1

Sturm Liouville Problem

1

1) 1) D.E.2) D.E. B.V.P. 2) [ Textbook 3]

(i) Linear first-order equation: 0' kyy , k is a constant. General solution: kxecy 1

(ii) Linear second-order equation: 0" yy , 0 .

General solution: xcxcy sincos 21

(iii) Linear second-order equation: ,0" yy 0 . The general solution of this differential equation has two real forms. General solution:

xcxcy sinhcosh 21

xx ececy 21

It should be noted that often in practice the exponential form is used when the domain of x is an infinite or semi-infinite interval* and the hyperbolic form is used when the domain of is a finite interval.

(iv) Cauchy-Euler equation: 0'" 22 yxyyx General solution:

:0 xcxcy 21 :0 xccy ln21

(v) Parametric Bessel equation: ,0)('" 2222 ynxxyyx ...,2,1,0n General solution:

)()( 21 xYcxJcy nn It is important that you recognize Bessel differential equation when 0n :

0'" 2 xyyxy General solution:

)()( 0201 xYcxJcy

Recall that )( xYn as 0x .

(vi) Legendres differential equation: ,0)1('2")1( 2 ynnxyyx ...,2,1,0n Particular solutions are the Legendre polynomials ),(xPy n

where

2

,1)(0 xPy ,)(1 xxPy ),13(21)( 22 xxPy

2

Ex: B.V.P. D.E.: y + y = 0 B.C.: y(0) = 0y (L) = 0 = 0 < 0 > 0 :

(1) = 0 y + y = 0 y = 0 y = c1 x + c2

B.C. c1 = c2 = 0 Trivial Solution: y = 0

(2) < 0 y = c1 cosh x + c2 sinh x

B.C.: y (0) = 0 c1 = 0y = c2 sinh x

B.C.: y (L) = 0 0 = c2 sinh L sinh L 0 c2 =

0y = 0

(3) > 0 y = c1 cos x + c2 sin x

B.C.: y (0) = 0 c1 = 0y = c2 sin x B.C.: y (L) = 0 0 = c2 sin L .. (1) c2 = 0 Trivial Solution sin L = 0 .. (2)

(2) L = n = 222

Ln

n = 1, 2, 3, [ n 0]

y = c2 sin( )xLn D.E.{sin( )x

Ln

n =

1, 2, 3, }[0, L] ** Eigenvalue and Eigenfunction []

= 222

Ln

n = 1, 2, 3, ... [Characteristic Value]

[Eigenvalue]

y = sin( )xL

n [Characteristic Function]

[Eigenfunction]

3

Ex. B.V.P. : D.E.: y+ y = 0 B.C.: y (0) = 0y (L) = 0

Eigenvalue: = 222

Ln

n = 0, 1, 2, 3, ...

Eigenfunction : y = c1 cos( )xLn

c1 0

** { cos( )xL

nn = 0, 1, 2, 3, ... }[0, L]

H.W. B.V.P y+ y = 0 y (0) = 0y ( ) = 0

[79 ] [80 ] [85 ] [86 ]

n =22 1

2n

n = 0, 1, 2, 3, ...

yn (x) = sin xn2

12

H.W. B.V.P. y+ y = 0y (0) = y (2 ) = 0 [78 ]

n = 4

2n n = 0, 1, 2, 3, ...

yn = cos 2nx

H.W. B.V.P. y+ y = 0y (1) = y (1)y(1) = y (1) [85 ]

n = (n )2 n = 0, 1, 2, 3, ...

yn = An cos n x + Bn sin n x H.W. B.V.P. y + 4y + ( +1)y = 0y(0) = y (1) = 0 [73 ]

n 321

n = tan 3n n > 3

yn = e2xsin ( 3n x)

H.W. 21( )

d dyxy x dx dx

= n (n +1)n

(1) y (1) = y ( ) = 0 D.E. (2) y (0) = 1y (1) = D.E. [81 ] (1) y = x (n + 1)

(2)

4

H.W. n a < x < aa y (a) = 1

0)1(222

2 ynndxdyx

dxydx < y

5

1 1 2 2 1 D.E. Trivial Solution y = 0

Regular S.L.P.

1) .........321 n n n

2) Nonzero Multiples 3) Linearly Independent 4) [a, b] xp nm yy , nm ,

2..........0

1.........0

nnn

mmm

yxpxqyxrdxd

yxpxqyxrdxd

B.C.

