

− ∂2u(t, x)∂x2

+ (0.1)∂2u(t − 1, x)

∂x2= 0, t > 0, 0 < x < π,

u(t, x) = e−t sinx, −1 ≤ t ≤ 0, 0 ≤ x ≤ π,u(t, 0) = u(t, π) = 0, t ≥ 0,

(2.17)

for the delay parabolic differential equation.


The exact solution of this problem for t ∈ [n − 1, n], n = 0, 1, 2, . . . , x ∈ [0, π] is

u(t, x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

e−t sinx, −1 ≤ t ≤ 0,e−t{1 + (0.1) et} sinx, 0 ≤ t ≤ 1,

e−t{1 + (0.1) et +

[(0.1)e(t − 1)]2

2!

}sinx, 1 ≤ t ≤ 2,

...

e−t{1 + (0.1) et +

[(0.1)e(t − 1)]2

2!+ · · · + [(0.1)e(t − n)](n+1)

(n + 1)!

}sinx, n ≤ t ≤ n + 1,

...(2.18)

For the approximate solution of delay parabolic equation (2.17), consider the set of grid points

[−1,∞]τ × [0, π]h = {(tk, xn) : tk = kτ, −N ≤ k ≤ ∞, xn = nh, 0 ≤ n ≤M, Mh = π}.(2.19)

Using difference scheme accurate to first order for the approximate solutions of the initial-boundary-value problem for the delay parabolic equation (2.17), we get the following systemof equations:

ukn − uk−1n

τ− ukn+1 − 2ukn + u

kn−1

h2+ (0.1)

uk−Nn+1 − 2uk−Nn + uk−Nn−1h2

= 0,

mN + 1 ≤ k ≤ (m + 1)N, m = 0, 1, . . . , 1 ≤ n ≤M − 1,

ukn = e−tk sinxn, −N ≤ k ≤ 0, 0 ≤ n ≤M,

uk0 = ukM = 0, k ≥ 0.

(2.20)

In this first step, applying difference scheme accurate to first order, we obtain a system ofequations in matrix form

AUmn+1 + BU

mn + CUm

n−1 = Rϕmn , 1 ≤ n ≤M − 1, m = 0, 1, . . . ,

Um0 = 0, Um

M = 0,(2.21)


where A, B, C are (N + 1) × (N + 1) matrices defined by

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 · · · 0 0 0 00 a 0 0 0 · · · 0 0 0 00 0 a 0 0 · · · 0 0 0 00 0 0 a 0 · · · 0 0 0 00 0 0 0 a · · · 0 0 0 0· · · · · · · · · · · ·0 0 0 0 0 · · · a 0 0 00 0 0 0 0 · · · 0 a 0 00 0 0 0 0 · · · 0 0 a 00 0 0 0 0 · · · 0 0 0 a

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 · · · 0 · · · 0 · · · 0 0b c 0 · · · 0 · · · 0 · · · 0 00 b c · · · 0 · · · 0 · · · 0 00 0 b · · · 0 · · · 0 · · · 0 00 0 0 · · · 0 · · · 0 · · · 0 00 0 0 · · · 0 · · · 0 · · · 0 00 0 0 · · · 0 · · · 0 · · · 0 0· · · · · · · · · · · · · · · ·0 0 0 · · · 0 · · · 0 · · · 0 00 0 0 · · · 0 · · · 0 · · · c 00 0 0 · · · 0 · · · 0 · · · b c

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

.

(2.22)

C = A, R is (N + 1) × (N + 1) identity matrix and ϕmn ,Ums are (N + 1) × 1 column vectors as

ϕmn =

⎡⎢⎢⎢⎢⎢⎢⎣

ϕmNn

ϕmN+1n...

ϕ(m+1)Nn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×(1)

, Ums =

⎡⎢⎢⎢⎢⎢⎢⎣

UmNs

UmN+1s...

U(m+1)Ns

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×(1)

for s = n ± 1, n, (2.23)

where umNn is given for anym = 0, 1, . . .,

ϕkn = −(0.1)uk−Nn+1 − 2uk−Nn + uk−Nn−1

h2,

mN + 1 ≤ k ≤ (m + 1)N, m = 0, 1, . . . , 1 ≤ n ≤M − 1,

ukn = e−tk sinxn, −N ≤ k ≤ 0.

(2.24)

Here, we denote

a = − 1h2, b = − 1

τ, c =

1τ+

2h2. (2.25)


So, we have second-order difference equation (2.21) with matrix coefficients. To solve thisdifference equation, we have applied a procedure of modified Gauss elimination method.Hence, we obtain a solution of the matrix equation in the following form:

Umj = αj+1Um

j+1 + βmj+1, j =M − 1, . . . , 2, 1,

UmM = 0,

(2.26)

where αj (j = 1, . . . ,M) are (N + 1) × (N + 1) square matrices and βmj (j = 1, . . . ,M) are(N + 1) × 1 column matrices defined by

αj+1 = −(B + Cαj)−1

A,

βmj+1 =(B + Cαj

)−1(Rϕmj − Cβj

),

(2.27)

where j = 1, . . . ,M − 1, α1 is the (N + 1) × (N + 1) zero matrix, and βm1 is the (N + 1) × 1 zeromatrix.

Second, using the second order of accuracy difference scheme for the approximatesolutions of problem (2.17) and applying formulae

2u(0) − 5u(h) + 4u(2h) − u(3h)h2

− u′′(0) = O(h2),

2u(1) − 5u(1 − h) + 4u(1 − 2h) − u(1 − 3h)h2

− u′′(1) = O(h2),

(2.28)

we obtain the following system of equations:

ukn − uk−1n

τ− ukn+1 − 2ukn + u

kn−1

h2+τ

2

(ukn+2 − 4ukn+1 + 6ukn − 4ukn−1 + u

kn−2

h4

)

+(0.1)

{uk−Nn+1 − 2uk−Nn + uk−Nn−1

2h2+uk−1−Nn+1 − 2uk−1−Nn + uk−1−Nn−1

2h2

−τ2

[uk−Nn+2 − 4uk−Nn+1 + 6uk−Nn − 4uk−Nn−1 + uk−Nn−2

2h4

+uk−1−Nn+2 − 4uk−1−Nn+1 + 6uk−1−Nn − 4uk−1−Nn−1 + uk−1−Nn−2

2h4

]}= 0

mN + 1 ≤ k ≤ (m + 1)N, m = 0, 1, . . . , 2 ≤ n ≤M − 2,

ukn = e−tk sinxn, −N ≤ k ≤ 0, 0 ≤ n ≤M,

uk1 =45uk2 −

15uk3 , k ≥ 0,

ukM−1 =45ukM−2 −

15ukM−3, k ≥ 0,

uk0 = ukM = 0, k ≥ 0.

(2.29)


In the second step, we apply second-order difference scheme to get the system of linear equa-tions in matrix form

AUmn+2 + BU

mn+1 + CU

mn +DUm

n−1 + EUmn−2 = Rϕ

mn ,

m = 0, 1, . . . , 2 ≤ n ≤M − 2,

Um0 = 0, Um

M = 0,

Um1 =

45Um

2 − 15Um

3 ,

UmM−1 =

45UmM−2 −

15UmM−3,

(2.30)

where A, B, C are (N + 1) × (N + 1) matrices defined by

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 · · · 0 0 0 00 x 0 0 0 · · · 0 0 0 00 0 x 0 0 · · · 0 0 0 00 0 0 x 0 · · · 0 0 0 00 0 0 0 x · · · 0 0 0 0· · · · · · · · · · · ·0 0 0 0 0 · · · x 0 0 00 0 0 0 0 · · · 0 x 0 00 0 0 0 0 · · · 0 0 x 00 0 0 0 0 · · · 0 0 0 x

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 · · · 0 0 0 00 y 0 0 0 · · · 0 0 0 00 0 y 0 0 · · · 0 0 0 00 0 0 y 0 · · · 0 0 0 00 0 0 0 y · · · 0 0 0 0· · · · · · · · · · · ·0 0 0 0 0 · · · y 0 0 00 0 0 0 0 · · · 0 y 0 00 0 0 0 0 · · · 0 0 y 00 0 0 0 0 · · · 0 0 0 y

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 · · · 0 0 0 0z t 0 0 0 · · · 0 0 0 00 z t 0 0 · · · 0 0 0 00 0 z t 0 · · · 0 0 0 00 0 0 z t · · · 0 0 0 0· · · · · · · · · · · ·0 0 0 0 0 · · · z t 0 00 0 0 0 0 · · · 0 z t 00 0 0 0 0 · · · 0 0 z t

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

(2.31)


B = D, E = A, R is (N + 1) × (N + 1) identity matrix, and ϕmn , Ums are (N + 1) × 1 column

vectors as

ϕmn =

⎡⎢⎢⎢⎢⎢⎢⎣

ϕmNn

ϕmN+1n...

ϕ(m+1)Nn

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×(1)

, Ums =

⎡⎢⎢⎢⎢⎢⎢⎣

UmNs

UmN+1s...

U(m+1)Ns

⎤⎥⎥⎥⎥⎥⎥⎦

(N+1)×(1)

for s = n ± 1, n ± 2, n, (2.32)

where umNn is given for anym = 0, 1, . . .,

ϕkn = −(0.1){uk−Nn+1 − 2uk−Nn + uk−Nn−1

2h2+uk−1−Nn+1 − 2uk−1−Nn + uk−1−Nn−1

2h2

−τ2

[uk−Nn+2 − 4uk−Nn+1 + 6uk−Nn − 4uk−Nn−1 + uk−Nn−2

2h4

+uk−1−Nn+2 − 4uk−1−Nn+1 + 6uk−1−Nn − 4uk−1−Nn−1 + uk−1−Nn−2

2h4

]},

mN + 1 ≤ k ≤ (m + 1)N, m = 0, 1, . . . , 1 ≤ n ≤M − 1,

ukn = e−tk sinxn, −N ≤ k ≤ 0.

(2.33)

Here, we denote

x =τ

2h4, y = − 1

h2− 2τh4,

z = − 1τ, t =

1τ+

2h2

+3τh4,

(2.34)

Hence, we have second-order difference equation (2.30) with matrix coefficients. For thesolution of this matrix equation, we use the modified Gauss elimination method. We seeka solution of the matrix equation by the following form:

Umj = αj+1Um

j+1 + βj+1Umj+2 + γ

mj+1, j =M − 2, . . . , 2, 1, 0

UmM = 0,

UmM−1 =

[(βM−2 + 5I

) − 4(I − αM−2)αM−1]−1[4(I − αM−2)γmM−1 − γmM−2

],

(2.35)

where αj (j = 2, . . . ,M− 2) and βj (j = 2, . . . ,M− 2) are (N + 1)× (N + 1) square matrices andγmj (j = 2, . . . ,M − 2) are column matrices defined by

αj+1 = −(C +Dαj + Eβj−1 + Eαj−1αj)−1(

B +Dβj + Eαj−1βj),

βj+1 = −(C +Dαj + Eβj−1 + Eαj−1αj)−1(A),

γmj+1 = −(C +Dαj + Eβj−1 + Eαj−1αj)−1(

Rϕmj −Dγmj − Eαj−1γmj − Eγmj−1),

(2.36)


Table 1: Comparison of the errors of different difference schemes in t ∈ [0, 1].

Method N =M = 20 N =M = 40 N =M = 80Difference scheme (2.20) 0.00599088 0.00286092 0.00139629Difference scheme (2.29) 0.00076265 0.00020317 0.00005148







where j = 2, . . . ,M − 2, α1 is (N + 1) × (N + 1) zero matrix, and β1 is (N + 1) × (N + 1) zeromatrix, γm1 and γm2 are (N + 1) × 1 zero matrices.

We give the results of the numerical analysis. The numerical solutions are recordedfor different values ofN andM and ukn represent the numerical solutions of these differenceschemes at (tk, xn). Tables 1, 2, 3, and 4 are constructed forN = M = 20, 40, 80 in t ∈ [0, 1],t ∈ [1, 2], t ∈ [2, 3], t ∈ [3, 4], respectively, and the error is computed by the following formula:

ENM = max−N≤k≤N1≤n≤M−1

∣∣∣u(tk, xn) − ukn∣∣∣. (2.37)

Thus, by using the second order of accuracy difference scheme, the accuracy of solutionincreases faster than the first order of accuracy difference scheme.

3. Homotopy Analysis Method

In this section, we consider homotopy analysis method for the solution of problem (1.1). Westudy the initial-boundary-value problem for the delay parabolic equation (1.1). To illustratethe basic idea of homotopy analysis method (HAM) developed by Liao (see, e.g., [29–35]),the following differential equation is considered:

N[u(t, x)] = f(t, x), (3.1)


whereN is a linear operator for problem (1.1), t and x denote independent variables, u(t, x) isan unknown function, and f(t, x) is a known analytical function. Liao constructs the so-calledzero-order deformation equation

(1 − q)L[φ(t, x; q) − u0(t, x)

]= q�

{N[φ(t, x; q

)] − f(t, x)}, (3.2)

where q ∈ [0, 1] is an embedding parameter, � is a nonzero auxiliary parameter, L is anauxiliary linear operator, u0(t, x) is an initial guess of u(t, x), and φ(t, x; q) is an unknownfunction. When q = 0 and q = 1, it holds

φ(t, x; 0) = u0(t, x), φ(t, x; 1) = u(t, x), (3.3)

respectively. As q increases from 0 to 1, the solution φ(t, x; q) varies from the initial guessu0(t, x) to the solution u(t, x). Expanding φ(t, x; q) in Taylor series with respect to q, we get

φ(t, x; q

)= u0(t, x) +

+∞∑

m=1

um(t, x)qm, (3.4)

where

um(t, x) =1m!

∂mφ(t, x; q

)

∂qm

∣∣∣∣∣q=0

, (3.5)

when the initial guess u0(t, x), the auxiliary linear operator L and the auxiliary parameter �

are chosen properly, the series (3.4) converges at q = 1. We get

u(t, x) = u0(t, x) ++∞∑

m=1

um(t, x). (3.6)

Then define the vectors

−→un = {u0(t, x), u1(t, x), . . . , un(t, x)}. (3.7)

Differentiating the zero-order deformation equation (3.2)m times with respect to the embed-ding parameter q and dividing them bym!, we obtain themth-order deformation equation

L[um(t, x) − χmum−1(t, x)

]= �m(�um−1), (3.8)

where

m(�um−1) =1

(m − 1)!∂m−1(N

[φ(t, x; q

)] − f(t, x))

∂qm−1

∣∣∣∣∣q=0

,

χm =

{0, m ≤ 1,1, m > 1.

(3.9)


with the initial condition

um(0, x) = 0, m ≥ 1. (3.10)

High-order deformation equation (3.8) is governed by the linear operator L.m(�um−1) can berepresented by u1(t, x), um(t, x), u2(t, x), . . . , um−1(t, x) and high-order deformation equationcan be solved consecutively. TheNth-order approximation of u(t, x) is given by

u(t, x) ≈ u0(t, x) +N∑

m=1

um(t, x). (3.11)

3.1. Homotopy Analysis Solution

For the approximate solution of the delay parabolic differential equation with the Dirichletcondition, we consider the delay parabolic equation (2.17) and rewrite the equation for t ∈[0, 1] in the following form:

∂u(t, x)∂t

− ∂2u(t, x)∂x2

= (0.1) e−(t−1) sinx, 0 < t ≤ 1, 0 < x < π,

u(0, x) = sinx, 0 ≤ x ≤ πu(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1.

(3.12)

To solve the initial-boundary-value problem (3.12) by means of HAM, we choose the initialapproximation

u0(t, x) = sinx, (3.13)

and the linear operator

L[φ(t, x; q

)]=∂φ(t, x; q

)

∂t, (3.14)

with the property

L[c] = 0, (3.15)

where c is constant of integration. From (3.12), we define a linear operator as

N[φ(t, x; q

)]=∂φ(t, x; q

)

∂t− ∂2φ

(t, x; q

)

∂x2. (3.16)

Firstly, we construct the zero-order deformation equation

(1 − q)L[φ(t, x; q) − u0(t, x)

]= q�

{N[φ(t, x; q

)] − f(t, x)}, (3.17)


when q = 0 and q = 1,

φ(t, x; 0) = u0(t, x) = u(0, x), φ(t, x; 1) = u(t, x). (3.18)

Then, we getmth-order deformation equations (3.8) form ≥ 1 with the initial conditions

um(0, x) = 0, (3.19)

where

m(�um−1) =∂um−1(t, x)

∂t− ∂2um−1(t, x)

∂x2− (1 − χm

)(0.1)e−(t−1) sinx. (3.20)

The solution of themth-order deformation equations (3.20) form ≥ 1 is

um(t, x) = χmum−1(t, x) + �L−1[m(�um−1)]. (3.21)

From (3.12) and (3.21), we obtain

u0(t, x) = sinx,

u1(t, x) = −�(0.1)e∞∑

k=1

(−1)k+1tkk!

sinx + �t sinx,

u2(t, x) = �(� + 1)t sinx − �(� + 1)∞∑

k=2

(−1)ktk−1(k − 1)!

sinx

+ �2(0.1)e

∞∑

k=2

(−1)k+1tkk!

sinx + �2 t

2

2!sinx,

u3(t, x) = �(� + 1)t sinx + 3�2(� + 1)t2

2!sinx − �(� + 1)2

∞∑

k=3

(−1)k+1tk−2(k − 2)!

sinx

− 2�2(� + 1)∞∑

k=3

(−1)k+1tk−1(k − 1)!

sinx − �3(0.1)e

∞∑

k=3

(−1)k+1tkk!

sinx + �3 t

3

3!sinx,

...

un(t, x) = f1(� + 1, t, x) + (−1)n�n(0.1)e∞∑

k=n

(−1)k+1tkk!

sinx + �n t

n

n!sinx,

...

(3.22)


and so on. Then for � = −1, we get

u0(t, x) = sinx,

u1(t, x) = (0.1)e∞∑

k=1

(−1)k+1tkk!

sinx − t sinx,

u2(t, x) = (0.1)e∞∑

k=2

(−1)k+1tkk!

sinx +t2

2!sinx,

u3(t, x) = (0.1)e∞∑

k=3

(−1)k+1tkk!

sinx − t3

3!sinx,

...

un(t, x) = (0.1)e∞∑

k=n

(−1)k+1tkk!

+(−1)ntnn!

sinx,

...

(3.23)

and so on.From (3.6), when we take � = −1, the solution of (3.12) can be obtained as

u(t, x) = ((0.1)et + 1)∞∑

k=0

(−1)ktkk!

sinx. (3.24)

Equation (3.24) has the closed form

u(t, x) = ((0.1)et + 1)e−t sinx, (3.25)

which is the exact solution of (3.12).Second, we consider the solution of the delay parabolic equation (2.17) for t ∈ [1, 2]

and rewrite this equation in the following form:

∂u(t, x)∂t

− ∂2u(t, x)∂x2

= (0.1)e−(t−1)[(0.1)e(t − 1) + 1] sinx, 1 < t ≤ 2, 0 < x < π,

u(1, x) = e−1[(0.1)e + 1] sinx, 0 ≤ x ≤ πu(t, 0) = u(t, π) = 0, 1 ≤ t ≤ 2.

