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HUT DEPARTMENT OF MATH.
APPLIED--------------------------------------------------------------------------------------------------------
--IDE MODERNIZED PROGRAM
PARTIAL DIFFERENTIAL EQUATIONS (PDE)
ORTHOGONAL FUNCTIONS - GENERALEXPANSION PROBLEM (PHOTO, P. 552)
PhD. NGUYEN QUO C LAN 12/2009)
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CONTENTS-------------------------------------------------------------------------------------------------------------------------------
3. Orthogonal with respect to weight function.
Generalizedfourier series
1. Recall: Heat problem. Orthogonal feature of basic solution2.
Example: Heat problem with radiating end
4. Sturn Liouville Theorem5. Application to the Heat problem
with radiating end
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RECALL: SEPARATION FOR PDE (HEAT
PROBLEM)-----------------------------------------------------------------------------------------------------------------------------------
Heat problem: ( ) ( ) ( ) ( )xuxututux
ua
t
u02
2
0,,0,,0, ===
=
Separation: ( ) ( ) ( ) ( ) 1,sin,, 2 == nnxebtxutTxXtxu tannn(
) ( )
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EXAMPLE: HEAT PROBLEM WITH A RADIATING
END-----------------------------------------------------------------------------------------------------------------------------------
Heat problem: ( ) ( ) ( )...,,,0,0,2
2
thutx
utu
x
ua
t
u =
=
=
Separation: ( ) ( ) ( ) ( ) ( ) huXXXtTxXtxu === /// &0,( )
0cossinsin&0,2 =+=>= hxBxX
?Root:1
&,tantan ====== z
h
zzz
hh
nnn
zz =rootsofnumberInfinite
( )
=
=1 sin,
2
nn
ta
n xebtxu
n
( ) ( ) ?sin0,1
===
=n
n
nn bxfxbxu z
zy =
zy tan=
2
2
3
Is the system {sinnx} orthogonal?
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ORTHGONAL SYSTEM. APPLICATION: FOURIER
SERIES-----------------------------------------------------------------------------------------------------------------------------------
Fourier series: Given a orthogonal system {n} over (a, b) and( )
( ) ( )
200
,,
n
nn
i
b
a
ni
b
a
n
i
ii
fafbaxxaxf
===
=
=
( ) ( ) tscoefficienFouriersincos,21:2,0, 222 ==== mxnxba
Definition: A system {n(x)} is called orthogonal over (a, b)
if:
Example: The trigonometric system {cos0x = 1, cosx, cos2x,
cosnx, , sinx, , sinmx, } is orthogonal over (0, 2) as
( ) ( ) ( ) ndxxmndxxx n
b
a
nmn
b
a
mn 0&,0,22 ===
1,0sincossin1cos12
0
2
0
2
0
=== mnmxdxnxmxdxnxdx
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ORTHGONAL WITH RESPECT TP THE WEIGHT
FUNCTION-------------------------------------------------------------------------------------------------------------------------------------------
Generalized Fourier series: If the system {n} is orthogonal
withrespect to the weight function p(x) over (a, b) and we have
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
:,......2
00
=+++= b
a
n
b
a
n
nnn
dxxxp
dxxxpxf
abaxxaxaf
Generalized Fourier coefficients
Definition: A system {n(x)} is called orthogonal with respect
tothe weight function p(x) > 0 over (a, b) if:
( ) ( ) ( )
=
= mn
mndxxxxp
b
a
mn,0
,0
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STURM LIOUVILLE
THEOREM-------------------------------------------------------------------------------------------------------------------------------------------
Separation for a lot of boundary problems with 2nd order
linearPDE gives y// + y = 0, y(a) = y(b) = 0: r(x) 1 > 0, q(x) =
0, p(x) 1, n1 = n2 = 0 By Sturm Liouville: Solution {yn} is
orthogonal
Consider the differential equation with the boundary
conditions:
If 1, 2 n are distinct values of the parameter forwhich this
problem has nontrivial solutions y1, y2 yn thenthe system {yn} is
orthogonal with respect to the weight p(x)
( )[ ] ( ) ( )[ ] ( ) ( ) ( )
( ) ( )( )barrqpnmnm
bynbym
aynaymbaxyxpxqyxr
dx
d
,over0;continuous,,;0,0where0
0&,,0
22
22
21
21
/
22
/11/
>>+>+
=
==++
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HEAT PROBLEM WITH A RADIATING
END-----------------------------------------------------------------------------------------------------------------------------------
We have already the solution: ( )
=
=1
sin,2
n
nta
n xebtxu n
From the initial condition: ( ) ( ) ( ) 2,0,sin0,1
==
=
xxfxbxu
n
nn
As {sinnx} is orthogonal: ( ) xxxfb nnn 2sinsin,=
Return to the heat problem with radiating end. Separation
gives:
By Sturn Liouville, the basic solution {Xn(x)} = {sinnx}
isorthogonal with respect to the weight function p(x) = 1. So
thesystem {sinnx} is simple orthogonal.
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) !1,,0,1,1,0,01with
Liouville-Sturm:0,00,0
2211
/2//
======>=
=+==+
nhmnmxpxqxr
XhXXxXxX