1 1

1 1

0 .................... 3

0 .................... 4m m

n n

y a y a

y a y a

2 2

2 2

0 ..................... 5

0 ..................... 6m m

n n

y b y b

y b y b

(1) ny 2 my

mnnmnmnm yxrdxdyyxr

dxdyyyxp

ax bx

............... 7

b

m n m na

m n n m m n n m

p x y y dx

r b y b y b y b y b r a y a y a y a y a

1 1, 2 2, 3456

)()( ayay nm 0)()( ayay mn .................. (8)

)()( byby nm 0)()( byby mn ................... (9)

(8)(9)(7)

6

0)( dxyyxpab

nm nm .................. (10)

Ex. B.V.P.:

0)1()1(,0)0(0

:.C.B:.E.D

yyyyy

Regular S.L.P. 1)( xr 0)( xq 1)( xp 11 01 12 12

0 Eigenvalue 2nn x nx tan 0x x n = 1, 2, 3, ...

Eigenfunction xy nsin n = 1, 2, 3, ...

......,3,2,1,sin nxn 1)( xp 0, 1 2 0nx tan 0x x D.E. Trivial Solution 0y =

0 Eigenvalue < 0 Eigenvalue Ex. B.V.P. 0)1(2 2 yyy 0)()0( yy

xexw 2)( 86

n nxecy xnn sin

0

0)( dxyyxw mn n m

H.W. B.V.P. 04 yyy 0)0( y 0)( y 87

22 1 4

2nn

ny 2 1sin

2nnc x

n = 0, 1, 2, 3, ...

H.W. B.V.P. 044 2 yyy 0)1( y 0)2(2)2( yy

21

1 41

2 87

1 xxecy 2 2

xexcy 2

1

2 )3(

H.W. Sturm-Liouville Problem 71 71

85

7

-Periodic Sturm-Liouville Problem

0)( xr ( ) 0p x bxa Two-Point

D.E. dxd [ yxr )( ] + [ )()( xpxq ] y = 0 ............. (1)

B.C. )()( byay )()( byay .............. (2) Periodic S.L.P.

Ex. B.V.P.: )()(),()(

,0:.C.B:.E.D

yyx

yyyy

0 > 0 < 0 i) 0 Eigenvalue 0 Eigenfuction by b ii) > 0

xay cos + xb sin ..................... (1)

B.C. )()( yy )sin(2 b = 0 ..................... (2)

B.C. )()( yy a2 )sin( = 0 ..................... (3)

> 0 (2)(3)

sina = b sin = 0 .................... (4)

sin 0a = b = 0 Trivial Solution y = 0

sin = 0a b Eigenvalue = 2n n = 1, 2, 3, ... .................... (5) Eigenfunction )(xyn = na )cos(nx + nb )sin(nx .................... (6)

na nb iii) < 0

y = xae + xbe .................... (7)

B.C. y )( = y )(

xae + be = ae + be .................... (8) B.C. y )( = y )(

ae be = ae be ........... (9)

(8) a =b (9)

2 ae = 2 ae .................... (10)

> 0

8

aeae (11). a = 0 b = 0 Trivial Solution [ Periodic S.L.P. Eigenvalue]

H.W. D.E. 2(1 ) '' ' 0, [ 1, 1]x y xy y x

(1) D.E. RegularPeriodic Singular S.L.P. (2) D.E. S.L. Form (3) D.E. Eigenfunction )(xn [87 ]

(1) D.E. Singular Sturm-Liouville Problem

(2) 01

'12

2

y

xyx

(3) [1, 1] )(xn 21( )

1p x

x

1

1( ) ( ) ( ) 0 ,m np x x x dx m n

H.W. B.V.P. '' 0, (0) ( ), '( ) '(0)y y y y L y L y [87 ]

2

02 220 1 ; cos sinn n n n nx nxn y y A BL LL

H.W. B.V.P. '' 0, ( 3 ) (3 ), '( 3 ) (3 )y y y y y y [85 ]

(1) 00 constant00 cy

(2) p

nn

2

n = 123

(3) cos sin3 3n n n

nx nxy A B

- [Singular Sturm-Liouville Problem]

)(xr > 0 )(xp > 0 a < x

9

2) 0)( br B.C.