(3.26)

Now, we choose the initial approximation

u0(t, x) = e−1[(0.1)e + 1] sinx. (3.27)

We take the linear operator (3.14)with the property (3.15), and we define the operator (3.16)from (3.26).


Firstly, we construct the zero-order deformation equation (3.2) and then, we obtainmth-order deformation equations (3.8) form ≥ 1 with the initial conditions

um(1, x) = 0, (3.28)

where

m(�um−1) =∂um−1(t, x)

∂t− ∂2um−1(t, x)

∂x2− (1 − χm

)(0.1)e−(t−1)[(0.1)e(t − 1) + 1] sinx. (3.29)


um(t, x) = χmum−1(t, x) + �L−1[m(�um−1)]. (3.30)

From (3.26) and themth-order deformation equations (3.30), we get

u0(t, x) = e−1[(0.1)e + 1] sinx,

u1(t, x) = �e−1[(0.1)e + 1] sinx(t − 1)

− �(0.1)∞∑

k=1

(−1)k+1(t − 1)k

k!− �(0.1)2e

∞∑

k=1

(−1)k+1(t − 1)k+1

(k + 1)(k − 1)!sinx,

u2(t, x) = �(� + 1)e−1[(0.1)e + 1](t − 1) sinx

− �(� + 1)(0.1)∞∑

k=2

(−1)k(t − 1)k−1

(k − 1)!sinx − �(� + 1)(0.1)2e

∞∑

k=2

(−1)k(t − 1)k

k(k − 2)!sinx

+ �2e−1[(0.1)e + 1] sinx

(t − 1)2

2!+ �

2(0.1)∞∑

k=2

(−1)k+1(t − 1)k

k!sinx

+ �2(0.1)2e

∞∑

k=2

(−1)k+1(t − 1)k+1

(k + 1)k(k − 2)!sinx,

u3(t, x) = �(� + 1)2e−1[(0.1)e + 1] sinx(t − 1)

+ 2�2(� + 1)e−1[(0.1)e + 1] sinx(t − 1)2

2!

+ �(� + 1)2(0.1)∞∑

k=3

(−1)k+1(t − 1)k−2

(k − 2)!sinx − �(� + 1)2(0.1)2e

∞∑

k=3

(−1)k+1(t − 1)k

(k − 1)(k − 3)!sinx

− 2�2(� + 1)(0.1)∞∑

k=3

(−1)k(t − 1)k−1

(k − 1)!sinx − �(� + 1)(0.1)2e

∞∑

k=3

(−1)k(t − 1)k

k(k − 1)(k − 3)!sinx

+ �2(� + 1)(0.1)2e

∞∑

k=3

(−1)k(t − 1)k

k(k − 1)(k − 2)!sinx + �

3e−1[(0.1)e + 1] sinx(t − 1)3

3!


− �3(0.1)

∞∑

k=3

(−1)k+1(t − 1)k

k!sinx − �

3(0.1)2e∞∑

k=3

(−1)k+1(t − 1)k+1

(k + 1)k(k − 1)(k − 3)!sinx,

...

un(t, x) = f2(� + 1, t, x) + �ne−1[(0.1)e + 1] sinx

(t − 1)n

n!

− �n(0.1)

∞∑

k=n

(−1)k+1(t − 1)k

k!sinx

+ (−1)n�n(0.1)2e∞∑

k=n

(−1)k+1(t − 1)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx,

...

(3.31)

and so on. When we choose � = −1, we obtain

u0(t, x) = e−1[(0.1)e + 1] sinx,

u1(t, x) = −e−1[(0.1)e + 1] sinx(t − 1)

+ (0.1)∞∑

k=1

(−1)k+1(t − 1)k

k!sinx + (0.1)2e

∞∑

k=1

(−1)k+1(t − 1)k+1

(k + 1)(k − 1)!sinx,

u2(t, x) = e−1[(0.1)e + 1] sinx(t − 1)2

2!

+ (0.1)∞∑

k=2

(−1)k+1(t − 1)k

k!sinx + (0.1)2e

∞∑

k=2

(−1)k+1(t − 1)k+1

(k + 1)k(k − 2)!sinx,

u3(t, x) = −e−1[(0.1)e + 1] sinx(t − 1)3

3!

+ (0.1)∞∑

k=3

(−1)k+1(t − 1)k

k!sinx + (0.1)2e

∞∑

k=3

(−1)k+1(t − 1)k+1

(k + 1)k(k − 1)(k − 3)!sinx,

...

un(t, x) = (−1)ne−1[(0.1)e + 1] sinx(t − 1)n

n!

+ (0.1)∞∑

k=n

(−1)k+1(t − 1)k

k!sinx

+ (0.1)2e∞∑

k=n

(−1)k+1(t − 1)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx,

...

(3.32)

and so on.


From (3.6), the solution of (3.26) for � = −1 can be obtained as

u(t, x) =

([(0.1)e(t − 1)]2

2!+ (0.1)et + 1

) ∞∑

k=0

(−1)ktkk!

sinx. (3.33)

Equation (3.33) has the closed form

u(t, x) =

([(0.1)e(t − 1)]2

2!+ (0.1)et + 1

)e−t sinx, (3.34)

which is the exact solution of the (3.26).Now, we consider the solution of the delay parabolic equation (2.17) for t ∈ [2, 3] and

rewrite the equation in the following form:

∂u(t, x)∂t

− ∂2u(t, x)∂x2

= (0.1)e−(t−1)[[(0.1)e(t − 2)]2

2!+ (0.1)e(t − 1) + 1

]sinx,

2 < t ≤ 3, 0 < x < π,

u(2, x) = e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx, 0 ≤ x ≤ π

u(t, 0) = u(t, π) = 0, 2 ≤ t ≤ 3.

(3.35)

The initial approximation is

u0(t, x) = e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx. (3.36)

We take the linear operator (3.14) with the property (3.15). From parabolic equation(3.35), we define a linear operator (3.16) and obtain the zero-order deformation equation(3.2). Thus, we get the mth-order deformation equations (3.8) for m ≥ 1 with the initialconditions

um(2, x) = 0, (3.37)

where

m(�um−1) =∂um−1(t, x)

∂t− ∂2um−1(t, x)

∂x2− (1 − χm

)

× (0.1)e−(t−1)[[(0.1)e(t − 2)]2

2!+ (0.1)e(t − 1) + 1

]sinx.

(3.38)


um(t, x) = χmum−1(t, x) + �L−1[m(�um−1)]. (3.39)


From (3.35) and (3.39), we obtain

u0(t, x) = e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx,

u1(t, x) = �e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx(t − 2)

− �e−1(0.1)∞∑

k=1

(−1)k+1(t − 2)k

k!sinx − �(0.1)2

∞∑

k=1

(−1)k+1(t − 2)k+1

(k + 1)(k − 1)!sinx

− �(0.1)2∞∑

k=1

(−1)k+1(t − 2)k

k!sinx − �e

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 2)k+2

(k + 2)(k − 1)!sinx,

u2(t, x) = �(� + 1)e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx(t − 2)

− �(� + 1)e−1(0.1)∞∑

k=2

(−1)k(t − 2)k−1

(k − 1)!sinx − �(� + 1)(0.1)2

∞∑

k=2

(−1)k(t − 2)k−1

(k − 1)!sinx

− �(� + 1)(0.1)2∞∑

k=2

(−1)k(t − 2)k

k(k − 2)!sinx − �(� + 1)

∞∑

k=2

(−1)k(t − 2)k+1

(k + 1)(k − 2)!sinx

+ �2e−2

[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)2

2!

+ �2e−1(0.1)

∞∑

k=2

(−1)k+1(t − 2)k

k!sinx + �

2(0.1)2∞∑

k=2

(−1)k+1(t − 2)k+1

(k + 1)k(k − 2)!sinx

+ �2(0.1)2

∞∑

k=2

(−1)k+1(t − 2)k

k!sinx + �

2e(0.1)3

2!

∞∑

k=2

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)(k − 2)!sinx,

u3(t, x) = �(� + 1)2e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx(t − 2)

+ 2�2(� + 1)e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)2

2!

− �(� + 1)2(0.1)e−1∞∑

k=3

(−1)k+1(t − 2)k−2

(k − 2)!sinx

− �(� + 1)2(0.1)2∞∑

k=3

(−1)k+1(t − 2)k−2

(k − 2)!sinx

− 3�(� + 1)(0.1)2∞∑

k=3

(−1)k+1(t − 2)k−1

(k − 1)!sinx


− �2(� + 1)(0.1)2

∞∑

k=3

(−1)k+1(t − 2)k−1

(k − 1)!sinx

− 2�2(� + 1)e−1(0.1)∞∑

k=3

(−1)k+1(t − 2)k−1

(k − 1)!sinx

− �(� + 1)2(0.1)3

2!e

∞∑

k=3

(−1)k+1(t − 2)k

k(k − 3)!sinx

− 2�2(� + 1)(0.1)2∞∑

k=3

(−1)k+1(t − 2)k

k(k − 1)(k − 3)!sinx

− 2�2(� + 1)(0.1)3

2!e

∞∑

k=3

(−1)k+1(t − 2)k+1

(k + 1)k(k − 3)!sinx

+ �3e−2

[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)3

3!

− �3e−1(0.1)

∞∑

k=3

(−1)k+1(t − 2)k

k!sinx − �

3(0.1)2∞∑

k=3

(−1)k+1(t − 2)k+1

(k + 1)k(k − 1)(k − 3)!sinx

− �3(0.1)2

∞∑

k=3

(−1)k+1(t − 2)k

k!sinx − �

3e(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)k(k − 3)!sinx,

...

un(t, x) = f3(� + 1, t, x) + �ne−2

[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)n

n!

+ (−1)n�ne−1(0.1)∞∑

k=n

(−1)k+1(t − 2)k

k!sinx

+ (−1)n�n(0.1)2∞∑

k=n

(−1)k+1(t − 2)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx

+ (−1)n�n(0.1)2∞∑

k=n

(−1)k+1(t − 2)k

k!sinx

+ (−1)n�ne (0.1)3

2!

∞∑

k=n

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)k · · · (k − (n − 3))(k − n)! sinx,

...

(3.40)


and so on. When we choose � = −1, we obtain

u0(t, x) = e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx,

u1(t, x) = −e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx(t − 2)

+ e−1(0.1)∞∑

k=1

(−1)k+1(t − 2)k

k!sinx + (0.1)2

∞∑

k=1

(−1)k+1(t − 2)k+1

(k + 1)(k − 1)!sinx

+ (0.1)2∞∑

k=1

(−1)k+1(t − 2)k

k!sinx + e

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 2)k+2

(k + 2)(k − 1)!sinx,

u2(t, x) = e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)2

2!

+ e−1(0.1)∞∑

k=2

(−1)k+1(t − 2)k

k!sinx + (0.1)2

∞∑

k=2

(−1)k+1(t − 2)k+1

(k + 1)k(k − 2)!sinx

+ (0.1)2∞∑

k=2

(−1)k+1(t − 2)k

k!sinx + e

(0.1)3

2!

∞∑

k=2

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)(k − 2)!sinx,

u3(t, x) = −e−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)3

3!

+ e−1(0.1)∞∑

k=3

(−1)k+1(t − 2)k

k!sinx + (0.1)2

∞∑

k=3

(−1)k+1(t − 2)k+1

(k + 1)k(k − 1)(k − 3)!sinx

+ (0.1)2∞∑

k=3

(−1)k+1(t − 2)k

k!sinx + e

(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)k(k − 3)!sinx,

...

un(t, x) = (−1)ne−2[[(0.1)e]2

2!+ 2(0.1)e + 1

]sinx

(t − 2)n

n!

+ e−1(0.1)∞∑

k=n

(−1)k+1(t − 2)k

k!sinx

+ (0.1)2∞∑

k=n

(−1)k+1(t − 2)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx

+ (0.1)2∞∑

k=n

(−1)k+1(t − 2)k

k!sinx

+ e(0.1)3

2!

∞∑

k=n

(−1)k+1(t − 2)k+2

(k + 2)(k + 1)k · · · (k − (n − 3))(k − n)! sinx,

...(3.41)

and so on.


From (3.6), the solution of (3.35) for � = −1 is

u(t, x) =

([(0.1)e(t − 2)]3

3!+[(0.1)e(t − 1)]2

2!+ (0.1)et + 1

) ∞∑

k=0

(−1)ktkk!

sinx. (3.42)

This series has the closed form

u(t, x) =

([(0.1)e(t − 2)]3

3!+[(0.1)e(t − 1)]2

2!+ (0.1)et + 1

)e−t sinx, (3.43)

which is the exact solution of the (3.35).Finally, we consider the solution of the delay parabolic equation (2.17) for t ∈ [3, 4]

and rewrite this equation in the following form:

∂u(t, x)∂t

− ∂2u(t, x)∂x2

= (0.1)e−(t−1)[[(0.1)e(t − 3)]3

3!+[(0.1)e(t − 2)]2

2!+ (0.1)e(t − 1) + 1

]sinx,

3 < t ≤ 4, 0 < x < π,

u(3, x) = e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx, 0 ≤ x ≤ π,

u(t, 0) = u(t, π) = 0, 3 ≤ t ≤ 4.(3.44)

We take the initial approximation

u0(t, x) = e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx (3.45)

and the linear operator (3.14)with the property (3.15). From (3.44), we define linear operator(3.16).

We construct the zero-order deformation equation (3.2) and the mth-order deforma-tion equations (3.8) form ≥ 1 with the initial conditions

um(3, x) = 0, (3.46)

where

m(�um−1) =∂um−1(t, x)

∂t− ∂2um−1(t, x)

∂x2− (1 − χm

)

× (0.1)e−(t−1)(

[(0.1)e(t − 3)]3

3!+[(0.1)e(t − 2)]2

2!+ (0.1)e(t − 1) + 1

)sinx.

(3.47)



um(t, x) = χmum−1(t, x) + �L−1[m(�um−1)]. (3.48)

From (3.44) and (3.48), we get

u0(t, x) = e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx,

u1(t, x) = �e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx(t − 3)

− �e−2(0.1)∞∑

k=1

(−1)k+1(t − 3)k

k!sinx − �e−1(0.1)2

∞∑

k=1

(−1)k+1(t − 3)k+1

(k + 1)(k − 1)!sinx

− 2�e−1(0.1)2∞∑

k=1

(−1)k+1(t − 3)k

k!sinx − �

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 3)k+2

(k + 2)(k − 1)!sinx

− �(0.1)3∞∑

k=1

(−1)k+1(t − 3)k+1

(k + 1)(k − 1)!sinx − �

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 3)k

k!sinx

− �e(0.1)4

3!

∞∑

k=1

(−1)k+1(t − 3)k+3

(k + 3)(k − 1)!sinx,

u2(t, x) = �(� + 1)e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx(t − 3)

− �(� + 1)(0.1)∞∑

k=2

(−1)k(t − 3)k−1

(k − 2)!sinx − �(� + 1)(0.1)2e−1

∞∑

k=2

(−1)k(t − 3)k

k!sinx

− 2�(� + 1)(0.1)2e−1∞∑

k=2

(−1)k(t − 3)k−1

(k − 1)!sinx

− �(� + 1)(0.1)3

2!

∞∑

k=2

(−1)k(t − 3)k−1

(k − 1)!sinx

− �(� + 1)(0.1)3∞∑

k=2

(−1)k(t − 3)k

k(k − 2)!sinx − �(� + 1)(0.1)3

∞∑

k=2

(−1)k(t − 3)k+1

(k + 1)(k − 2)!sinx

− �(� + 1)(0.1)4

3!e

∞∑

k=2

(−1)k(t − 3)k+2

(k + 2)(k − 2)!sinx

+ �2e−3

[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)2

2!

+ �2e−2(0.1)

∞∑

k=2

(−1)k+1(t − 3)k

k!sinx + �

2e−1(0.1)2∞∑

k=2

(−1)k+1(t − 3)k+1

(k + 1)k(k − 2)!sinx


+ 2�2e−1(0.1)2∞∑

k=2

(−1)k+1(t − 3)k

k!sinx + �

2 (0.1)3

2!

∞∑

k=2

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)(k − 2)!sinx

+ �2(0.1)3

∞∑

k=2

(−1)k+1(t − 3)k+1

(k + 1)k(k − 2)!sinx + �

2 (0.1)3

2!

∞∑

k=2

(−1)k+1(t − 3)k

k!sinx

+ �2e

(0.1)4

3!

∞∑

k=2

(−1)k+1(t − 3)k+3

(k + 3)(k + 2)(k − 2)!sinx,

u3(t, x) = �(� + 1)2e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx(t − 3)

− �(� + 1)2(0.1)e−2∞∑

k=3

(−1)k+1(t − 3)k−2

(k − 2)!sinx

− �(� + 1)2(0.1)2e−1∞∑

k=3

(−1)k+1(t − 3)k−1

(k − 1)(k − 3)!sinx

− 2�(� + 1)2(0.1)2e−1∞∑

k=3

(−1)k+1(t − 3)k−2

(k − 2)!sinx

− �(� + 1)2(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k

k(k − 3)!sinx

− �(� + 1)2(0.1)3∞∑

k=3

(−1)k+1(t − 3)k−1

(k − 1)(k − 3)!sinx

− �(� + 1)2(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k−2

(k − 2)!sinx

− �(� + 1)2e(0.1)4

3!

∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)(k − 3)!sinx

+ 2�2(� + 1)e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)2

2!

− 2�2(� + 1)e−2(0.1)∞∑

k=3

(−1)k+1(t − 3)k−1

(k − 1)!sinx

− �2(� + 1)e−1(0.1)2

∞∑

k=3

(−1)k+1(t − 3)k

k(k − 1)(k − 3)!sinx

− 4�2(� + 1)e−1(0.1)2∞∑

k=3

(−1)k+1(t − 3)k−1

(k − 1)!sinx


− 2�2(� + 1)(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)k(k − 3)!sinx

− 2�2(� + 1)(0.1)3∞∑

k=3

(−1)k+1(t − 3)k

k(k − 1)(k − 3)!sinx

− 2�2(� + 1)(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k−1

(k − 1)!sinx

− 2�2(� + 1)e(0.1)4

3!

∞∑

k=3

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)(k − 3)!sinx

+ �3e−3

[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)3

3!

− �3e−2(0.1)

∞∑

k=3

(−1)k+1(t − 3)k

k!sinx − �

3e−1(0.1)2∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1)(k − 3)!sinx

− 2�3e−1(0.1)2∞∑

k=3

(−1)k+1(t − 3)k

k!sinx − �

3 (0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)k(k − 3)!sinx

− �3(0.1)3

∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1)(k − 3)!sinx − �

3 (0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k

k!sinx

− �3e

(0.1)4

3!

∞∑

k=3

(−1)k+1(t − 3)k+3

(k + 3)(k + 2)(k + 1)(k − 3)!sinx,

...

un(t, x) = f4(� + 1, t, x) + �ne−3

[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)n

n!

+ (−1)n�ne−2(0.1)∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+ (−1)n�ne−1(0.1)2∞∑

k=n

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx

+ (−1)n�n2e−1(0.1)2∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+ (−1)n�n (0.1)3

2!

∞∑

k=n

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)k(k − 3) · · · (k − (n − 3))(k − n)! sinx

+ (−1)n�n(0.1)3∞∑

k=n

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx


+ (−1)n�n (0.1)3

2!

∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+ (−1)n�ne (0.1)4

3!

∞∑

k=n

(−1)k+1(t − 3)k+3

(k + 2)(k + 1)k · · · (k − (n − 4))(k − n)! sinx,

...

(3.49)

and so on. For � = −1, we get

u0(t, x) = e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx,

u1(t, x) = −e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx(t − 3)

+ e−2(0.1)∞∑

k=1

(−1)k+1(t − 3)k

k!sinx + e−1(0.1)2

∞∑

k=1

(−1)k+1(t − 3)k+1

(k + 1)(k − 1)!sinx

+ 2e−1(0.1)2∞∑

k=1

(−1)k+1(t − 3)k

k!sinx +

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 3)k+2

(k + 2)(k − 1)!sinx

+ (0.1)3∞∑

k=1

(−1)k+1(t − 3)k+1

(k + 1)(k − 1)!sinx +

(0.1)3

2!

∞∑

k=1

(−1)k+1(t − 3)k

k!sinx

+ e(0.1)4

3!

∞∑

k=1

(−1)k+1(t − 3)k+3

(k + 3)(k − 1)!sinx,

u2(t, x) = e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)2

2!

+ e−2(0.1)∞∑

k=2

(−1)k+1(t − 3)k

k!sinx + e−1(0.1)2

∞∑

k=2

(−1)k+1(t − 3)k+1

(k + 1)k(k − 2)!sinx

+ 2e−1(0.1)2∞∑

k=2

(−1)k+1(t − 3)k

k!sinx +

(0.1)3

2!

∞∑

k=2

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)(k − 2)!sinx

+ (0.1)3∞∑

k=2

(−1)k+1(t − 3)k+1

(k + 1)k(k − 2)!sinx +

(0.1)3

2!

∞∑

k=2

(−1)k+1(t − 3)k

k!sinx

+ e(0.1)4

3!

∞∑

k=2

(−1)k+1(t − 3)k+3

(k + 3)(k + 2)(k − 2)!sinx,

u3(t, x) = −e−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)3

3!

+ e−2(0.1)∞∑

k=3

(−1)k+1(t − 3)k

k!sinx + e−1(0.1)2

∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1)(k − 3)!sinx


+ 2e−1(0.1)2∞∑

k=3

(−1)k+1(t − 3)k

k!sinx +

(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)k(k − 3)!sinx

+ (0.1)3∞∑

k=3

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1)(k − 3)!sinx +

(0.1)3

2!

∞∑

k=3

(−1)k+1(t − 3)k

k!sinx

+ e(0.1)4

3!

∞∑

k=3

(−1)k+1(t − 3)k+3

(k + 3)(k + 2)(k + 1)(k − 3)!sinx,

...

un(t, x) = (−1)ne−3[[(0.1)e]3

3!+[2(0.1)e]2

2!+ 3(0.1)e + 1

]sinx

(t − 3)n

n!

+ e−2(0.1)∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+ e−1(0.1)2∞∑

k=n

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx

+ 2e−1(0.1)2∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+(0.1)3

2!

∞∑

k=n

(−1)k+1(t − 3)k+2

(k + 2)(k + 1)k(k − 3) · · · (k − (n − 3))(k − n)! sinx

+ (0.1)3∞∑

k=n

(−1)k+1(t − 3)k+1

(k + 1)k(k − 1) · · · (k − (n − 2))(k − n)! sinx

+(0.1)3

2!

∞∑

k=n

(−1)k+1(t − 3)k

k!sinx

+ e(0.1)4

3!

∞∑

k=n

(−1)k+1(t − 3)k+3

(k + 2)(k + 1)k · · · (k − (n − 4))(k − n)! sinx,

...

(3.50)

and so on.From (3.6), the solution of (3.44) is obtained as

u(t, x) =

([(0.1)e(t − 3)]4

4!+[(0.1)e(t − 2)]3

3!+[(0.1)e(t − 1)]2

2!+ (0.1)et + 1

) ∞∑

k=0

(−1)ktkk!

sinx.

(3.51)


Table 5: The absolute error at x = π/2 when � = −1.t uexact uapp uerr

0.5 0.68896672324764 0.68718517400418 0.001781549243461.5 0.31617066069370 0.31528438731741 0.000886273376282.5 0.14472535694258 0.14432199982347 0.000403357119103.5 0.06624155329591 0.06619299448877 0.00004855880714

Table 6: Comparison of the absolute error in HAM when � = −1 and the errors of different differenceschemes at (0.5, π/2).

Method uapp uerr

HAM forN = 3 0.68718517400418 0.00178154924346Difference scheme (2.20) forN =M = 48 0.69069236574147 0.00172564249383Difference scheme (2.29) forN =M = 10 0.69067805057383 0.00171132732619


Method uapp uerr


This series has the closed form

u(t, x) =

([(0.1)e(t − 3)]4

4!+[(0.1)e(t − 2)]3

3!+[(0.1)e(t − 1)]2

2!+ (0.1)et + 1

)e−t sinx,

(3.52)

which is the exact solution of (3.44).We give the HAM solutions in t ∈ [0, 1], t ∈ [1, 2], t ∈ [2, 3], t ∈ [3, 4]. We use four

terms for evaluating the approximate solution uapp =∑3

k=0 uk(t, x). According to the �-curveof uxt(0, 0), the solution series is convergent when −1.48 ≤ � ≤ 0.48, −1.41 ≤ � ≤ 2.10, −1.19 ≤� ≤ 0.12, and −1.02 ≤ � ≤ 0, respectively, in t ∈ [0, 1], t ∈ [1, 2], t ∈ [2, 3], t ∈ [3, 4]. Wetake � = −1 to determine how much the approximate solution is accurate and compute theabsolute errors uerr = |uexact −uapp| at the points (0.5, π/2), (1.5, π/2), (2.5, π/2), (3.5, π/2) inTable 5.

4. Conclusion

The numerical solutions of first order of difference scheme (2.20) and second order of differ-ence scheme (2.29) for different values ofN andM and the approximate solutions obtainedby HAM forN = 3 in (3.11) when � = −1 are given at the same points (0.5, π/2), (1.5, π/2),(2.5, π/2), (3.5, π/2) in Tables 6, 7, 8, and 9, respectively. The absolute errors computed showthat, with homotopy analysis method, the results are more accurate for the parabolic delayequation (2.17).

Although HAM seems to be more rapid than finite difference method, the series solu-tions obtained by HAM are convergence only for the regions determined by convergence



Method uapp uerr



Method uapp uerr



Method uapp uerr


Table 11: Comparison of the absolute error in HAM when � = −2.1 and the errors of different differenceschemes at (1.5, π/2).

Method uapp uerr

HAM forN = 3 −0.71640688651704 1.0325774721074Difference scheme (2.20) forN =M = 4 0.26625468031687 0.04991598037683Difference scheme (2.29) forN =M = 4 0.26908021797566 0.04709044271804

Table 12: Comparison of the absolute error in HAM when � = 1.5 and the errors of different differenceschemes at (2.5, π/2).

Method uapp uerr


control parameter �. So, convergence region is limited for HAM. The comparison of twomethods of finite difference and homotopy analysis shows that latter is more rapid andmore accurate in the cases that series solutions are convergence. When we take � out ofthe convergence region determined by � curves, it is shown that finite difference methodis faster and more accurate than HAM. The approximate solutions obtained by HAM fordifferent values of � chosen from out of the convergence region of the series solutions andthe numerical solutions of first and second order of difference schemes (2.20) and (2.29) forN = M = 4 are given in Tables 10, 11, 12, and 13, respectively at the same points (0.5, π/2),(1.5, π/2), (2.5, π/2), (3.5, π/2).


Table 13: Comparison of the absolute error in HAM when � = 2 and the errors of different differenceschemes at (3.5, π/2).

Method uapp uerr

HAM for N = 3 1.33954603011737 1.27330447682146Difference scheme (2.20) forN =M = 4 0.02904127716967 0.03720027612623Difference scheme (2.29) forN =M = 4 0.04842804202239 0.01781351127352

Despite HAM, by finite difference method, we can guarantee the convergence in thewhole domain that (2.17) is defined in. Therefore finite difference method is more efficientthan HAM.

Acknowledgments

This work is supported by Trakya University Scientific Research Projects Unit (Projectnumber: 2010-91). The author is grateful to Professor Allaberen Ashyralyev (Fatih University,Turkey) for his valuable suggestions for the improvement of this paper.

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Research ArticleExistence Results for Solutions ofNonlinear Fractional Differential Equations

Ali Yakar1 and Mehmet Emir Koksal2

1 Department of Statistics, Gaziosmanpasa University, 60250 Tokat, Turkey2 Department of Primary Mathematics Education, Mevlana University, 34528 Konya, Turkey

Correspondence should be addressed to Ali Yakar, [email protected]



Copyright q 2012 A. Yakar and M. E. Koksal. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This paper deals with theoretical and constructive existence results for solutions of nonlinearfractional differential equations using the method of upper and lower solutions which generatea closed set. The existence of solutions for nonlinear fractional differential equations involvingRiemann-Liouville differential operator in a closed set is obtained by utilizing various types ofcoupled upper and lower solutions. Furthermore, these results are extended to the finite systemsof nonlinear fractional differential equations leading to more general results.

1. Introduction

Fractional derivative, introduced around the 17th century, was developed almost until the19th century. Although the introduction of the concept of fractional calculus involvingfractional differentiation and integral is a few centuries old, it was realized only a fewdecades ago that these functional operations play an important role in various fields ofscience and engineering [1–8]. As a reason, since the significance of the fractional calculushas been more clearly perceived, many quality researches have been put forward on thisbranch of mathematical analysis in the literature (see [9–11] and the references therein),and many physical phenomena, chemical processes, biological systems, and so forth havedescribed with fractional derivatives. In this framework, fractional differential equationshave been gaining much interest and attracting the attention of many researchers. Somerecent contributions on fractional differential equations can be seen in [9–20] and thereferences within.

On the other hand, the study for solutions of fractional ordinary and partial differentialequations has received great interest by scientists. Especially, in the last decade, there are


noteworthy works on the analytical and numerical solutions of fractional partial differentialequations (see [21–28] and the references there in).

The attention drawn to basic theoretical concepts like the theory of existence anduniqueness of solutions to nonlinear fractional-order differential equations is obvious.Recently, there have been many paper investigating the existence and uniqueness offractional-order differential equations [29–34].

An interesting and fruitful technique for providing existence results for nonlinearproblems is the method of upper and lower solutions. This technique permits us to establishthe existence results in a closed set, namely, the ordered interval, generated by upper andlower solutions. Thus, in this context, we are concerned with the existence of solutions of thefollowing nonlinear fractional-order initial value problem (IVP):

Dqx(t) = F(t, x), t ∈ J = [t0, T], x(t)(t − t0)1−q∣∣∣t=t0

= x0, (1.1)

where F ∈ C[J × R,R] and Dq is Riemann-Liouville (R-L) fractional derivative of order q,0 < q < 1.

The corresponding Volterra fractional integral equation of (1.1) is defined as

x(t) =x0(t − t0)q−1

Γ(q) +

1Γ(q)∫ t

t0

(t − s)q−1F(s, x(s))ds. (1.2)

In recent years, Lakshmikantham and Vatsala investigated the existence theory andestablished a Peanos type local existence theorem for (1.1) by using integral inequalitiesand perturbation techniques [35]. McRae also studied an important existence result utilizingthe method of upper and lower solutions [36], by means of which, monotone iterative andquasilinearization techniques are developed to fractional differential equations [37–40].

In this paper, we utilize the technique of upper and lower solutions and establish someexistence results in terms of various types of coupled upper and lower solutions. Thenwewillextend this idea to the finite systems of nonlinear fractional differential equations.

The organization of this paper is as follows. In Section 2, we provide necessarybackground. In Section 3, we focus on the existence of solutions of nonlinear fractionaldifferential equations in a sector. Finally, in Section 4, we extend our results to the finitesystems of fractional differential equations.

2. Preliminaries

Now, we present some basic definitions and theorems which are used throughout the paper.

Definition 2.1. Let p = 1 − q, then a function σ(t) is said to be a Cp function if σ ∈ Cp where

Cp[J,R] ={u ∈ C[(t0, T],R] : u(t)(t − t0)p ∈ C[J,R]

}. (2.1)


If we replace F(t, x) in (1.1) by the sum of two functions such that F = f + g wheref, g ∈ C[J × R,R], then problem (1.1) takes the following form:

Dqx(t) = f(t, x) + g(t, x), x(t)(t − t0)1−q∣∣∣t=t0

= x0. (2.2)

We give a variety of possible definitions of upper and lower solutions relative to (2.2).

Definition 2.2. Let v, w ∈ Cp[J,R], p = 1 − q, 0 < q < 1 be locally Holder continuous withexponent λ > q,Dqv, Dqw exist, and f, g ∈ C[J×R,R], then v andw are said to be as follows:

(i) natural upper and lower solutions of (2.2), respectively, if

Dqv ≤ f(t, v) + g(t, v), v0 ≤ x0,

Dqw ≥ f(t,w) + g(t,w), w0 ≥ x0, t ∈ J,(2.3)

(ii) coupled upper and lower solutions of type I of (2.2), respectively, if

Dqv ≤ f(t, v) + g(t,w), v0 ≤ x0,

Dqw ≥ f(t,w) + g(t, v), w0 ≥ x0, t ∈ J,(2.4)

(iii) coupled upper and lower solutions of type II of (2.2), respectively, if

Dqv ≤ f(t,w) + g(t, v), v0 ≤ x0,

Dqw ≥ f(t, v) + g(t,w), w0 ≥ x0, t ∈ J,(2.5)

(iv) coupled upper and lower solutions of type III of (2.2), respectively, if

Dqv ≤ f(t,w) + g(t,w), v0 ≤ x0,

Dqw ≥ f(t, v) + g(t, v), w0 ≥ x0, t ∈ J,(2.6)

where v0 = v(t)(t − t0)1−q|t=t0 and w0 = w(t)(t − t0)1−q|t=t0 .

Lemma 2.3. Let m ∈ Cp([t0, T], R) be locally Holder continuous with exponent λ > q and for anyt1 ∈ (t0, T], and one has

m(t1) = 0, m(t) ≤ 0 for t0 ≤ t ≤ t1, (2.7)

then it follows that

Dqm(t1) ≥ 0. (2.8)

Proof. For the proof, see [16].


Remark 2.4. A dual result for Lemma 2.3 is valid.The explicit solution of the following nonhomogeneous linear fractional differential

equation

Dqx = αx + f(t), x0 = x(t)(t − t0)1−q∣∣∣t=t0

, (2.9)

involving R-L fractional differential operator of order q (0 < q < 1), is necessary for furtherdevelopment of our main results. In (2.9), α is a real number and f ∈ Cp([t0, T],R).

When we apply the method of successive approximations [16] to find the solutionx(t) = x(t, t0, x0) explicitly for the given nonhomogeneous IVP (2.9), we obtain

x(t) = x0(t − t0)q−1Eq,q(α(t − t0)q

)+∫ t

t0

(t − s)q−1Eq,q(α(t − s)q)f(s)ds, t ∈ [t0, T], (2.10)

where Eq,q denotes the two-parameter Mittag-Leffler function and Eq,q(tq) =∑∞

k=0 tqk/Γ(q(k+

1)), q > 0.If f(t) ≡ 0, we get

x(t) = x0(t − t0)q−1Eq,q(α(t − t0)q

), t ∈ [t0, T], (2.11)

as the solution of the corresponding homogeneous IVP for (2.9).We next give a Peano’s type local existence result and then an existence result in a

special closed set generated by upper and lower solutions.

Theorem 2.5. Assume that F ∈ C(R0,Rn) and |F(t, x)| ≤ M on R0 where R0 = {(t, x) : t0 ≤ t ≤

t0 + a and |x − x0(t)| ≤ b} and x0(t) = x0(t − t0)q−1/Γ(q), then IVP (1.1) possesses at least onesolution x(t) on [t0, t0 + α] where α = min{a, ((b/M)Γ(q + 1))1/q}.

For the proof of the theorem, see [35].If the existence of upper and lower solutions w,v such that v(t) ≤ w(t), t ∈ J for IVP

(1.1) is known, the existence of solutions can be proved in the closed set

Ω = [(t, x) : v(t) ≤ x(t) ≤ w(t), t ∈ [t0, T]]. (2.12)

Theorem 2.6. Let v and w ∈ Cp[[t0, T],R] be natural upper and lower solutions of IVP (1.1),which are locally Holder continuous with exponent λ > q such that v(t) ≤ w(t) on J = [t0, T] andf ∈ C(Ω,R), then there exists a solution x(t) of IVP (1.1) satisfying v(t) ≤ x(t) ≤ w(t), t ∈ [t0, T].

For the detailed proof of the above theorem, see [36].

3. Existence Theorems

We are in position to give existence of solutions in the closed set Ω.


Theorem 3.1. Let v and w ∈ Cp[[t0, T],R] be coupled upper and lower solutions of type I of (2.2)such that f, g ∈ C(Ω,R) and v(t) ≤ w(t), t ∈ [t0, T]. Moreover, assume that g(t, x) is nonincreasingin x for each t, then there exists a solution x(t) of (2.2) satisfying v(t) ≤ x(t) ≤ w(t) on [t0, T].

Proof. Let p : [t0, T] × R → R be defined by

p(t, x) = min[w(t),max(x(t), v(t))]. (3.1)

Then f(t, p(t, x)) + g(t, p(t, x)) defines a continuous extension of f + g to [t0, T] × R

which is also bounded since f + g is bounded on Ω. Employing Theorem 2.5, we get thefollowing equation:

Dqx(t) = f(t, p(t, x)

)+ g

(t, p(t, x)

), x(t)(t − t0)1−q

∣∣∣t=t0

= x0, (3.2)

having a solution x(t) on [t0, T]. We wish to prove that v(t) ≤ x(t) ≤ w(t) on [t0, T]. For thispurpose, consider the following equations:

wε(t) = w(t) + εγ(t), vε(t) = v(t) − εγ(t), (3.3)

where γ(t) = (t − t0)q−1Eq,q((t − t0)q) and ε > 0. This implies that

wε(t)(t − t0)1−q∣∣∣t=t0

= w0ε = w(t)(t − t0)1−q

∣∣∣t=t0

+εγ(t)(t − t0)1−q∣∣∣t=t0

,

vε(t)(t − t0)1−q∣∣∣t=t0

= v0ε = v(t)(t − t0)1−q

∣∣∣t=t0

−εγ(t)(t − t0)1−q∣∣∣t=t0

,(3.4)

which gives w0ε = w0 + εγ0, v0

ε = v0 − εγ0 where γ0 > 0. It follows that v0ε < x

0 < w0ε in view

of the upper and lower definitions of w(t) and v(t). It is enough to show that

vε(t) < x(t) < wε(t) on [t0, T], (3.5)

which proves the claim of the theorem as ε → 0. First, suppose that the inequality x(t) <wε(t) on [t0, T] is not true, then there would exist a t1 ∈ (t0, T] such that

x(t1) = wε(t1), x(t) < wε(t) on [t0, t1). (3.6)


Hence, x(t1) > w(t1) ≥ v(t1), therefore p(t1, x(t1)) = w(t1) and v(t1) ≤ p(t1, x(t1)) ≤ w(t1). Ifwe construct m(t) = x(t) − wε(t), we get m(t1) = 0 and m(t) ≤ 0, t0 ≤ t ≤ t1. EmployingLemma 2.3,we obtain Dqm(t1) ≥ 0 which gives a contradiction

f(t1, w(t1)) + g(t1, w(t1)) = f(t1, p(t1, x(t1))

)+ g

(t1, p(t1, x(t1))

)

= Dqx(t1)

≥ Dqwε(t1)

= Dqw(t1) + εγ(t1)

> Dqw(t1)

≥ f(t1, w(t1)) + g(t1, v(t1))

≥ f(t1, w(t1)) + g(t1, w(t1)).