0)()( 11 ayay . (3)

11 , 011 D.E. bx bounded

3) 0)()( brar ax bx B.C.D.E. ba, Bounded

Ex. Sturm-Liouville Problem 0)( yx

yx 0)()1( eyy

D.E 2 0x y xy y

Eq. 02 m 1) 0 21 ln cxcy

021 cc 0y Trivial Sol.

2) 0 1 2y c x c x

021 cc 0y .. Trivial Sol.

3) 0 )lnsin()lncos( 21 xcxcy

01 c sin2c = 0 2)( k 1, 2, 3,k Eigenvalues!

Ex. S.L.P. 0 yy 0)0()0( yy 0)1( y

1) 0 xx ececy 21

021 cc 0y .. Trivial Sol.

2) 0 xccy 21 21 cc

)1()( 10 xcxy Eigenfunction! (n = 0 CASE)

3) 0 xcxcy sincos 21

0sincos0

21

21

cccc

tan .. (1)

10

Eigenualue n (1) 4)12( 2

n

n 1n

Eigenfunctions 0,1],[sin AnxxAy nnn

Self-Adjoint Form Sturm-Liouville Form

0d r x y q x p x ydx

------------ (1)

Self-Adjoint Form

(1) D.E.

( ) ( ) 0y R x y Q x P x y ------------ (2)

dxxRe )( ------------ (3)

(2)(1)

dxxR

exr)(

)( ------------ (4)

)()()()()(

xrxQexQxqdxxR ------------ (5)

( )( ) ( ) ( ) ( )R x dxp x P x e P x r x ------------ (6)

(2) D.E. 0)()()()( yxdxcyxbyxa ------------ (7)

)()(

1 xFexa

dxab ------------ (8)

(7)(1) )()()( xFxaxr ------------ (9)

( )( ) ( ) ( ) ( )( )

c xq x r x c x F xa x

------------ (10)

)()()()()()( xFxdxr

xaxdxp ------------ (11)

** D.E. [(1)(2)(7)]a, b

** B.V.P. = D.E. + B.C. [ ax bx ] ** I.V.P. = D.E. + I.C. [] B.V.P.B.V.P.

11

Ex. D.E. 02 2 yxxyx

y Self-Adjoint Form

x

xR 2)( 2)( xxQ P(x) = x

r (x) = 22

xedx

x

4)()()( xxrxQxq p(x) = P(x) r (x) = x3

D.E. 0342 yxxyxdxd

Ex. Bessel Equation 0)( 2222 ynxyxyx n = 0123 Self-Adjoint

Form 222 )(,)(,)(,)( xxdnxcxxbxxa [ 2 ]

xe

xaxF

dxab 1)(

1)(/

r (x) = a (x) F (x) = x

q (x) = c (x) F (x)=x

n2

xxFxdxp )()()( D.E. Self-Adjoint Form

02

2

y

xnxyx

dxd

(1) Bessel D.E. xYcxJc nn 21

(2) x = 0 r (x) = 0 xJn xYn xJn 0x Bounded x = b B.C.

022 bJbJ nn Chain Rule

bnJxJdxd

n 022 B.C. bx ii

Eigenvalue bxii ...,3,2,1i

(3) ...,3,2,1, ixJ in [0, b] p (x) = x

0 dxxJxJx jb

a nin ji

...,3,2,1, ii B.C. Eigenvalues

12

Ex. Legendre D.E. ......,2,1,0,0121 2 nynnyxyx Self-Adjoint Form

1)(,0)(,2)(,1)( 2 xdxcxxbxxa

1

)(1)(

dxabe

xaxF [** 1 nn ]

21)()()( xxFxaxr 0)()()( xFxcxq 1)()()( xFxdxp Self-Adjoint Form

011 2 ynnyxdxd

(1) Legendre D.E. ......,2,1,0),( nxPn

(2) 011 rr 1x x = 1 B.C.