(3.7)

Here, we have used the nonincreasing property of g(t, x) in x for each t and the fact thatγ(t1) > 0.

Similarly, it can be proved that the other side of the inequality (3.5) is valid for t0 ≤ t ≤T . To do this, wemust show that vε(t) < x(t) on [t0, T]. Suppose that it is not true and so thereexists a t1 such that vε(t1) = x(t1) and vε(t) < x(t) for t0 ≤ t < t1, then x(t1) < v(t1) ≤ w(t1)and p(t1, x(t1)) = v(t1). Hence, v(t1) ≤ p(t1, x(t1)) ≤ w(t1). If we put m(t) = vε(t) − x(t), wegetm(t1) = 0 andm(t) ≤ 0, t0 ≤ t ≤ t1. Employing Lemma 2.3, we find Dqm(t1) ≥ 0. Since thenonincreasing property of g(t, x) in x for each t and the fact that γ(t1) > 0, we arrive at thecontradiction

f(t1, v(t1)) + g(t1, v(t1)) = f(t1, p(t1, x(t1))

)+ g

(t1, p(t1, x(t1))

)

= Dqx(t1)

≤ Dqvε(t1)

= Dqv(t1) − εγ(t1)< Dqv(t1)

≤ f(t1, v(t1)) + g(t1, w(t1))

≤ f(t1, v(t1)) + g(t1, v(t1)).

(3.8)

Consequently, we have vε(t) < x(t) < wε(t) on [t0, T], and letting ε → 0, we get v(t) ≤ x(t) ≤w(t) on [t0, T] proving the theorem.

Theorem 3.2. Let v and w ∈ Cp[[t0, T],R] be coupled upper and lower solutions of type II of(2.2) such that f, g ∈ C(Ω,R) and v(t) ≤ w(t), t ∈ [t0, T]. Moreover, assume that f(t, x) isnonincreasing in x for each t, then there exists a solution x(t) of (2.2) satisfying v(t) ≤ x(t) ≤ w(t)on [t0, T].

Proof. Let p : [t0, T] × R → R be defined by

p(t, x) = min[w(t),max(x(t), v(t))]. (3.9)


Then f(t, p(t, x)) + g(t, p(t, x)) defines a continuous extension of f + g to [t0, T] × R

which is also bounded since f + g is bounded on Ω. Therefore, by Theorem 2.5,

Dqx(t) = f(t, p(t, x)

)+ g

(t, p(t, x)

), x(t)(t − t0)1−q

∣∣∣t=t0

= x0 (3.10)

has a solution x(t) on [t0, T]. We intend to show that v(t) ≤ x(t) ≤ w(t) on [t0, T]. For thispurpose, consider the following equations:

wε(t) = w(t) + εγ(t), vε(t) = v(t) − εγ(t), (3.11)

where γ(t) and ε are defined as before. It follows that v0ε < x

0 < w0ε . It is enough to show that

vε(t) < x(t) < wε(t) on [t0, T]. (3.12)

Suppose that it is not true. Thus, there would exist a t1 ∈ (t0, T] such that

x(t1) = wε(t1), vε(t) < x(t) < wε(t) on [t0, t1). (3.13)

Hence, x(t1) > w(t1) ≥ v(t1); therefore, we get p(t1, x(t1)) = w(t1) and v(t1) ≤ p(t1, x(t1)) ≤w(t1). Setting m(t) = x(t) − wε(t), we have m(t1) = 0 and m(t) ≤ 0, t0 ≤ t ≤ t1. EmployingLemma 2.3, we obtain Dqm(t1) ≥ 0 which yields a contradiction

f(t1, w(t1)) + g(t1, w(t1)) = f(t1, p(t1, x(t1))

)+ g

(t1, p(t1, x(t1))

)

= Dqx(t1)

≥ Dqwε(t1)

= Dqw(t1) + εγ(t1)

> Dqw(t1)

≥ f(t1, v(t1)) + g(t1, w(t1))

≥ f(t1, w(t1)) + g(t1, w(t1)).

(3.14)

Here, we have used the nonincreasing property of f(t, x) in x for each t and the factthat γ(t1) > 0. Thus, we get x(t) < wε(t) on [t0, T].

After utilizing the similar procedure, the other case can be proved easily. Conse-quently, we have vε(t) < x(t) < wε(t) on [t0, T], and letting ε → 0, we get v(t) ≤ x(t) ≤ w(t)on [t0, T]which proves the theorem.

Theorem 3.3. Let v and w ∈ Cp[[t0, T],R] be coupled upper and lower solutions of type III of(2.2) such that f, g ∈ C(Ω,R) and v(t) ≤ w(t), t ∈ [t0, T]. Moreover, assume that both f(t, x)and g(t, x) are nonincreasing in x for each t, then there exists a solution x(t) of (2.2) satisfyingv(t) ≤ x(t) ≤ w(t) on [t0, T].

Proof. In a similar manner in previous theorems, the existence of the solution can be proved.Thus, we omit the details.


4. Extensions to the Systems of Differential Equations

We can generalize the result of Theorem 2.6 to finite systems of fractional differential equa-tions. Consider the following fractional differential system:

Dqx(t) = F(t, x), x(t)(t − t0)1−q∣∣∣t=t0

= x0, (4.1)

where F ∈ C[[t0, T]×Rn,Rn], andDqx is the fractional derivative of x ∈ R

n and 0 < q < 1, p =1 − q.

At this point, we shall need an important property, known as quasimonotone non-decreasing relative to systems of inequalities.

Definition 4.1. A function F ∈ C[J × Rn,Rn] is said to possess quasimonotone nondecreasing

property if u ≤ v, ui = vi for some i, 1 ≤ i ≤ n, then Fi(t, u) ≤ Fi(t, v).Here, we shall be using vectorial inequalities, which are understood to mean the same

inequalities hold between their corresponding components.Next, we give the following existence result for systems of differential equations.

Theorem 4.2. Let v andw ∈ Cp[[t0, T],Rn] be upper and lower solutions of (4.1), respectively, suchthat F ∈ C(Ω,Rn) and v(t) ≤ w(t), t ∈ [t0, T] where Ω = [(t, x) : v(t) ≤ x(t) ≤ w(t), t ∈ [t0, T]].Moreover, assume that F(t, x) is quasimonotone nondecreasing in x for each t, then there exists asolution x(t) of (4.1) satisfying v(t) ≤ x(t) ≤ w(t) on [t0, T].

The proof of this theorem is a special case of the following theorem in which we chooseF not to be quasimonotone nondecreasing in x provided that we strengthen the notion ofupper and lower solutions of (4.1) as follows:

For each i, 1 ≤ i ≤ n,

Dqvi(t) ≤ Fi(t, ρ

) ∀ρ such that vi(t) = ρi(t), v(t) ≤ x(t) ≤ w(t) on [t0, T],

Dqwi(t) ≤ Fi(t, ρ

) ∀ρ such that wi(t) = ρi(t), v(t) ≤ x(t) ≤ w(t) on [t0, T].(4.2)

We state and prove the following existence result relative to the definition of upper and lowersolutions in (4.2).

Theorem 4.3. Let v and w ∈ Cp[[t0, T],Rn] be upper and lower solutions of (4.1), respectively,satisfying the relations given in (4.2), which are also locally Holder continuous with exponent λ > qsuch that v(t) ≤ w(t) and F ∈ C(Ω,Rn), then there exists a solution x(t) of (4.1) satisfying v(t) ≤x(t) ≤ w(t) on [t0, T].

Proof. Let p : [t0, T] × Rn → R

n given by

p(t, x) = min[w(t),max(x(t), v(t))] for each i, (4.3)


then F(t, p(t, x)) defines a continuous extension of F to [t0, T] × Rn which is also bounded

since f + g is bounded on Ω. Therefore, by Theorem 2.5,

Dqx(t) = F(t, p(t, x)

), x(t)(t − t0)1−q

∣∣∣t=t0

= x0 (4.4)

has a solution x(t) on [t0, T]. For ε > 0 and e = (1, 1, . . . , 1), consider wε(t) = w(t) + εγ(t)eand vε(t) = v(t) − εγ(t)e where γ(t) = (t − t0)q−1Eq,q((t − t0)q). It is clear that v0

ε < x0 < w0

ε . Wewish to show that vε(t) < x(t) < wε(t) on [t0, T]. Suppose that it is not true, then there existsan index j, 1 ≤ j ≤ n and a t1 ∈ (t0, T] such that

xj(t1) = wεj(t1), x(t) ≤ wε(t), t0 ≤ t ≤ t1, xi(t1) ≤ wεi(t1) for i /= j. (4.5)

Thus, we have v(t1) ≤ p(t1, x(t1)) ≤ w(t1) and pj(t1, x(t1)) = wj(t1). Setting mj(t) = vj(t) −wj(t), it follows that

mj(t1) = 0, mj(t) ≤ 0, t ∈ [t0, t1]; mi(t1) ≤ 0, i /= j. (4.6)

Applying Lemma 2.3 to the component mj(t), we get Dqmj(t1) ≥ 0 or Dqxj(t1) ≥Dqwεj(t1)which yields a contradiction

Fj(t1, w(t1)) = Fj(t1, p(t1, x(t1))

)= Dqxj(t1)

≥ Dqwεj(t1)

= Dqwj(t1) + εγ(t1)

> Dqwj(t1)

≥ Fj(t1, w(t1)).

(4.7)

Now, letting ε → 0, we arrive at v(t) ≤ x(t) ≤ w(t) on [t0, T] which proves the conclusion ofthe theorem.

Sometimes, we can have arbitrary coupling relative to upper and lower solutions. Letpi and qi be nonnegative integers for each i, 1 ≤ i ≤ n, so that we can split the vector x into(xi, [x]pi , [x]qi). Then the system (4.1) can be written as

Dqxi(t) = Fi(t, xi, [x]pi , [x]qi

), x(t)(t − t0)1−q

∣∣∣t=t0

= x0, (4.8)

where F ∈ C[[t0, T] × Rn,Rn].

Definition 4.4. A function F ∈ C[[t0, T] × Rn,Rn] is said to possess a mixed quasimonotone

property if for each i, Fi(t, xi, [x]pi , [x]qi) is monotone nondecreasing in [x]pi and monotonenonincreasing in [x]qi .


Definition 4.5. The functions v andw ∈ Cp[[t0, T],Rn] are said to be coupled upper and lowerquasisolutions of (4.8) if they satisfy

Dqvi ≤ Fi(t, vi, [v]pi, [v]qi

), v0 ≤ x0,

Dqwi ≥ Fi(t,wi, [w]pi, [w]qi

), w0 ≥ x0,

(4.9)

for each i, 1 ≤ i ≤ n.Next, we give an existence result which is also a special case of Theorem 4.3.

Theorem 4.6. Let v,w ∈ Cp[[t0, T],Rn] be coupled upper and lower quasisolutions of (4.8) andF ∈ C[[t0, T] × R

n,Rn]. If F(t, x) possesses a mixed quasimonotone property, then there exists asolution x(t) of (4.8) such that v(t) ≤ x(t) ≤ w(t) on [t0, T].

It should be noted that if F satisfies a mixed quasimonotone property, then (4.2) holdsfor coupled upper and lower quasisolutions given by (4.9). Therefore, Theorem 4.3 includesTheorem 4.6 as a special case.

5. Conclusion

In this work, some existence theorems have been established for nonlinear fractional-orderdifferential equations relative to coupled upper and lower solutions. The differential operatoris taken in the Riemann-Liouville sense. For the further developments in applications ofdynamical systems, we have generalized these results to the finite systems of nonlinearfractional differential equations. Being defined by a suitable differential operator, the processof finding a solution between upper and lower solutions generating a closed set could beapplied to various types of linear and nonlinear fractional partial differential equations as afuture work.

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Research ArticleOn the Global Well-Posedness of the ViscousTwo-Component Camassa-Holm System

Xiuming Li

School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China

Correspondence should be addressed to Xiuming Li, [email protected]

Received 3 January 2012; Accepted 15 February 2012


Copyright q 2012 Xiuming Li. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We establish the local well-posedness for the viscous two-component Camassa-Holm system.Moreover, applying the energy identity, we obtain a global existence result for the system with(u0, η0) ∈ H1(R) × L2(R).

1. Introduction

We are interested in the global well-pose dness of the initial value problem associated to theviscous version of the two-component Camassa-Holm shallow water system [1–3], namely,

mt + umx + 2uxm −Aux + ρρx = 0, t > 0, x ∈ R,

m = u − uxx,, t > 0, x ∈ R,

ρt +(uρ)x = 0, t > 0, x ∈ R,

(1.1)

where the variable u(t, x) represents the horizontal velocity of the fluid or the radial stretchrelated to a prestressed state, and ρ(t, x) is related to the free surface elevation fromequilibrium or scalar density with the boundary assumptions, u → 0 and ρ → 1 as |x| → ∞.The parameter A > 0 characterizes a linear underlying shear flow, so that (1.1)models wave-current interactions [4–6]. All of those are measured in dimensionless units.


Set p(x) := (1/2)e−|x|, x ∈ R. Then (1 − ∂2x)−1f = p ∗f for all f ∈ L2(R), where ∗ denotesthe spatial convolution. Let η = ρ−1, (1.1) can be rewritten as a quasilinear nonlocal evolutionsystem of the type

ut + uux = −∂x(1 − ∂2x

)−1(u2 +

12u2x −Au +

12η2 + η

), t > 0, x ∈ R,

ηt + uηx + ηux + ux = 0, t > 0, x ∈ R.

(1.2)

The system (1.1) without vorticity, that is, A = 0, was also rigorously justifiedby Constantin and Ivanov [1] to approximate the governing equations for shallow waterwaves. The multipeakon solutions of the same system have been constructed by Popivanovand Slavova [7], and the corresponding integral surface is partially ruled. Chen et al. [8]established a reciprocal transformation between the two-component Camassa-Holm systemand the first negative flow of AKNS hierarchy. More recently, Holm et al. [9] proposeda modified two-component Camassa-Holm system which possesses singular solutions incomponent ρ. Mathematical properties of (1.1) with A = 0 have been also studied furtherin many works. For example, Escher et al. [10] investigated local well-posedness for thetwo-component Camassa-Holm system with initial data (u0, ρ0) ∈ Hs × Hs−1 with s ≥ 2and derived some precise blow-up scenarios for strong solutions to the system. Constantinand Ivanov [1] provided some conditions of wave breaking and small global solutions. Guiand Liu [11] recently obtained results of local well-posedness in the Besov spaces and wavebreaking for certain initial profiles. More recently, Gui and Liu [12] studied global existenceand wave-breaking criteria for the system (1.2)with initial data (u0, ρ0 − 1) ∈ Hs ×Hs−1 withs > (3/2).

In this paper, we consider the global well-posedness of the viscous two-componentCamassa-Holm system

ut + uux − uxx = −∂x(1 − ∂2x

)−1(u2 +

12u2x −Au +

12η2 + η

), t > 0, x ∈ R,

ηt + uηx + ηux + ux = 0, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R,

η(0, x) = η0(x), x ∈ R.

(1.3)

The goal of the present paper is to study global existence of solutions for (1.3) to betterunderstand the properties of the two-component Camassa-Holm system (1.2). We state themain result as follows.

Theorem 1.1. For (u0, η0) ∈ H1(R) × L2(R), there exists a unique global solution (u, η) of (1.3)such that

u(t, x) ∈ C([0,∞);L2

x(R))∩ C((0,∞);H1

x(R)),

η(t, x) ∈ C([0,∞);L2

x(R)).

(1.4)


To proof Theorem 1.1, we will first establish global well-posedness of the followingregularized two-component system with ε > 0 given:

ut + uux − uxx = −∂x(1 − ∂2x

)−1(u2 +

12u2x −Au +

12η2 + η

), t > 0, x ∈ R,

ηt − εηxx + uηx + ηux + ux = 0, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R,

η(0, x) = η0(x), x ∈ R,

(1.5)

that is,

mt −mxx + umx + 2uxm −Aux + ηηx + ηx = 0, t > 0, x ∈ R,

ηt − εηxx + uηx + ηux + ux = 0, t > 0, x ∈ R,

m = u − uxx, t ≥ 0, x ∈ R,

u(0, x) = u0(x), η(0, x) = η0(x), x ∈ R.

(1.6)

Due to the Duhamel’s principle, we can also rewrite (1.6) as an integral equation

u(t, x) = et∂2xu0 +

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)dτ,

η(t, x) = eεt∂2xη0 +

∫ t

0eε(t−τ)∂

2xg(u, ∂xu, η, ∂xη

)dτ,

(1.7)

where g(u, ∂xu, η, ∂xη) = −u∂xη − η∂xu − ∂xu,

f(u, ∂xu, η

)= −∂x

(1 − ∂2x

)−1(u2 +

12(∂xu)

2 −Au +12η2 + η

)− 12∂x(u2), (1.8)

et∂2xu0 = (e−4π

2tξu0(ξ))∨, eεt∂

2xη0 = (e−4π

2εtξη0(ξ))∨, here and in what follows, we denote the

Fourier (or inverse Fourier) transform of a function f by f (or f∨).The remainder of the paper is organized as follows. In Section 2, we will set up

and introduce some estimates for the nonlinear part of (1.5). In Section 3, we will getthe local well-posedness of (1.3) by constructing the global well-posedness of (1.5) usingthe contraction argument and energy identity. The last section is devoted to the proof ofTheorem 1.1.

2. Preliminaries

We will list some lemmas needed in Section 3. First, we state the following lemma whichconsists of the crucial inequality involving the operator ∂x(1 − ∂2x)−1.


Lemma 2.1 (see [13]). For g, h ∈ L2(R),

∥∥∥∥∂x(1 − ∂2x

)−1(gh)∥∥∥∥

L2x

≤ c∥∥g∥∥L2x‖h‖L2

x, (2.1)

or more generally

∥∥∥∥|∂x|s(1 − ∂2x

)−1(gh)∥∥∥∥

L2x

≤ c∥∥g∥∥L2x‖h‖L2

x, (2.2)

for all s < 3/2.

The next two lemmas are regarding the nonlinear part of (1.5).

Lemma 2.2. Consider the following:

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2

TL2x+∥∥η∥∥2L2TL

2x+ T1/2

(‖u‖L2

TL2x+∥∥η∥∥L2TL

2x

)),

(2.3)

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤ C(‖∂xu‖2L2

TL2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

).

(2.4)

Proof. Let us prove (2.3) firstly. Thanks to Lemma 2.1, the Sobolev embedding theoremH1

x(R) ↪→ L∞x (R), and the Holder’s inequality, we have

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤ supt∈[0,T]

∫ t

0

∥∥∥e(t−τ)∂2xf(u, ∂xu, η

)(τ, x)

∥∥∥L2x

dτ

≤∫T

0‖u‖2

L2xdt +

12

∫T

0‖∂xu‖2L2

xdt +A

∫T

0‖u‖L2

xdt +

12

∫T

0

∥∥η∥∥2L2xdt

+∫T

0

∥∥η∥∥L2xdt +

∫T

0‖u∂xu‖L2

xdt,

(2.5)


which yields that

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤∫T

0‖u‖2

L2xdt +

12

∫T

0‖∂xu‖2L2

xdt +AT1/2

(∫T

0‖u‖2

L2xdt

)1/2

+12

∫T

0


+ T1/2

(∫T

0


)1/2

+∫T

0‖u‖H1

x‖∂xu‖L2

xdt

≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

)).

(2.6)

Similarly, we can get

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤ supt∈[0,T]

∫ t

0

∥∥∥eε(t−τ)∂2xg(u, ∂xu, η, ∂xη

)(τ, x)

∥∥∥L2x

dτ

≤∫T

0

∥∥u∂xη∥∥L2xdt +

∫T

0

∥∥η∂xu∥∥L2xdt +

∫T

0‖∂xu‖L2

xdt

≤∫T

0‖u‖L∞

x

∥∥∂xη∥∥L2xdt +

∫T

0

∥∥η∥∥L∞x‖∂xu‖L2

xdt + T1/2

(∫T

0‖∂xu‖2L2

xdt

)1/2

,

(2.7)

and then

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x) dτ

∥∥∥∥∥L∞T L

2x

≤∫T

0‖u‖H1

x


∫T

0

∥∥η∥∥H1

x‖∂xu‖L2

xdt + T1/2

(∫T

0‖∂xu‖2L2

xdt

)1/2

≤∫T

0‖∂xu‖L2

x


∫T

0

∥∥∂xη∥∥L2x‖∂xu‖L2

xdt + T1/2

(∫T

0‖∂xu‖2L2

xdt

)1/2

≤ 2∫T

0‖∂xu‖2L2

xdt + 2

∫T

0

∥∥∂xη∥∥2L2xdt + T1/2

(∫T

0‖∂xu‖2L2

xdt

)1/2

,

(2.8)

which implies (2.4).



∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤ CT1/2(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

)),

(2.9)

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤ CT1/2(‖∂xu‖2L2

TL2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

).

(2.10)

Proof. We mainly prove (2.9). For this, we have that

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤

⎛⎜⎝∫T

0

∥∥∥∥∥∥

∫

R

(∫ t

0e(t−τ)∂

2xf(τ, x)dτ

)2

dx

∥∥∥∥∥∥L∞T

dt

⎞⎟⎠

1/2

≤ T1/2

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(τ, x)dτ

∥∥∥∥∥L∞T L

2x

.

(2.11)

Therefore, applying Lemma 2.2, we can easily obtain (2.9).Similarly, we can also obtain (2.10).

Let us state the following lemma, which was obtained in [13] (up to a slight modifica-tion).

Lemma 2.4. For any u0 ∈ H1(R) and δ > 0, there exists T1 = T1(u0) > 0 such that

∥∥∥∂xet∂2xu0∥∥∥L2TL

2x

=

(∫T1

0

∫

R

∣∣∣∂xet∂2xu0∣∣∣2dx dt

)1/2

≤ δ. (2.12)

For any η0 ∈ L2(R), ε > 0, and δ > 0, there exists T2 = T2(η0, ε) > 0 such that

∥∥∥∂xeεt∂2xη0∥∥∥L2TL

2x

=

(∫T2

0

∫

R

∣∣∣∂xeεt∂2xη0∣∣∣2dx dt

)1/2

≤ δ. (2.13)


Next, we consider the nonlinear part of (1.7). When written as v = u − et∂2xu0, μ =

η − eεt∂2xη0, then we have

∂tv = ∂2xv + f(u, ∂xu, η

),

∂tμ = ε∂2xμ + g(u, ∂xu, η, ∂xη

),

v(x, 0) = 0,

μ(x, 0) = 0,

(2.14)

that is,

v(x, t) =∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ,

μ(x, t) =∫ t

0eε(t−τ)∂


)(τ, x)dτ.

(2.15)

First, we have the following basic estimates.


∥∥f∥∥L1TL

2x≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

)), (2.16)

∥∥g∥∥L1TL

2x≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

). (2.17)

Proof. First, let us prove (2.16),

∥∥f∥∥L1TL

2x=∥∥∥∥12∂x(u2)+ ∂x

(1 − ∂2x

)−1(u2 +

12(∂xu)

2 −Au +12η2 + η

) ∥∥∥∥L1TL

2x

≤∫T

0

∥∥∥∥12∂x(u2) + ∂x

(1 − ∂2x

)−1(u2 +

12(∂xu)

2 −Au +12η2 + η

)∥∥∥∥L2x

dt

≤ 12

∫T

0

∥∥∥∥∂x(1 − ∂2x

)−1(∂xu)

2∥∥∥∥L2x

dt +∫T

0

∥∥∥∂x(1 − ∂2x)−1(u2)∥∥∥L2x

dt

+∫T

0

∥∥∥∥∂x(1 − ∂2x

)−1(Au)

∥∥∥∥L2x

dt +12

∫T

0

∥∥∥∥∂x(1 − ∂2x

)−1(η2)∥∥∥∥

L2x

dt

+∫T

0‖u∂xu‖L2

xdt +

∫T

0

∥∥∥∥∂x(1 − ∂2x

)−1(η)∥∥∥∥

L2x

dt,

(2.18)


which implies

∥∥f∥∥L1TL

2x≤ 1

2

∫T

0‖∂xu‖2L2

xdt +

∫T

0‖u‖2

L2xdt +

∫T

0‖u‖L∞

x‖∂xu‖L2

xdt +A

∫T

0‖u‖L2

xdt

+12

∫T

0

∥∥η∥∥2L2xdt +

∫T

0

∥∥η∥∥L2xdt,

(2.19)

then we can get that

∥∥f∥∥L1TL

2x≤ 1

2

∫T

0‖∂xu‖2L2

xdt + C

∫T

0‖u‖2

L2xdt +

∫T

0‖u‖H1

x‖∂xu‖L2

xdt

+12

∫T

0

∥∥η∥∥2L2xdt + T1/2

(∫T

0


)1/2

+AT1/2

(∫T

0‖u‖2

L2xdt

)1/2

≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

)),

(2.20)

where we applied Lemma 2.1, Sobolev embedding theorem H1x(R) ↪→ L∞

x (R), and Holder’sinequality. This proves (2.16).

Next, we prove (2.17),

∥∥g∥∥L1TL

2x=∥∥−u∂xη − η∂xu − ∂xu

∥∥L1TL

2x

≤∫T

0

∥∥u∂xη + η∂xu + ∂xu∥∥L2xdt

≤∫T

0

∥∥u∂xη∥∥L2xdt +

∫T

0

∥∥η∂xu∥∥L2xdt +

∫T

0‖∂xu‖L2

xdt

≤∫T

0‖u‖L∞

x


∫T

0

∥∥η∥∥L∞x‖∂xu‖L2

xdt +

∫T

0‖∂xu‖L2

xdt,

(2.21)

which yields that

∥∥g∥∥L1TL

2x≤∫T

0‖u‖H1

x


∫T

0

∥∥η∥∥H1

x‖∂xu‖L2

xdt +

∫T

0‖∂xu‖L2

xdt

≤ C(∫T

0‖u‖2

L2xdt +

∫T

0‖∂xu‖2L2

xdt +

∫T

0

∥∥η∥∥2L2xdt +

∫T

0

∥∥∂xη∥∥2L2xdt +

∫T

0‖∂xu‖L2

xdt

)

≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

),

(2.22)

and this ends the proof of (2.17).


Then we have some estimates for ∂xv and ∂xμ.


‖∂xv‖L2TL

2x≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

)), (2.23)

∥∥∂xμ∥∥L2TL

2x≤ Cε−1/2

(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

). (2.24)

Proof. We mainly prove (2.24). We have that

∥∥μ∥∥L∞T L

2x=

∥∥∥∥∥

∫ t

0eε(t−τ)∂

2xg(u, ∂xu, η, ∂xη)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤∫T

0

∥∥g∥∥L2xdt =

∥∥g∥∥L1TL

2x.

(2.25)

Multiply μ to the second equation of (2.14) and integrate with respect to x over R.After integration by parts, we have

d

dt

∫

R

μ2

2dx + ε

∫

R

(∂xμ)2dx =

∫

R

μgdx,

12

∫

R

μ2(T)dx − 12

∫

R

μ2(0)dx + ε∫T

0

∫

R

(∂xμ)2dx dt =

∫T

0

∫

R

μgdx dt,

(2.26)

for any ε > 0, which implies

ε

∫T

0

∫

R

(∂xμ)2dx dt ≤

∫T

0

∫

R

μgdx dt,

∥∥∂xμ∥∥2L2TL

2x≤ 1ε

∫T

0

∥∥μ∥∥L2x

∥∥g∥∥L2xdt ≤ 1

ε

∥∥μ∥∥L∞T L

2x

∥∥g∥∥L1TL

2x.

(2.27)

By (2.25), together with (2.27), we have that

∥∥∂xμ∥∥L2TL

2x≤ 1ε1/2∥∥g∥∥L1TL

2x. (2.28)

By Lemma 2.5, (2.24) follows.Similarly, we can also obtain (2.23).


3. Local Well-Posedness

Let z :=( uη), A(z) :=

(et∂

2x u0

eεt∂2x η0

),

B(z) :=

⎛⎜⎜⎜⎝

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∫ t

0eε(t−τ)∂


)(τ, x)dτ

⎞⎟⎟⎟⎠,

D1 := C([0, T);L2

x(R))∩ C((0, T);H1

x(R)), D2 := C

([0, T);L2

x(R)),

XTa :=

{z ∈ D1 ×D2 : �z� = ‖z −A(z)‖L∞

T L2x+ ‖∂xz‖L2

TL2x+ ‖z‖L2

TL2x≤ a},

(3.1)

and define the mapping Φ : XTa → XT

a by

Φ(z) = A(z) + B(z). (3.2)

Theorem 3.1. For any ε > 0, there exist T = Tε > 0 and a > 0 such that Φ(XTa ) ⊆ XT

a . In addition,Φ : XT

a → XTa is a contraction mapping.

Proof. We first need to show that the map is well defined for some appropriate a and T . Letz ∈ XT

a , then we have

�Φz� = ‖Φz −A(z)‖L∞T L

2x+ ‖Φz‖L2

TL2x+ ‖∂x(Φz)‖L2

TL2x. (3.3)

Considering the terms in (3.3) one by one, from Lemma 2.2, the first term in (3.3) canbe estimated as follows:

‖Φz −A(z)‖L∞T L

2x=

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

+

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L∞T L

2x

≤ C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+∥∥∂xη

∥∥2L2TL

2x

)

+ CT1/2(‖u‖L2

TL2x+ ‖∂xu‖L2


2x

)

≤ C�z�2 + CT1/2�z�.

(3.4)


From Lemma 2.3, the second term in (3.3) can be estimated as follows:

‖Φz‖L2TL

2x=

∥∥∥∥∥et∂2xu0 +

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

+

∥∥∥∥∥eεt∂2xη0 +

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤∥∥∥et∂

2xu0∥∥∥L2TL

2x

+

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

+∥∥∥eεt∂

2xη0∥∥∥L2TL

2x

+

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L2TL

2x

,

(3.5)

which implies

‖Φz‖L2TL

2x≤ T1/2‖u0‖L2

x+

∥∥∥∥∥

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

+ T1/2∥∥η0∥∥L2x+

∥∥∥∥∥

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤ T1/2‖z0‖L2x+ CT1/2�z�2 + CT�z�.

(3.6)

From Lemma 2.6, the third term in (3.3) can be estimated as follows:

‖∂x(Φz)‖L2TL

2x=

∥∥∥∥∥∂xet∂2xu0 + ∂x

∫ t

0e(t−τ)∂

2xf(u, ∂xu, η

)(τ, x)dτ

∥∥∥∥∥L2TL

2x

+

∥∥∥∥∥∂xeεt∂2xη0 + ∂x

∫ t

0eε(t−τ)∂


)(τ, x)dτ

∥∥∥∥∥L2TL

2x

≤∥∥∥∂xet∂

2xu0∥∥∥L2TL

2x

+ ‖∂xv‖L2TL

2x+∥∥∥∂xeεt∂

2xη0∥∥∥L2TL

2x

+∥∥∂xμ

∥∥L2TL

2x

≤ 2δ + C(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+ T1/2

(‖u‖L2


2x

))

+C

ε1/2

(‖u‖2

L2TL

2x+ ‖∂xu‖2L2


2x+∥∥∂xη

∥∥2L2TL

2x+ T1/2‖∂xu‖L2

TL2x

)

≤ 2δ + C(1 +

1ε1/2

)�z�2 + CT1/2

(1 +

1ε1/2

)�z�.

(3.7)


Combining (3.4)–(3.7), we have that

�Φz� ≤ 2δ + T1/2‖z0‖L2x+ C�z�2 + CT1/2�z�2 + C

(1 +

1ε1/2

)�z�2

+ CT�z� + CT1/2(2 +

1ε1/2

)�z�

≤ 2δ + T1/2‖z0‖L2x+ C(2 + T1/2

)(1 +

1ε1/2

)a2

+ CTa + CT1/2(2 +

1ε1/2

)a.

(3.8)

With appropriate values of δ, a, and T , we are able to have that �Φz� ≤ a, that is,Φ : XT

a → XTa is well defined.

Similar to the above argument, we can show that Φ : XTa → XT

a is a contractionmapping,

�Φz1 −Φz2� ≤ C′�z1 − z2�, (3.9)

where C′ = C′(T, a, ε, ‖z1‖L2TL

2x, ‖z2‖L2

TL2x, ‖∂xz1‖L2

TL2x, ‖∂xz2‖L2

TL2x) can be chosen as 0 < C′ < 1

with appropriate values of T and a.

Theorem 3.2. For any ε > 0 and (u0, η0) ∈ H1(R) × L2(R), there exist a T = T(u0, η0, ε) > 0 and aunique solution (uε, ηε) of (1.5) such that

uε(x, t) ∈ C([0, T);L2

x(R))∩ C((0, T);H1

x(R)),

ηε(x, t) ∈ C([0, T);L2

x(R)).

(3.10)

Proof. Theorem 3.2 is merely Theorem 3.1 with a standard uniqueness argument.

Theorem 3.3. For any ε > 0 and (u0, η0) ∈ H1(R) × L2(R), there exists a unique global solution(uε, ηε) of (1.5) such that

uε(x, t) ∈ C([0,∞);L2

x(R))∩ C((0,∞);H1

x(R)),

ηε(x, t) ∈ C([0,∞);L2

x(R)).

(3.11)

Proof. To prove Theorem 3.3, we need only to establish the a priori energy identity. Multiply-ing the first equation in (1.6) by u and integrating by parts (with respect to x over R), we havethat

12d

dt

∫

R

(u2 + u2x

)dx +

∫

R

(u2x + u

2xx

)dx − 1

2

∫

R

η2uxdx +∫

R

uηxdx = 0, (3.12)


where we used the relation m = u − uxx and∫Ru2uxdx = 0. Multiplying the first equation in

(1.6) by η and integrating by parts (with respect to x over R), we have that

12d

dt

∫

R

η2dx + ε∫

R

η2xdx +12

∫

R

η2uxdx −∫

R

uηxdx = 0. (3.13)

From (3.12) and (3.13), we obtain the energy identity

12d

dt

∫

R

(u2 + u2x + η

2)dx +

∫

R

(u2x + u

2xx

)dx + ε

∫

R

η2xdx = 0, (3.14)

which gives rise to the following inequality independent of ε and T :

supt∈[0,T)

(‖u(t, ·)‖2H1 +

∥∥η(t, ·)∥∥2L2

)+ 2‖ux‖2L2

T(H1) ≤ ‖u0‖2H1 +∥∥η0∥∥2L2 . (3.15)

According to Theorem 3.2 and the energy inequality (3.15), one can extend the local solutionto the global one by a standard contradiction argument, which completes the proof ofTheorem 3.3.

From Theorem 3.3, one has the local well-posedness of system (1.3).

Theorem 3.4. For (u0, η0) ∈ H1(R) × L2(R), there exist a T = T(u0, η0) > 0 and a unique localsolution (u, η) of (1.3) such that

u(x, t) ∈ C([0, T);L2

x(R))∩ C((0, T);H1

x(R)),

η(x, t) ∈ C([0, T);L2

x(R)).

(3.16)

Similar to the proof of Theorem 4.1 in [12] (up to a slight modification), we may getthe following.

Theorem 3.5. Let z0 = (u0, η0) ∈ Hs ×Hs−1, s ≥ 1, and let z = (u, η) be the corresponding solutionto (1.3). Assume that T ∗

z0 > 0 is the maximal time of existence, then

T ∗z0 <∞ =⇒

∫T∗z0

0‖∂xu(τ)‖L∞dτ = ∞. (3.17)



Proof of Theorem 1.1. From Theorem 3.4, we have got the local solution of (1.3). On the otherhand, according to the second equation of (1.3), we have

12d

dt

∥∥η∥∥2L2x≤ 2‖∂xu‖L∞

x

∥∥η∥∥2L2x+ ‖∂xu‖L2

x

∥∥η∥∥L2x

≤ 4‖∂xu‖L∞x

∥∥η∥∥2L2x+ ‖∂xu‖L2

x

∥∥η∥∥L2x

≤ 4‖∂xu‖L∞x

∥∥η∥∥2L2x+∥∥η∥∥2L2x+14‖∂xu‖2L2

x

=(4‖∂xu‖L∞

x+ 1)∥∥η∥∥2L2x+14‖∂xu‖2L2

x.

(4.1)

An application of Gronwall’s inequality yields

∥∥η∥∥2L2x≤(∥∥η0

∥∥2L2x+14‖∂xu‖2L2

TL2x

)e∫ t0(1+4‖∂xu‖L∞x )dτ . (4.2)

Similar to the proof of Theorem 3.3, we may get the energy identity

12d

dt

∫

R

(u2 + u2x + η

2)dx +

d

dt

∫

R

(u2x + u

2xx

)dx = 0, (4.3)

which implies

supt∈[0,T)

(‖u(t, ·)‖2H1 +

∥∥η(t, ·)∥∥2L2

)+ 2‖ux‖2L2

T(H1) ≤ ‖u0‖2H1 +∥∥η0∥∥2L2 . (4.4)

Due to the Sobolev embedding theorem H1(R) ↪→ L∞(R) and (4.4), we obtain that for anyT < +∞,

∫T

0‖∂xu‖L∞

xdt ≤

∫T

0‖∂xu‖H1

xdt ≤ T1/2‖∂xu‖L2

T (H1x) < +∞. (4.5)

Therefore, from Theorem 3.5, we deduce that themaximal existence time T = +∞. This provesTheorem 1.1.