(3) ......,2,1,0),( nxPn [11] p (x) =1

1

10)()( dxxPxP nm nm

H.W. D.E. 02 yyxyx Self-Adjoint Form Sturm-Liouville

r (x) = x q (x) = 0 p (x) =x1 [77 ]

4 [Bessel and Legendre Series]

- [Fourier-Bessel Series] [A] (1) Bessel D.E. 0)(" 2222 nxyxyx n012 ----------- (1)

02

2

y

xnxyx

dxd n012

xYcxJcy nn 21 )( ----------- (2) (2)

xJxxJxdxd

nn

nn

1 ----------- (3)

13

xJxxJxdxd

nn

nn

1 ----------- (4)

(3) [0, b] r (0) = 0 B.C. 022 bJbJ nn ------------ (5)

i B.C. Eigenvalue ,3,2,1, ixJ in p (x) = x

00

dxxJxJx jnb

in ji ------------ (6)

(4) (0b) f (x)

xJcxf ini

i

1

------------ (7)

dxxJx

dxxfxJxc b

in

b

in

i

0

2

0)(

------------ (8)

f (x) Fourier-Bessel Series [F.B.S.] (5) [Square Norm]

2 20

bn i n iJ x xJ x dx Eigenvalue i

xJy n (1) 2xy

022222 ydxdnxyx

dxd

x = 0 x = b

bb ynxyxdxxy00

2222222 )(][2 xJy n n > 00 nJ n = 0 x = 0

2222 yxyx

222220

2222 )(2 bJnbbJbdxxJx nb

nn ------------ (9) [ )( xJy n ](9) B.C.

[CASE I]:

2 = 1 2 0 B.C. 100 bJn

ix Eigenvalue bxii / 00 nJ ...,3,2,1n 100 J n 0 [ 0

n = 123 Trivial Function n = 0 0 (10)]

14

(4) 11...............1 xJxxJnxJx nnn

(9)(10)

122

21

22 bJbxJ inin

[CASE II]:

2 = h 0 2 b(5) B.C. 130 bJbbhJ nn

n = 123(13) ix bxii / 0 n = 123(9)

14bJhbJb inini xJ in Square Norm

152

22

22222 bJhnbxJ in

i

iin

[CASE III]:

(13)h = 0 n = 0 i

1600 bJ

0 (4) n = 0

1700 10 bJbJ

01 x 0 0 1J 1 0 (15)

0h 0n 01 Square Nom Square Nom

182

12

0

2 bxdxb

(15), ,0,00 nhi

192

20

22

0 bJbxJ ii

[B] - [Fourier-Bassel Series]

(0b) xf Fourier-Bassel Series :

(1) 201

xJcxf ini

i

212

021

2 dxxfxJxbJbc i

b

nin

i

:i 220 bJn

15

(2) 231

xJcxf ini

i

2

2 2 2 2 2 0

2 24bi

i n ii n i

c xJ x f x dxb n h J b

:i

25i n nhJ b bJ b

(3) 2602

1 xJccxf ii

i

272021

dxxfxb

cb

2 2 0

2 28b

io

c x f x dxb J b

:i

2900 bJ

H.W. 3xxf 0 < x < 1 xf Bessel )(3 xJ Fourier-Bessel

[73]

xJAxf nn

n 32

nnnN

n JxJxJxfA

42

3

3 2,

[C] - [Convergence of Fourier-Bessel Series]

(0b) f f f - f xf f

2

xfxf

-[Fourier-Legendre Series] [A]

1. Legendre D.E. ......,2,1,0,0)1('2")1( 2 nynnxyyx (1)

Self-adjoint Form:

......,2,1,0,0)1(])1[( 2 nynnyxdxd (2)

16

)(xPy n ......,2,1,0, n (3) 2. 0)()()( 1 xPxPxxnP nnn .....,3,2,1,0, n (4)

)]()([121)( 11 xPxPnxP nnn

.....,3,2,1,0, n (5)

)(1)(112)( 11 xPn

nxxPnnxP nnn .....,3,2,1,0, n (6)

3. [1, 1] r (1) = r (1) = 0 B.C.

4. Legendre Polynomial { ......},2,1,0),( nxPn [1, 1] p(x) = 1

1

10)()( dxxPxP nm m n (7)

5. 1

1

22

122)()( ndxxPxP nn (8)

H.W. xxxf ,)( 1 )(xf Fourier Legendre [85 ]

)()()()( 4163285021 xPxPxPxf

H.W. 22)( 23 xxxf

0

)()(n

nn xPcxf )(xPn Legendre Polynomial

10 , cc = ? [86 ] 5310 ,3 cc [B] -[Fourier Legendre Series]

(1, 1) )(xf Fourier Legendre Series

0

)()(n

nn xPcxf ... (9)

1

1212 )()( dxxPxfc nnn ... (10)

** Fourier Legendre Series

cosx dxd sin (9)(10)

0

)(cos)(n

nnPcf ... (11)

0

sin)(cos)(212 dPFnc nn ... (12)

)(cosf )(F

17

[C] Fourier-Legendre Series f 'f (1, 1)(9) Fourier-Legendre Series

)(xf [ )()( xfxf ]/2

Ex.