Acknowledgments

The work of the author is supported in part by the NSF of China under Grant no. 11001111and no. 11141003. The author would like to thank the referees for constructive suggestionsand comments.


References

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Research ArticleThe Local and Global Existence of Solutions fora Generalized Camassa-Holm Equation

Nan Li, Shaoyong Lai, Shuang Li, and Meng Wu

Department of Applied Mathematics, Southwestern University of Finance and Economics,Chengdu 610074, China

Correspondence should be addressed to Shaoyong Lai, [email protected]

Received 13 October 2011; Accepted 13 January 2012


Copyright q 2012 Nan Li et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A nonlinear generalization of the Camassa-Holm equation is investigated. By making use of thepseudoparabolic regularization technique, its local well posedness in Sobolev space HS(R) withs > 3/2 is established via a limiting procedure. Provided that the initial value u0 satisfies the signcondition and u0 ∈ Hs(R) (s > 3/2), it is shown that there exists a unique global solution for theequation in space C([0,∞);Hs(R)) ∩ C1([0,∞);Hs−1(R)).

1. Introduction

Camassa and Holm [1] employed the Hamiltonian method to derive a completely integrableshallow water wave model

ut − utxx + 2kux + 3uux = 2uxuxx + uuxxx, (1.1)

which was alternatively established as a water wave equation in [2–4]. Equation (1.1) alsomodels wave current interaction [5], while Dai [6] derived it as a model in elasticity (see[7]). In addition, it was pointed out in Lakshmanan [8] that the Camassa-Holm equation(1.1) could be relevant to the modeling of tsunami waves (see Constantin and Johnson [9]).

After the birth of the Camassa-Holm equation (1.1), many works have been carriedout to probe its dynamic properties. For k = 0, (1.1) has travelling wave solutions of theform ce−|x−ct|, called peakons, which describes an essential feature of the travelling wavesof largest amplitude (see [10–14]). For k > 0, its solitary waves are stable solitons [15]. Itis shown in [16–18] that the inverse spectral or scattering approach is a powerful tool tohandle the Camassa-Holm equation and analyze its dynamics. It is worthwhile to mention


that (1.1) gives rise to geodesic flow of a certain invariant metric on the Bott-Virasoro group[19–21], and this geometric illustration leads to a proof that the least action principle holds.Xin and Zhang [22] proved the global existence of the weak solution in the energy spaceH1(R) without any sign conditions on the initial value, and the uniqueness of this weaksolution is obtained under some assumptions on the solution [23]. Coclite et al. [24] extendedthe analysis presented in [22, 23] and obtained many useful dynamic properties to otherpartial differential equations (see [25–28] for an alternative approach). Li and Olver [29]established the local well posedness in the Sobolev space Hs(R) with s > 3/2 for (1.1) andgave conditions on the initial data that lead to finite time blowup of certain solutions. It isshown in Constantin and Escher [30] that the blowup occurs in the form of breaking waves,namely, the solution remains bounded but its slope becomes unbounded in finite time. Forother methods to handle the problems relating to various dynamic properties of the Camassa-Holm equation and other shallow water equations, the reader is referred to [31–39] and thereferences therein.

Motivated by the work in Hakkaev and Kirchev [33] to investigate the generalizationforms of the Camassa-Holm equation with high-order nonlinear terms, we study thefollowing generalized Camassa-Holm equation:

ut − utxx + kumux + (m + 3)um+1ux = (m + 2)umuxuxx + um+1uxxx, (1.2)

where m ≥ 0 is a natural number and k ≥ 0. Obviously, (1.2) reduces to (1.1) if we set m = 0.As the Camassa-Holm equation (1.1) has been discussed by many mathematicians, we let thenatural numberm ≥ 1 in this paper.

The objective of this paper is to study (1.2). Its local well posedness of solutionsin the Sobolev space Hs(R) with s > 3/2 is developed by using the pseudoparabolicregularization method. Provided that (1 − ∂2x)u0 + k/2(m + 1) ≥ 0 and u0 ∈ Hs (s >3/2), the existence and uniqueness of the global solutions are established in spaceC([0,∞);Hs(R))

⋂C1([0,∞);Hs−1(R)). It should be mentioned that the existence and

uniqueness of global strong solutions for the nonlinear generalized Camassa-Holm modelslike (1.2) have never been investigated in the literatures.

2. Main Results

The space of all infinitely differentiable functions φ(t, x)with compact support in [0,+∞)×Ris denoted by C∞

0 . Lp = Lp(R)(1 ≤ p < +∞) is the space of all measurable functions h suchthat ‖h‖pLp =

∫R |h(t, x)|pdx < ∞. We define L∞ = L∞(R) with the standard norm ‖h‖L∞ =

infm(e)=0supx∈R\e|h(t, x)|. For any real number s,Hs = Hs(R) denotes the Sobolev space withthe norm defined by

‖h‖Hs =(∫

R

(1 + |ξ|2

)s∣∣∣h(t, ξ)∣∣∣2dξ

)1/2

<∞, (2.1)

where h(t, ξ) =∫R e

−ixξh(t, x)dx.For T > 0 and nonnegative number s, C([0, T);Hs(R)) denotes the Frechet space of all

continuous Hs-valued functions on [0, T). We set Λ = (1 − ∂2x)1/2. For simplicity, throughoutthis paper, we let c denote any positive constant which is independent of parameter ε.


We consider the Cauchy problem of (1.2), which has the equivalent form

ut − utxx = − k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

− (m + 1)∂x(umu2x

)+ umuxuxx, k ≥ 0, m ≥ 1,

u(0, x) = u0(x).

(2.2)

Now, we give our main results for problem (2.2).

Theorem 2.1. Suppose that the initial function u0(x) belongs to the Sobolev space Hs(R) with s >3/2. Then there is a T > 0, which depends on ‖u0‖Hs , such that there exists a unique solution u(t, x)of the problem (2.2) and

u(t, x) ∈ C([0, T];Hs(R))⋂C1

([0, T];Hs−1(R)

). (2.3)

Theorem 2.2. Let u0(x) ∈ Hs, s > 3/2 and (1 − ∂2x)u0 + k/2(m + 1) ≥ 0 for all x ∈ R. Thenproblem (2.2) has a unique solution satisfying that

u(t, x) ∈ C([0,∞);Hs(R))⋂C1

([0,∞);Hs−1(R)

). (2.4)

3. Local Well-Posedness

In order to prove Theorem 2.1, we consider the associated regularized problem

ut − utxx + εutxxxx = − k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

− (m + 1)∂x(umu2x

)+ umuxuxx,

u(0, x) = u0(x),

(3.1)

where the parameter ε satisfies 0 < ε < 1/4.

Lemma 3.1. Let r and q be real numbers such that −r < q ≤ r. Then

‖uv‖Hq ≤ c‖u‖Hr‖v‖Hq , if r >12,

‖uv‖Hr+q−1/2 ≤ c‖u‖Hr‖v‖Hq , if r <12.

(3.2)

This lemma can be found in [34, 40].

Lemma 3.2. Let u0(x) ∈ Hs(R) with s > 3/2. Then the Cauchy problem (3.1) has a unique solutionu(t, x) ∈ C([0, T];Hs(R)) where T > 0 depends on ‖u0‖Hs(R). If s ≥ 2, the solution u ∈C([0,+∞);Hs) exists for all time.


Proof. Assuming that D = (1 − ∂2x + ε∂4x)−1, we know thatD : Hs → Hs+4 is a bounded linearoperator. Applying the operatorD on both sides of the first equation of system (3.1) and thenintegrating the resultant equation with respect to t over the interval (0, t) lead to

u(t, x) = u0(x) +∫ t

0D

[− k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

−(m + 1)∂x(umu2x

)+ umuxuxx

]dt.

(3.3)

Suppose that both u and v are in the closed ball BM0(0) of radiusM0 about the zero function inC([0, T];Hs(R)) and A is the operator in the right-hand side of (3.3). For any fixed t ∈ [0, T],we get the following:

∥∥∥∥∥

∫ t

0D

[− k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

−(m + 1)∂x(umu2x

)+ umuxuxx

]dt

−∫ t

0D

[− k

m + 1

(vm+1

)

x− m + 3m + 2

(vm+2

)

x+

1m + 2

∂3x

(vm+2

)

−(m + 1)∂x(vmv2

x

)+ vmvxvxx

]dt

∥∥∥∥Hs

≤ TC1

(sup0≤t≤T

∥∥∥um+1 − vm+1∥∥∥Hs

+ sup0≤t≤T

∥∥∥um+2 − vm+2∥∥∥Hs

+ sup0≤t≤T

∥∥∥D∂x[umu2x − vmv2

x

]∥∥∥Hs

+ sup0≤t≤T

‖D[umuxuxx − vmvxvxx]‖Hs

),

(3.4)

where C1 may depend on ε. The algebraic property ofHs0(R) with s0 > 1/2 derives

∥∥∥um+2 − vm+2∥∥∥Hs

=∥∥∥(u − v)

(um+1 + umv + · · · + uvm + vm+1

)∥∥∥Hs

≤ ‖(u − v)‖Hs

m+1∑

j=0‖u‖m+1−j

Hs ‖v‖jHs

≤Mm+10 ‖(u − v)‖Hs,

(3.5)

∥∥∥um+1 − vm+1∥∥∥Hs

≤Mm0 ‖(u − v)‖Hs,

∥∥∥D∂x(umu2x − vmv2

x

)∥∥∥Hs

≤∥∥∥D∂x

[um

(u2x − v2

x

)]∥∥∥Hs

+∥∥∥D∂x

[v2x(u

m − vm)]∥∥∥

Hs

≤ C(∥∥∥um

(u2x − v2

x

)∥∥∥Hs−1

+∥∥∥v2

x(um − vm)

∥∥∥Hs−1

)

≤ CMm+10 ‖u − v‖Hs.

(3.6)


Using the first inequality of Lemma 3.1, we have

‖D[umuxuxx − vmvxvxx]‖Hs =∥∥∥∥12D

[um

(u2x

)

x− vm

(v2x

)

x

]∥∥∥∥Hs

≤ 12

(∥∥∥D[um

(u2x − v2

x

)

x

]∥∥∥Hs

+∥∥∥D

[(v2x

)

x(um − vm)

]∥∥∥Hs

)

≤ C(∥∥∥um

(u2x − v2

x

)

x

∥∥∥Hs−2

+∥∥∥(v2x

)

x(um − vm)

∥∥∥Hs−2

)

≤ C(‖um‖Hs

∥∥∥u2x − v2x

∥∥∥Hs−1

+∥∥∥v2

x

∥∥∥Hs−1

‖um − vm‖Hs

)

≤ CMm+10 ‖u − v‖Hs,

(3.7)

where C may depend on ε. From (3.5)–(3.7), we obtain that

‖Au −Av‖Hs ≤ θ‖u − v‖Hs, (3.8)

where θ = TC2(Mm0 + Mm+1

0 ) and C2 is independent of 0 < t < T . Choosing T sufficientlysmall such that θ < 1, we know that A is a contraction. Applying the above inequality yieldsthat

‖Au‖Hs ≤ ‖u0‖Hs + θ‖u‖Hs. (3.9)

Choosing T sufficiently small such that θM0 + ‖u0‖Hs < M0, we deduce that Amaps BM0(0)to itself. It follows from the contraction-mapping principle that the mapping A has a uniquefixed-point u in BM0(0).

For s ≥ 2, using the first equation of system (3.1) derives

d

dt

∫

R

(u2 + u2x + εu

2xx

)dx = 0, (3.10)

from which we have the conservation law

∫

R

(u2 + u2x + εu

2xx

)dx =

∫

R

(u20 + u

20x + εu

20xx

)dx. (3.11)

The proof of the global existence result is a routine argument by using (3.11) (see Xin andZhang [22]).

Lemma 3.3 (Kato and Ponce [41]). If r ≥ 0, thenHr⋂L∞ is an algebra. Moreover

‖uv‖r ≤ c(‖u‖L∞‖v‖r + ‖u‖r‖v‖L∞), (3.12)

where c is a constant depending only on r.


Lemma 3.4 (Kato and Ponce [41]). Let r > 0. If u ∈ Hr⋂W1,∞ and v ∈ Hr−1 ⋂L∞, then

‖[Λr , u]v‖L2 ≤ c(‖∂xu‖L∞

∥∥∥Λr−1v∥∥∥L2

+ ‖Λru‖L2‖v‖L∞

). (3.13)

Lemma 3.5. Let s ≥ 2, and the function u(t, x) is a solution of problem (3.1) and the initial datau0(x) ∈ Hs. Then the following inequality holds:

‖u‖2H1 ≤∫

R

(u2 + u2x + εu

2xx

)dx

=∫

R

(u20 + u

20x + εu

20xx

)dx.

(3.14)

For q ∈ (0, s − 1], there is a constant c independent of ε such that

∫

R

(Λq+1u

)2dx ≤

∫

R

[(Λq+1u0

)2+ ε(Λqu0xx)

2]dx

+ c∫ t

0‖u‖2Hq+1

((‖u‖m−1

L∞ + ‖u‖mL∞

)‖ux‖L∞ + ‖u‖m−1

L∞ ‖ux‖2L∞

)dτ.

(3.15)

For q ∈ [0, s − 1], there is a constant c independent of ε such that

(1 − 2ε)‖ut‖Hq ≤ c‖u‖Hq+1

((‖u‖m−1

L∞ + ‖u‖mL∞

)‖u‖H1 + ‖u‖mL∞‖ux‖L∞ + ‖‖u‖m−1

L∞ ‖ux‖2L∞

).

(3.16)

Proof. The inequality ‖u‖2H1 ≤∫R(u

2 + u2x)dx and (3.11) derives (3.14).Using ∂2x = −Λ2 + 1 and the Parseval equality gives rise to

∫

R

ΛquΛq∂3x

(um+2

)dx = −(m + 2)

∫

R

(Λq+1u

)Λq+1

(um+1ux

)dx

+ (m + 2)∫

R

(Λqu)Λq(um+1ux

)dx.

(3.17)

For q ∈ (0, s − 1], applying (Λqu)Λq to both sides of the first equation of system (3.1)and integrating with respect to x by parts, we have the identity

12d

dt

∫

R

((Λqu)2 + (Λqux)

2 + ε(Λquxx)2)dx

= − k

m + 1

∫

R

(Λqu)Λq(um+1

)

xdx − (m + 2)

∫

R

(Λqu)Λq(um+1ux

)dx

−∫

R

(Λq+1u

)Λq+1

(um+1ux

)dx + (m + 1)

∫

R

(Λqux)Λq(umu2x

)dx

+∫

R

ΛquΛq(umuxuxx)dx.

(3.18)


We will estimate the terms on the right-hand side of (3.18) separately. For the second term,by using the Cauchy-Schwartz inequality and Lemmas 3.3 and 3.4, we have

∣∣∣∣

∫

R

(Λqu)Λq(um+1ux

)dx

∣∣∣∣ =∣∣∣∣

∫

R

(Λqu)[Λq

(um+1ux

)− um+1Λqux

]dx +

∫

R

(Λqu)um+1Λquxdx

∣∣∣∣

≤ c‖u‖Hq

((m + 1)‖u‖mL∞‖ux‖L∞‖u‖Hq + ‖ux‖L∞‖u‖mL∞‖u‖Hq

)

+m + 12

‖u‖mL∞‖ux‖L∞‖Λqu‖2L2

≤ c‖u‖2Hq‖u‖mL∞‖ux‖L∞ .

(3.19)

Similarly, for the first term in (3.18), we have

∣∣∣∣

∫

R

(Λqu)Λq(umux)dx∣∣∣∣ ≤ c‖u‖2Hq‖u‖m−1

L∞ ‖ux‖L∞ . (3.20)

Using the above estimate to the third term yields that

∣∣∣∣

∫

R

(Λq+1u

)Λq+1

(um+1ux

)dx

∣∣∣∣ ≤ c‖u‖2Hq+1‖u‖mL∞‖ux‖L∞ . (3.21)

For the fourth term, using the Cauchy-Schwartz inequality and Lemma 3.3, we obtain that

∣∣∣∣

∫

R

(Λqux)Λq(umu2x

)dx

∣∣∣∣ ≤ ‖Λqux‖L2

∥∥∥Λq(umu2x

)∥∥∥L2

≤ c‖u‖Hq+1(‖umux‖L∞‖ux‖Hq + ‖ux‖L∞‖umux‖Hq)

≤ c‖u‖2Hq+1‖ux‖L∞‖u‖mL∞ ,

(3.22)

in which we have used ‖umux‖Hq ≤ c‖(um+1)x‖Hq ≤ c‖u‖mL∞‖u‖Hq+1 .For the last term in (3.18), using um(u2x)x = (umu2x)x − (um)xu

2x results in

∣∣∣∣

∫

R

(Λqu)Λq(umuxuxx)dx∣∣∣∣ ≤

∣∣∣∣

∫

R

ΛquxΛq(umu2x

)dx

∣∣∣∣ +∫

R

ΛquΛq[(um)xu

2x

]dx

= K1 +K2.

(3.23)

For K1, it follows from (3.22) that

K1 ≤ c‖u‖2Hq+1‖ux‖L∞‖u‖mL∞ . (3.24)


For K2, applying Lemma 3.3 derives

K2 ≤ c‖u‖Hq

∥∥∥(um)xu2x

∥∥∥Hq

≤ c‖u‖Hq

(‖(um)x‖L∞

∥∥∥u2x∥∥∥Hq

+ ‖(um)x‖Hq

∥∥∥u2x∥∥∥L∞

)

≤ c‖u‖2Hq+1

(‖u‖m−1

L∞ ‖ux‖2L∞

).

(3.25)

It follows from (3.19)–(3.25) that there exists a constant c depending only onm such that

12d

dt

∫

R

[(Λqu)2 + (Λqux)

2 + ε(Λquxx)2]dx ≤ c‖u‖2Hq+1

(‖ux‖L∞‖u‖mL∞ + ‖u‖m−1

L∞ ‖ux‖2L∞

).(3.26)

Integrating both sides of the above inequality with respect to t results in (3.15).To estimate the norm of ut, we apply the operator (1 − ∂2x)−1 to both sides of the first

equation of system (3.1) to obtain the equation

(1 − ε)ut − εutxx =(1 − ∂2x

)−1[−εut − kumux − m + 3m + 2

(um+2

)

x

+1

m + 2∂3x

(um+2

)− (m + 1)∂x

(umu2x

)+ umuxuxx

].

(3.27)

Applying (Λqut)Λq to both sides of (3.27) for q ∈ [0, s − 1] gives rise to

(1 − ε)∫

R

(Λqut)2dx + ε

∫

R

(Λquxt)2dx

=∫

R

(Λqut)Λq−2[−εut + ∂x

(− k

m + 1um+1 − m + 3

m + 2um+2 +

1m + 2

∂2xum+2 − (m + 1)umu2x

)

+umuxuxx]dτ.