10

)(xf 10,01,

xx

Fourier-Legendre Series

Legendre Polynomial Rodrigues Formula ])1[(!2

1)( 2 nnn

nn xdxd

nxP

)13()()(

1)(

221

2

1

0

xxPxxP

xP

)35()( 3213 xxxP

)157063()(

)33035()(35

81

5

2481

4

xxxxP

xxxP

(10)

1 1

0 01 0

1 1

1 11 0

1 1 1( ) ( ) 1 12 2 23 3 3( ) ( ) 12 2 4

c f x P x dx dx

c f x P x dx xdx

1 1 33 31 0

1 1 4 24 41 0

1 1 5 35 51 0

7 7 1 7( ) ( ) 1 (5 3 )2 2 2 169 9 1( ) ( ) 1 (35 30 3) 02 2 811 11 1 11( ) ( ) 1 (63 70 15 )2 2 8 32

c f x P x dx x x dx

c f x P x dx x x dx

c f x P x dx x x x dx

.........)()()()()( 532113167143021 xPxPxPxPxf H.W. 11,)( 3 xxxf )(xf Fourier Legendre [78 ] )()()( 352153 xPxPxf H.W. 11,)( 2 xxxf )()()( 232031 xPxPxf

18

5

[Solutions of Nonhomogeneous B.V.P.]

The Method of Eigenfunction Expansion

B.V.P.

1 1

2 2

D.E.: [ ( ) ] ( ) ( ), ( , ) ................. (1)

( ) ( ) 0 ............... (2 )B.C.:

( ) ( ) 0 ............... (2 )

d r x w q x w f x x a bdx

w a w a aw b w b b

1) B.V.P. Sturm-Liouville Problem

1 1

2 2

D.E.: [ ( ) ] [ ( ) ( )] 0, [ , ] .............. (3)

( ) ( ) 0 ............... (4 )B.C.:

( ) ( ) 0 ............... (4 )

d r x y q x p x y x a bdx

y a y a ay b y b b

],[,0)(,0)( baxxpxr )(),(),(),( xpxqxrxy

* p(x) Eigenfunction(3)(4) p(x) Eigenfunction }{ ny 2) (1)(2)(3)(4) Eigenfunction

nnn

ycxw

1

)( .. (5)

3) (1) y (3)( w) wpyyfyrwwry ][][

x = a x = b

b

a

b

a

ba

b

a

ba dxpwyyfdxywrywrdxywrwyr )(]|[]|[

( )[ ( ) ( ) ( ) ( )] ( )[ ( ) ( ) ( ) ( )] ( ) ............ (6)b

ar b y b w b y b w b r a w a y a w a y a f y pwy dx

(2)(4) 0,0 2211 nn yy , (6)

)7(............. b

a n

b

a nndxyfdxpwy

Fourier Series

b

a n

b

a nn

dxpy

dxpwyc

2 (8)

0

0

19

(7)(8)

b

a nn

b

a nn

dxpy

dxyfc

2 (9)

Eigenfunction yn p(x) Orthonormal Eigenvalue n (9)

b

a nn

n dxyfc 1 . (10)

B.V.P.

)(]1[)(1

xydxyfxw nn

b

a nn

. (11)

* B.V.P. (3)(4) Sturm-Liouville Problem Eigenvalue n

b

a ndttytf 0)()( ... (12)

(1)(2) B.V.P.

Ex. B.V.P.

)()1(:.C.B

11)(:.E.D

ewwx

wx

wx )13(..................

(1) Eigenfunction Expansion 1( ) ( ) ( )p x q x f xx

Sturm-Liouville problem

0)()1(,01)( eyyyx

yx )14(.....................

p.9 (14) 1* ......,3,2,1,1 22* nnnn

122 nn )15(.....................