(3.28)

For the right hand of (3.28), we have

∣∣∣∣

∫

R

(Λqut)Λq−2(−εut)dx∣∣∣∣ ≤ ε‖ut‖2Hq ,

∣∣∣∣

∫

R

(Λqut)(1 − ∂2x

)−1Λq∂x

(− k

m + 1um+1 − m + 3

m + 2um+2 − (m + 1)umu2x

)dx

∣∣∣∣

≤ c‖ut‖Hq

(∫

R

(1 + ξ2

)q−1[∫

R

[− k

m + 1um

(ξ − η)u(η) − m + 3

m + 2um+1

(ξ − η)u(η)

−(m + 1)umux(ξ − η)ux

(η)]dη

]2)1/2

≤ c‖ut‖Hq‖u‖H1‖u‖Hq+1

(‖u‖m−1

L∞ + ‖u‖mL∞

).

(3.29)


Since

∫(Λqut)

(1 − ∂2x

)−1Λq∂2x

(um+1ux

)dx = −

∫(Λqut)Λq

(um+1ux

)dx

+∫(Λqut)

(1 − ∂2x

)−1Λq

(um+1ux

)dx,

(3.30)

using Lemma 3.3, ‖um+1ux‖Hq ≤ c‖(um+2)x‖Hq ≤ c(m + 2)‖u‖m+1L∞ ‖u‖Hq+1 and ‖u‖L∞ ≤ ‖u‖H1 ,

we have

∣∣∣∣

∫(Λqut)Λq

(um+1ux

)dx

∣∣∣∣ ≤ c‖ut‖Hq

∥∥∥um+1ux∥∥∥Hq

≤ c‖ut‖Hq‖u‖mL∞‖u‖H1‖u‖Hq+1 ,∣∣∣∣

∫(Λqut)

(1 − ∂2x

)−1Λq

(um+1ux

)dx

∣∣∣∣ ≤ c‖ut‖Hq‖u‖mL∞‖u‖H1‖u‖Hq+1 .

(3.31)

Using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.3 yields that

∣∣∣∣

∫

R

(Λqut)(1 − ∂2x

)−1Λq(umuxuxx)dx

∣∣∣∣ ≤ c‖ut‖Hq‖umuxuxx‖Hq−2

≤ c‖ut‖Hq

∥∥∥um(u2x

)

x

∥∥∥Hq−2

≤ c‖ut‖Hq

∥∥∥[um

(u2x

)]

x− (um)xu

2x

∥∥∥Hq−2

≤ c‖ut‖Hq

(∥∥∥umu2x∥∥∥Hq−1 +

∥∥∥(um)xu2x

∥∥∥Hq−2

)

≤ c‖ut‖Hq

(∥∥∥umu2x∥∥∥Hq

+∥∥∥(um)xu

2x

∥∥∥Hq

)

≤ c‖ut‖Hq‖u‖Hq+1

(‖u‖mL∞‖ux‖L∞ + ‖u‖m−1

L∞ ‖ux‖2L∞

),

(3.32)

in which we have used (3.25).Applying (3.29)–(3.32) into (3.28) yields the inequality

(1 − 2ε)‖ut‖Hq ≤ c‖u‖Hq+1

((‖u‖m−1

L∞ + ‖u‖mL∞

)‖u‖H1

+‖u‖mL∞‖ux‖L∞ + ‖u‖m−1L∞ ‖ux‖2L∞

) (3.33)

for a constant c > 0. This completes the proof of Lemma 3.5.

Remark 3.6. In fact, letting ε = 0 in problem (3.1), (3.14), (3.15), and (3.16) are still valid.


Setting φε(x) = ε−1/4φ(ε−1/4x) with 0 < ε < 1/4 and uε0 = φε u0, we know thatuε0 ∈ C∞ for any u0 ∈ Hs, s > 0. From Lemma 3.2, it derives that the Cauchy problem

ut − utxx + εutxxxx = − k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

− (m + 1)∂x(umu2x

)+ umuxuxx,

u(0, x) = uε0(x), x ∈ R

(3.34)

has a unique solution uε(t, x) ∈ C∞([0,∞);H∞).Furthermore, we have the following.

Lemma 3.7. For s > 0, u0 ∈ Hs, it holds that

‖uε0x‖L∞ ≤ c‖u0x‖L∞ , (3.35)

‖uε0‖Hq ≤ c if q ≤ s, (3.36)

‖uε0‖Hq ≤ cε(s−q)/4 if q > s, (3.37)

‖uε0 − u0‖Hq ≤ cε(s−q)/4 if q ≤ s, (3.38)

‖uε0 − u0‖Hs = o(1), (3.39)

where c is a constant independent of ε.

The proof of Lemma 3.7 can be found in [38].

Remark 3.8. For s ≥ 1, using ‖uε‖L∞ ≤ c‖uε‖H1/2+ ≤ c‖uε‖H1 , ‖uε‖2H1 ≤ c∫R(u

2ε + u

2εx)dx, (3.14),

(3.36), and (3.37), we know that

‖uε‖2L∞ ≤ c‖uε‖H1 ≤ c∫

R

(u2ε0 + u

2ε0x + εu

2ε0xx

)dx

≤ c(‖uε0‖2H1 + ε‖uε0‖2H2

)

≤ c(c + cε × ε(s−2)/2

)

≤ c0,

(3.40)

where c0 is independent of ε.

Lemma 3.9. If u0(x) ∈ Hs(R) with s ≥ 1 such that ‖u0x‖L∞ < ∞. Let uε0 be defined as in system(3.34). Then there exist two positive constants T and c, which are independent of ε, such that thesolution uε of problem (3.34) satisfies ‖uεx‖L∞ ≤ c for any t ∈ [0, T).


Proof. Using notation u = uε and differentiating both sides of the first equation of problem(3.34) or (3.27)with respect to x give rise to

(1 − ε)utx − εutxxx + 1m + 2

∂2(um+2

)−(m +

12

)(umu2x

)

=k

m + 1um+1 + um+2 −Λ−2

[εutx +

k

m + 1um+1 + um+2 +

(m +

12

)(umu2x

)

+12∂x

[(um)xu

2x

]].

(3.41)

Letting p > 0 be an integer and multiplying the above equation by (ux)2p+1 and then

integrating the resulting equation with respect to x yield the equality

1 − ε2p + 2

d

dt

∫

R

(ux)2p+2dx − ε∫

R

(ux)2p+1utxxxdx +p −m2p + 2

∫

R

(ux)2p+3umdx

=∫

R

(ux)2p+1(

k

m + 1um+1 + um+2

)dx

−∫

R

(ux)2p+1Λ−2[εutx +

k

m + 1um+1 + um+2 +

(m +

12

)(umu2x

)+12∂x

[(um)xu

2x

]]dx.

(3.42)

Applying the Holder’s inequality yields that

1 − ε2p + 2

d

dt

∫

R

(ux)2p+2dx ≤{ε

(∫

R

|utxxx|2p+2dx)1/(2p+2)

+(∫

R

∣∣∣um+1∣∣∣2p+2

dx

)1/(2p+2)

+(∫

R

∣∣∣um+2∣∣∣2p+2

dx

)1/(2p+2)

+(∫

R

|G|2p+2dx)1/(2p+2)

}

×(∫

R

|ux|2p+2dx)(2p+1)/(2p+2)

+∣∣∣∣p −m2p + 2

∣∣∣∣‖ux‖L∞‖u‖mL∞

∫

R

|ux|2p+2dx(3.43)

or

(1 − ε) ddt

(∫

R

(ux)2p+2dx)1/(2p+2)

≤{ε

(∫

R

|utxxx|2p+2dx)1/(2p+2)

+(∫

R

∣∣∣um+1∣∣∣2p+2

dx

)1/(2p+2)

+(∫

R

∣∣∣um+2∣∣∣2p+2

dx

)1/(2p+2)

+(∫

R

|G|2p+2dx)1/(2p+2)

}

+∣∣∣∣p −m2p + 2

∣∣∣∣‖ux‖L∞‖u‖mL∞

(∫

R

|ux|2p+2dx)1/(2p+2)

,

(3.44)


where

G = Λ−2[εutx +

k

m + 1um+1 + um+2 +

(m +

12

)(umu2x

)+12∂x

[(um)xu

2x

]]. (3.45)

Since ‖f‖Lp → ‖f‖L∞ as p → ∞ for any f ∈ L∞ ⋂L2, integrating both sides of (3.44) with

respect to t and taking the limit as p → ∞ result in the estimate

(1 − ε)‖ux‖L∞ ≤ (1 − ε)‖u0x‖L∞

+∫ t

0

[ε‖utxxx‖L∞ + c

(‖u‖m+1

L∞ + ‖u‖m+2L∞ + ‖G‖L∞

)+12‖u‖mL∞‖ux‖2L∞

]dτ.

(3.46)

Using the algebraic property ofHs0(R)with s0 > 1/2 and (3.40) yields that

‖u‖m+2L∞ ≤ c‖u‖m+2

H1 ≤ c, (3.47)

‖G‖L∞ ≤ c‖G‖H1/2+

= c∥∥∥∥Λ

−2[εutx + um+1 + um+2 +

(m +

12

)(umu2x

)+12∂x

[(um)xu

2x

]]∥∥∥∥H1/2+

≤ c(∥∥∥Λ−2uxt

∥∥∥H1/2+

+∥∥∥Λ−2

(umu2x

)∥∥∥H1/2+

+∥∥∥Λ−2∂x

[(um)xu

2x

]∥∥∥H1/2+

)+ c

≤ c(‖ut‖L2 +

∥∥∥umu2x∥∥∥H0

+∥∥∥(um)xu

2x

∥∥∥H0

)+ c

≤ c(‖ut‖L2 + ‖u‖mL∞‖ux‖L∞‖u‖H1 + ‖ux‖2L∞‖u‖m−1

L∞ ‖u‖H1

)+ c

≤ c(‖ut‖L2 + ‖ux‖L∞ + ‖ux‖2L∞

)+ c

≤ c(1 + ‖ux‖2L∞

),

(3.48)

where we have used (3.16) and (3.40). Using (3.48), we have

∫ t

0‖G‖L∞dτ ≤ c

∫ t

0

(1 + ‖ux‖2L∞

)dτ, (3.49)

where c is a constant independent of ε. Moreover, for any fixed r ∈ (1/2, 1), there exists aconstant cr such that ‖utxxx‖L∞ ≤ cr‖utxxx‖Hr ≤ cr‖ut‖Hr+3 . Using (3.16) and (3.40) yields that

‖utxxx‖L∞ ≤ c‖u‖Hr+4

(1 + ‖ux‖2L∞

). (3.50)

Making use of the Gronwall’s inequality to (3.15) with q = r + 3, u = uε and (3.40) gives riseto

‖u‖2Hr+4 ≤(∫

R

(Λr+4u0

)2+ ε

(Λr+3u0xx

)2)exp

[c

∫ t

0

(1 + ‖ux‖2L∞

)dτ

]. (3.51)


From (3.36), (3.37), (3.50), and (3.51), one has

‖utxxx‖L∞ ≤ cε(s−r−4)/4(1 + ‖ux‖2L∞

)exp

[c

∫ t

0

(1 + ‖ux‖2L∞

)dτ

]. (3.52)

For ε < 1/4, it follows from (3.46), (3.49), and (3.52) that

‖ux‖L∞ ≤ ‖u0x‖L∞

+ c∫ t

0

[ε(s−r)/4

(1 + ‖ux‖2L∞

)exp

(c

∫ τ

0

(1 + ‖ux‖2L∞

)dς

)+ 1 + ‖ux‖2L∞

]dτ.

(3.53)

It follows from the contraction mapping principle that there is a T > 0 such that theequation

‖W‖L∞ = ‖u0x‖L∞

+ c∫ t

0

[(1 + ‖W‖2L∞

)exp

(c

∫ τ

0

(1 + ‖W‖2L∞

)dς

)+ 1 + ‖W‖2L∞

]dτ

(3.54)

has a unique solutionW ∈ C[0, T]. Using the Theorem presented at page 51 in Li and Olver[29] or Theorem II in section I.1 presented in [42] yields that there are constants T > 0 andc > 0, which are independent of ε, such that ‖ux‖L∞ ≤ W(t) for arbitrary t ∈ [0, T], whichleads to the conclusion of Lemma 3.9.

Lemma 3.10 (Li and Olver [29]). If u and f are functions inHq+1 ∩ {‖ux‖L∞ <∞}, then

∣∣∣∣

∫

R

ΛquΛq(uf)xdx

∣∣∣∣ ≤

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

cq∥∥f

∥∥Hq+1‖u‖2Hq , q ∈

(12, 1

],

cq(∥∥f

∥∥Hq+1‖u‖Hq‖u‖L∞

+∥∥fx

∥∥L∞

‖u‖2Hq +∥∥f

∥∥Hq‖u‖Hq‖ux‖L∞

), q ∈ (0,∞).

(3.55)

Lemma 3.11. For u, v ∈ Hs(R) with s > 3/2,w = u−v, q > 1/2, and a natural number n, it holdsthat

∣∣∣∣

∫

R

ΛswΛs(un+1 − vn+1

)

xdx

∣∣∣∣ ≤ c(‖w‖Hs‖w‖Hq‖v‖Hs+1 + ‖w‖2Hs

). (3.56)

The proof of this Lemma can be found in [38].


Lemma 3.12. For problem (3.34), s > 3/2 and u0 ∈ Hs(R), there exist two positive constants c andM, which are independent of ε, such that the following inequalities hold for any sufficiently small εand t ∈ [0, T)

‖uε‖Hs ≤Mect, (3.57)

‖uε‖Hs+k1 ≤ ε−k1/4Mect, k1 > 0, (3.58)

‖uεt‖Hs+k1 ≤ ε−(k1+1)/4Mect, k1 > −1. (3.59)

Proof. If s > 3/2, u0 ∈ Hs, we obtain that

u0 ∈ Hs1 with 1 ≤ s1 ≤ 32,

‖u0x‖L∞ ≤ c‖u0x‖H1/2+ ≤ c‖u0‖Hs ≤ c.(3.60)

From Lemma 3.9, we know that there exist two constants T and c (both independent of ε)such that

‖uεx‖L∞ ≤ c for any t ∈ [0, T). (3.61)

Applying the inequality (3.15) with q + 1 = s and the bounded property of solution u(see (3.40) and (3.60)), we have

∫

R

(Λsuε)2dx ≤

∫

R

[(Λsuε0)

2 + ε(Λs−1uε0xx

)2]dx + c

∫ t

0‖uε‖2Hsdτ,

= A + c∫ t

0‖uε‖2Hsdτ,

(3.62)

where

A =∫

R

[(Λsuε0)

2 + ε(Λs−1uε0xx

)2]dx ≤ ‖uε0‖2Hs + ‖uε0‖2Hs+1

≤ c + cεε−1/2 ≤ 2c,(3.63)

in which we have used (3.36) and (3.37).From (3.61) and (3.62) and using the Gronwall’s inequality, we get the following:

‖uε‖Hs ≤ 2cect, (3.64)

from which we know that (3.57) holds.In a similar manner, for q + 1 = s + k1 and k1 > 0, applying (3.40) and (3.60) to (3.15),

we have

‖uε‖2Hs+k1 ≤(cε−k1/2 + cε−(k1+1)/2ε

)+ c

∫ t

0‖uε‖2Hs+k1dτ, (3.65)

which results in (3.58) by using Gronwall’s inequality.


From (3.16), for q = s + k1, we have

(1 − 2ε)‖uεt‖Hs+k1 ≤ c‖uε‖Hs+k1+1 , (3.66)

which leads to (3.59) by (3.58).

Lemma 3.13. If 1/2 < q < min{1, s − 1} and s > 3/2, then for any functions w, f defined on R, itholds that

∣∣∣∣

∫

R

ΛqwΛq−2(wf)xdx

∣∣∣∣ ≤ c‖w‖2Hq

∥∥f∥∥Hq , (3.67)

∣∣∣∣

∫

R

ΛqwΛq−2(wxfx)xdx

∣∣∣∣ ≤ c‖w‖2Hq

∥∥f∥∥Hs. (3.68)

The proof of this lemma can be found in [38].Our next step is to demonstrate that uε is a Cauchy sequence. Let uε and uδ be solutions

of problem (3.34), corresponding to the parameters ε and δ, respectively, with 0 < ε < δ < 1/4,and let w = uε − uδ. Then w satisfies the problem

(1 − ε)wt − εwxxt + (δ − ε)(uδt + uδxxt)

=(1 − ∂2x

)−1[ − εwt + (δ − ε)uδt − k

m + 1∂x

(um+1ε − um+1

δ

)− ∂x

(um+2ε − um+2

δ

)

− ∂x[∂x

(um+1ε

)∂xw + ∂x

(um+1ε − um+1

ε

)∂xuδ

]

+[umε uεxuεxx − umδ uδxuδxx

]] − 1m + 2

∂x(um+2ε − um+2

δ

),

(3.69)

w(x, 0) = w0(x) = uε0(x) − uδ0(x). (3.70)

Lemma 3.14. For s > 3/2, u0 ∈ Hs(R), there exists T > 0 such that the solution uε of (3.34) isa Cauchy sequence in C([0, T];Hs(R))

⋂C1([0, T];Hs−1(R)).

Proof. For qwith 1/2 < q < min{1, s− 1}, multiplying both sides of (3.69) by ΛqwΛq and thenintegrating with respect to x give rise to

12d

dt

∫

R

[(1 − ε)(Λqw)2 + ε(Λqwx)

2]dx

= (ε − δ)∫

R

(Λqw)[(Λquδt) + (Λquδxxt)]dx

− ε∫

R

ΛqwΛq−2wtdx + (δ − ε)∫

R

ΛqwΛq−2uδtdx

− 1m + 2

∫

R

(Λqw)Λq(um+2ε − um+2

δ

)

xdx

− k

m + 1

∫

R

ΛqwΛq−2(um+1ε − um+1

δ

)

xdx


−∫

R


δ

)

xdx

−∫

R

ΛqwΛq−2[∂x

(um+1ε

)∂xw

]

xdx

−∫

R

ΛqwΛq−2[∂x

(um+1ε − um+1

δ

)∂xuδ

]

xdx

+∫

R

ΛqwΛq−2[umε uεxuεxx − umδ uδxuδxx]dx.

(3.71)

It follows from the Schwarz inequality that

d

dt

∫[(1 − ε)(Λqw)2 + ε(Λqwx)

2]dx

≤ c{‖Λqw‖L2

[(δ − ε)(‖Λquδt‖L2 + ‖Λquδxxt‖L2)

+ε∥∥∥Λq−2wt

∥∥∥L2

+ (δ − ε)∥∥∥Λq−2uδt

∥∥∥L2

]

+∣∣∣∣

∫

R

ΛqwΛq(um+2ε − um+2

δ

)

xdx

∣∣∣∣

∣∣∣∣

∫ΛqwΛq−2

(um+1ε − um+1

δ

)

xdx

∣∣∣∣

+∣∣∣∣

∫ΛqwΛq−2

(um+2ε − um+2

δ

)

xdx

∣∣∣∣ +∣∣∣∣

∫

R

ΛqwΛq−2[∂x

(um+1ε

)∂xw

]

xdx

∣∣∣∣

+∣∣∣∣

∫

R

ΛqwΛq−2[∂x

(um+1ε − um+1

δ

)∂xuδ

]

xdx

∣∣∣∣

+∣∣∣∣

∫

R

ΛqwΛq−2[umε uεxuεxx − umδ uδxuδxx]dx

∣∣∣∣}.

(3.72)

Using the first inequality in Lemma 3.10, we have

∣∣∣∣

∫

R

ΛqwΛq(um+2ε − um+2

δ

)

xdx

∣∣∣∣ =∣∣∣∣

∫

R

ΛqwΛq(wgm+1)xdx

∣∣∣∣

≤ c‖w‖2Hq

∥∥gm+1∥∥Hq+1 ,

(3.73)

where gm+1 =∑m+1

j=0 um+1−jε u

j

δ. For the last three terms in (3.72), using Lemmas 3.1 and 3.13,

1/2 < q < min{1, s − 1}, s > 3/2, the algebra property of Hs0 with s0 > 1/2 and (3.40), wehave

∣∣∣∣

∫

R

ΛqwΛq−2(∂x

(um+1ε

)∂xw

)

xdx

∣∣∣∣ ≤ c‖w‖2Hq‖uε‖m+1Hs , (3.74)

∣∣∣∣

∫

R

ΛqwΛq−2(∂x

(um+1ε − um+1

δ

)∂xuδ

)

xdx

∣∣∣∣

≤ c‖w‖Hq‖uδ‖Hs

∥∥∥um+1ε − um+1

δ

∥∥∥Hq

≤ c‖w‖2Hq‖uδ‖Hs,

Abstract and Applied Analysis 17∣∣∣∣

∫

R

ΛqwΛq−2[umε uεxuεxx − umδ uδxuδxx]dx

∣∣∣∣

≤ c‖w‖Hq

∥∥∥(umε − umδ

)(u2εx

)

x+ umδ

[u2εx − u2δx

]

x

∥∥∥Hq−2

≤ c‖w‖Hq

(∥∥∥(umε − umδ

)(u2εx

)

x

∥∥∥Hq−1

+∥∥∥umδ

[u2εx − u2δx

]

x

∥∥∥Hq−2

)

≤ c‖w‖Hq

(∥∥umε − umδ∥∥Hq

∥∥∥(u2εx

)

x

∥∥∥Hq−1

+∥∥umδ

∥∥Hs

∥∥∥[u2εx − u2δx

]

x

∥∥∥Hq−2

)

≤ c‖w‖Hq

(‖w‖Hq

∥∥gm−1∥∥Hq‖u‖2Hs +

∥∥umδ∥∥Hs‖uεx + uδx‖Hq‖w‖Hq

)

≤ c‖w‖2Hq

(∥∥gm−1∥∥Hq‖u‖2Hs +

∥∥umδ∥∥Hs‖uεx + uδx‖Hq

).

(3.75)

Using (3.67), we derives that the inequality

∣∣∣∣

∫

R


δ

)

xdx

∣∣∣∣ =∣∣∣∣

∫

R

ΛqwΛq−2(wgm+1)xdx

∣∣∣∣

≤ c∥∥gm+1∥∥Hq‖w‖2Hq

(3.76)

holds for some constant c, where gm+1 =∑m+1

j=0 um+1−jε u

j

δ. Using the algebra property of Hq

with q > 1/2, q + 1 < s and Lemma 3.12, we have ‖gm‖Hq+1 ≤ c for t ∈ (0, T]. Then it followsfrom (3.57)–(3.59) and (3.73)–(3.76) that there is a constant c depending on T such that theestimate

d

dt

∫

R

[(1 − ε)(Λqw)2 + ε(Λqwx)

2]dx ≤ c

(δγ‖w‖Hq + ‖w‖2Hq

)(3.77)

holds for any t ∈ [0, T), where γ = 1 if s ≥ 3 + q and γ = (1 + s − q)/4 if s < 3 + q. Integrating(3.77)with respect to t, one obtains the estimate

12‖w‖2Hq =

12

∫

R

(Λqw)2dx

≤∫

R

[(1 − ε)(Λqw)2 + ε(Λqw)2

]dx

≤∫

R

[(Λqw0)

2 + ε(Λqw0x)2]dx + c

∫ t

0

(δγ‖w‖Hq + ‖w‖2Hq

)dτ.

(3.78)

Applying the Gronswall inequality, (3.37) and (3.39) yields that

‖u‖Hq ≤ cδ(s−q)/4ect + δγ(ect − 1), (3.79)

for any t ∈ [0, T).


Multiplying both sides of (3.69) by ΛswΛs and integrating the resultant equation withrespect to x, one obtains that

12d

dt

∫

R

[(1 − ε)(Λsw)2 + ε(Λswx)

2]dx

= (ε − δ)∫

R

(Λsw)[(Λsuδt) + (Λsuδxxt)]dx

− ε∫

R

ΛswΛs−2wtdx + (δ − ε)∫

R

ΛswΛs−2uδtdx

− k

m + 1

∫

R

(Λsw)Λs(um+1ε − um+1

δ

)

xdx

− 1m + 2

∫

R

(Λsw)Λs(um+2ε − um+2

δ

)

xdx

−∫

R

ΛswΛs−2(um+2ε − um+2

δ

)

xdx

−∫

R

ΛswΛs−2[∂x

(um+1ε

)∂xw

]

xdx

−∫

R

ΛswΛs−2[∂x

(um+1ε − um+1

δ

)∂xuδ

]

xdx

+∫

R

ΛswΛs−2[umε uεxuεxx − umδ uδxuδxx]dx.

(3.80)

From Lemma 3.13, we have

∣∣∣∣

∫

R

ΛswΛs−2(um+2ε − um+2

δ

)

xdx

∣∣∣∣ ≤ c3∥∥gm+1

∥∥Hs‖w‖2Hs. (3.81)

From Lemma 3.11, it holds that

∣∣∣∣

∫

R

ΛswΛs(um+2ε − um+2

δ

)

xdx

∣∣∣∣ ≤ c(‖w‖Hs‖w‖Hq‖uδ‖Hs+1 + ‖w‖2Hs

). (3.82)

Using the Cauchy-Schwartz inequality and the algebra property of Hs0 with s0 > 1/2, fors > 3/2, we have

∣∣∣∣

∫

R

ΛswΛs−2[∂x

(um+1ε

)∂xw

]

xdx

∣∣∣∣ =∣∣∣∣

∫

R

ΛqwΛs−2[∂x

(um+1ε

)∂xw

]

xdx

∣∣∣∣

≤ c‖Λsw‖L2

∥∥∥Λs−2[∂x

(um+1ε

)∂xw

]

x

∥∥∥L2

≤ c‖w‖Hq

∥∥∥∂x(um+1ε

)∂xw

∥∥∥Hs−1

≤ c‖uε‖m+1Hs ‖w‖2Hs

∣∣∣∣

∫

R

ΛswΛs−2[∂x

(um+1ε − um+1

δ

)∂xuδ

]

xdx

∣∣∣∣ ≤ c‖w‖Hs

∥∥∥Λs−2[∂x

(um+1ε − um+1

δ

)∂xuδ

]

x

∥∥∥L2

≤ c‖uδ‖Hs

∥∥gm∥∥Hs‖w‖2Hs,

(3.83)


∣∣∣∣

∫

R

ΛswΛs−2[umε uεxuεxx − umδ uδxuδxx]dx

∣∣∣∣

≤ c‖w‖Hs

(∥∥∥(umε − umδ

)(u2εx

)

x

∥∥∥Hs−2

+∥∥∥umδ

[u2εx − u2δx

]

x

∥∥∥Hs−2

)

≤ c‖w‖Hs

(∥∥(umε − umδ)∥∥

Hs

∥∥∥(u2εx

)

x

∥∥∥Hs−2

+∥∥umδ

∥∥Hs

∥∥∥[u2εx − u2δx

]

x

∥∥∥Hs−2

)

≤ c‖w‖2Hs,

(3.84)

in which we have used Lemma 3.1 and the bounded property of ‖uε‖Hs and ‖uδ‖Hs (seeLemma 3.12). It follows from (3.80)–(3.84) and (3.57)–(3.59) and (3.79) that there exists aconstant c depending onm such that

d

dt

∫

R

[(1 − ε)(Λsw)2 + ε(Λswx)

2]dx

≤ 2δ(‖uδt‖Hs + ‖uδxxt‖Hs +

∥∥∥Λs−2wt

∥∥∥L2

+∥∥∥Λs−2uδt

∥∥∥)‖w‖Hs

+ c(‖w‖2Hs + ‖w‖Hq‖w‖Hs‖uδ‖Hs+1

)

≤ c(δγ1‖w‖Hs + ‖w‖2Hs

),

(3.85)

where γ1 = min(1/4, (s−q−1)/4) > 0. Integrating (3.85)with respect to t leads to the estimate

12‖w‖2Hs ≤

∫

R

[(1 − ε)(Λsw)2 + ε(Λswx)

2]dx

≤∫

R

[(Λsw0)

2 + ε(Λsw0x)2]dx + c

∫ t

0

(δγ1‖w‖Hs + ‖w‖2Hs

)dτ.

(3.86)

It follows from the Gronwall inequality and (3.86) that

‖w‖Hs ≤(2∫

R

[(Λsw0)

2 + ε(Λsw0x)2]dx

)1/2

ect + δγ1(ect − 1

)

≤ c1(‖w0‖Hs + δ3/4

)ect + δγ1

(ect − 1

),

(3.87)

where c1 is independent of ε and δ.Then (3.39) and the above inequality show that

‖w‖Hs −→ 0 as ε −→ 0, δ → 0. (3.88)


Next, we consider the convergence of the sequence {uεt}. Multiplying both sides of (3.69) byΛs−1wtΛs−1 and integrating the resultant equation with respect to x, we obtain

(1 − ε)‖wt‖2Hs−1 +1

m + 2

∫

R

(Λs−1wt

)Λs−1

(um+2ε − um+2

δ

)

xdx

+∫

R

[−ε

(Λs−1wt

)(Λs−1wxxt

)+ (δ − ε)

(Λs−1wt

)Λs−1(uδt + uδxxt)

]dx

=∫

R

(Λs−1wt

)Λs−3

[−εwt + (δ − ε)uδt − k

m + 1∂x

(um+1ε − um+1

δ

)− ∂x

(um+2ε − um+2

δ

)

− ∂x[∂x

(um+1ε

)∂xw + ∂x

(um+1ε − um+1

ε

)∂xuδ

]

+[umε uεxuεxx − umδ uδxuδxx

]]dx.

(3.89)

It follows from (3.57)–(3.60) and the Schwartz inequality that there is a constant cdepending on T andm such that

(1 − ε)‖wt‖2Hs−1 ≤ c(δ1/2 + ‖w‖Hs + ‖w‖s−1

)‖wt‖Hs−1 + ε‖wt‖2Hs−1 . (3.90)

Hence,

12‖wt‖2Hs−1 ≤ (1 − 2ε)‖wt‖2Hs−1

≤ c(δ1/2 + ‖w‖Hs + ‖w‖Hs−1

)‖wt‖Hs−1 ,

(3.91)

which results in

12‖wt‖Hs−1 ≤ c

(δ1/2 + ‖w‖Hs + ‖w‖Hs−1

). (3.92)

It follows from (3.79) and (3.88) that wt → 0 as ε, δ → 0 in the Hs−1 norm. Thisimplies that uε is a Cauchy sequence in the spaces C([0, T);Hs(R)) and C([0, T);Hs−1(R)),respectively. The proof is completed.

Proof of Theorem 2.1. We consider the problem

(1 − ε)ut − εutxx =(1 − ∂2x

)−1[− k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

−(m + 1)∂x(umu2x

)+ umuxuxx

],

u(0, x) = uε0(x).

(3.93)


Letting u(t, x) be the limit of the sequence uε and taking the limit in problem (3.93) as ε → 0,from Lemma 3.14, it is easy to see that u is a solution of the problem

ut =(1 − ∂2x

)−1[− k

m + 1

(um+1

)

x− m + 3m + 2

(um+2

)

x+

1m + 2

∂3x

(um+2

)

−(m + 1)∂x(umu2x

)+ umuxuxx

],

u(0, x) = u0(x),

(3.94)

and hence u is a solution of problem (3.94) in the sense of distribution. In particular, if s ≥ 4,u is also a classical solution. Let u and v be two solutions of (3.94) corresponding to the sameinitial data u0 such that u, v ∈ C([0, T);Hs(R)). Then w = u − v satisfies the Cauchy problem

wt =(1 − ∂2x

)−1{∂x

[− k

m + 1wgm − m + 3

m + 2wgm+1 +

1m + 2

∂2x(wgm+1

)

−∂x(um+1

)∂xw − ∂x

(um+1 − vm+1

)∂xv

]

+umuxuxx − vmvxvxx},

w(0, x) = 0.

(3.95)

For any 1/2 < q < min{1, s−1}, applying the operatorΛqwΛq to both sides of equation(3.95) and integrating the resultant equation with respect to x, we obtain the equality

12d

dt‖w‖2Hq =

∫

R

(Λqw)Λq−2{∂x

[− k

m + 1wgm − m + 3

m + 2wgm+1 +

1m + 2

∂2x(wgm+1

)

−∂x(um+1

)∂xw − ∂x

(um+1 − vm+1

)∂xv

]

+umuxuxx − vmvxvxx}dx.

(3.96)

By the similar estimates presented in Lemma 3.14, we have

d

dt‖w‖2Hq ≤ c‖w‖2Hq . (3.97)

Using the Gronwall inequality leads to the conclusion that

‖w‖Hq ≤ 0 × ect = 0 (3.98)

for t ∈ [0, T). This completes the proof.


4. Global Existence of Strong Solutions

We study the differential equation

pt = um+1(t, p), t ∈ [0, T),

p(0, x) = x.(4.1)

Motivated by the Lagrangian viewpoint in fluid mechanics, by which one looks at themotion of individual fluid particles (see [43]), we state the following Lemma.

Lemma 4.1. Let u0 ∈ Hs, s ≥ 3 and let T > 0 be the maximal existence time of the solution toproblem (2.2). Then problem (4.1) has a unique solution p ∈ C1([0, T)×R). Moreover, the map p(t, .)is an increasing diffeomorphism of R with px(t, x) > 0 for (t, x) ∈ [0, T) × R.

Proof. From Theorem 2.1, we have u(t, x) ∈ C([0, T);Hs(R))⋂C1([0, T);Hs−1(R)) and

Hs(R) ∈ C1(R), where the Sobolev imbedding theorem is used. Thus, we conclude thatboth functions u(t, x) and ux(t, x) are bounded, Lipschitz in space and C1 in time. Using theexistence and uniqueness theorem of ordinary differential equations derives that problem(4.1) has a unique solution p ∈ C1([0, T) × R).

Differentiating (4.1) with respect to x yields that

d

dtpx = (m + 1)umux

(t, p

)px, t ∈ [0, T), b /= 0,

px(0, x) = 1,(4.2)

which leads to

px(t, x) = exp

(∫ t

0(m + 1)umux

(τ, p(τ, x)

)dτ

). (4.3)

For every T ′ < T , using the Sobolev imbedding theorem yields that

sup(τ,x)∈[0,T ′)×R

|ux(τ, x)| <∞. (4.4)

It is inferred that there exists a constant K0 > 0 such that px(t, x) ≥ e−K0t for (t, x) ∈[0, T) × R. It completes the proof.

The next Lemma is reminiscent of a strong invariance property of the Camassa-Holmequation (the conservation of momentum [44, 45]).

Lemma 4.2. Let u0 ∈ Hs with s ≥ 3, and let T > 0 be the maximal existence time of the problem(2.2), it holds that

y(t, p(t, x)

)p2x(t, x) = y0(x)e

∫ t0mu

muxdτ , (4.5)

where (t, x) ∈ [0, T) × R and y := u − uxx + k/2(m + 1).


Proof. We have

d

dt

[y(t, p(t, x)

)p2x(t, x)

]= ytp2x + 2ypxpxt + yxptp2x

= ytp2x + 2y(m + 1)umuxp2x + um+1yxp

2x

=[yt + kumux + (m + 2)umuxy + yxum+1

]p2x +mu

muxyp2x

=[ut − utxx + kumux + (m + 2)umux(u − uxx) + um+1(ux − uxxx)

]p2x

+mumuxyp2x= mumuxyp2x.

(4.6)

Using px(0, x) = 1 and solving the above equation, we complete the proof of this lemma.

Lemma 4.3. If u0 ∈ Hs, s ≥ 3/2, such that (1 − ∂2x)u0 + k/2(m + 1) ≥ 0, then the solution ofproblem (2.2) satisfies the following:

‖ux‖L∞ ≤ ‖u‖L∞ +k

2(m + 1)≤ c. (4.7)

Proof. Using u0−u0xx+k/2(m+1) ≥ 0, it follows from Lemma 4.2 that u−uxx+k/2(m+1) ≥ 0.Letting Y1 = u − uxx, we have

u =12e−x

∫x

−∞eηY1

(t, η

)dη +

12ex

∫∞

x

e−ηY1(t, η

)dη, (4.8)

from which we obtain that

∂xu(t, x) = −12

(e−x

∫x

−∞eηY1

(t, η

)dη + ex

∫∞

x

e−ηY1(t, η

)dη

)+ ex

∫∞

x

e−ηY1(t, η

)dη

= −u(t, x) + ex∫∞

x

e−ηY1(t, η

)dη

= −u(t, x) + ex∫∞

x

e−η(Y1

(t, η

)+

k

2(m + 1)

)dη − k

2(m + 1)ex

∫∞

x

e−ηdη

= −u(t, x) + ex∫∞

x

e−η(y(t, η

))dη − k

2(m + 1)

≥ −u(t, x) − k

2(m + 1).

(4.9)

On the other hand, we have

∂xu(t, x) =12

(e−x

∫x

−∞eηY1

(t, η

)dη + ex

∫∞

x

e−ηY1(t, η

)dη

)− e−x

∫x

−∞eηY1

(t, η

)dη

= u(t, x) − e−x∫x

−∞eηY1

(t, η

)dη


= u(t, x) − e−x∫x

−∞eη

(Y1

(t, η

)+

k

2(m + 1)

)dη +

k

2(m + 1)e−x

∫x

−∞eηdη

= u(t, x) − e−x∫x

−∞eηy

(t, η

)dη +

k

2(m + 1)

≤ u(t, x) + k

2(m + 1).

(4.10)

The inequalities (3.40), (4.9), and (4.10) derive that (4.7) is valid.

Proof of Theorem 2.2. Noting Remarks 3.6 and 3.8, ‖u‖H1 ≤ c and taking q + 1 = s in inequality(3.15), we have

‖u‖2Hs ≤ ‖u0‖2Hs + c∫ t

0‖u‖2Hs

(‖ux‖L∞ + ‖ux‖2L∞

)dτ, (4.11)

from which we obtain that

‖u‖Hs ≤ ‖u0‖Hsec∫ t0(‖ux‖L∞+‖ux‖2L∞ )dτ . (4.12)

Applying Lemma 4.3 derives

‖u‖Hs ≤ ‖u0‖Hse(c+c2)t. (4.13)

From Theorem 2.1 and (4.13), we know that the result of Theorem 2.2 holds.

Acknowledgment

This work is supported by the Applied and Basic Project of Sichuan Province (2012JY0020).

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