)lnsin( xnAyn )16(......................

dxx

nduxnu ln

2 21 0

1 1sin ( ln ) sine n

n x dx udux n

0

cos sin2

12

nu u un

20

2A (9)

cn

1

1 2 sin( ln )e

nn

c n x dxx

1

2 1 cos( ln )e

n

n xn

odd,

)1(22

even,0

22 nnn

n

(11)

0

3 ])12[(])12[(]ln)12sin[(4)(

n nnxnxw

(17)

H.W. 1 B.V.P. 0)()0(, yyxyy [

00sin xdxx ]

H.W. 2 B.V.P. 0)1()0(,4 2 yyxyy

1 23 212sin

]1)4

12[()2

12(

)]2

12()1(1[4)(

n

n

xnnn

n

xy

H.W. 3 B.V.P. 0)2()1(),sin(ln13)( yyxx

yx

yx

xnnn

xyn

n

ln2ln

sin])()2[(ln])()2(ln3[

)1()2(lnsin)2(ln2)(1

22222

H.W. 4 B.V.P. 2 cosy y x , (0) (1) 0y y

2

#2 2 2 21

( 1) cos1 1( ) sin

( 1)( 2)n

ny x n x

n n

H.W. 5 Consider the nonhomogeneous equation for spring-mass system: sink Ax x tm m

, where

sinA t is an external periodic force. Suppose that the object m is at its equilibrium

21

position initially and when t = 1 second. Using the method of eigenfunction expansions, find a solution to the nonhomogeneous value problem.

)(tx (1) n 1, 2, 3,n (3) nn )1(cos

1

222 sin)()1(sin2)(

n n

n

tnnn

kAtx

(6) . #

(2) n 1, 2, 3,n (3)

22sinsin)(

mnktnAtn

kAtx

np

(7) . #

22mnk 22mnk D.E. Eigenvalue

2 2 2 2

2 21 1 1 1 0nmn mn

k mn

H.W. 6 Using the method of eigenfunction expansions, solve the given nonhomogeneous boundary value problem:

2( ) / ln , (1) ( ) 0xy y x x y y e

2 2

4 2 21

16 ( 1) 2 ( 1) 1 cos ln #216 ( 4)

n n

nn

e e ny xn n

* (1)(2) Greens Solution

b

adttftxkxw )(),()( (18)

(11) x t

b

an n

nn dttfxytyxw )(])()([)(1

(19)

[Greens function] K(x, t)

1

)()(),(n n

nn xytytxK

.. (20)

D.E.

1.) D.E. xxFuxcuxbuxa , 1

, xa xcxb

2.) (1)

1 2x b

da x a

22

3

4

5

x bd

ap x e

c xq x p x

a x

F xf x p x

a x

(1) D.E.

6d dup x q x u f xdx dx

Self-Adjoint Form , xp q(x) f (x)

3.) D.E.

70

vxq

dxdvxp

dxd

xvxv 21 (7)(7)

82211 xvcxvcxv

21 cc ,21 xvxv

4.)

91221 dfvxvdfvxvxwxx

21 vv Bounded (9)

dfvxvdfvxv

xfxvxvxfxvxv

dfvxvdfvxvxw

xx

xx

1221

1221

1221'

10

p(x) x

xfxvxvxvxvxpqw

xfxvxvxvxvxp

fdvdxdvxp

dxdfdv

dxdvxp

dxd

dxdwxp

dxd

2121

1221

12

21

11

(7)

01221112212121 vpvvvpvvpdxdvvp

dxdxvxvxvxvxp

dxd

23

kxvxvxvxvxp 2121 12

(11) w(x) D.E.:

13xkfwxqdxdwxp

dxd

x 21 vv Bounded

140 ww

(13) k (6)

15kwu

(9) w (12) k

16,

dfxRxux

17, 21211221

xvxvxvxvxpvxvvxvxR

(6)(13)(14)~(16)(16) I.V.P.

buu

axxfuxqdxdup

dxd

180:.C.I

18,:.E.D

(17) ,xR ,xR x

(7) D.E. xRxR ,,

5.) ,xR I.V.P.

cpdx

dR

bR

axRxqdxdRxp

dxd

x

x

191

190:

)19(,0)(:.E.D

I.C.

,xR [Disturbance] u [Influence] R(x, ) Influence Function One-Side Greens Function

6) I.C. 0)(orand0)( uu (7)

(16)(18) I.V.P

)20()()()(),()( 2211 xvcxvcdfxRxux

24

x 0 )(),(),(),( 2211 xvxvxvxv

Ex. I.V.P. :

0)0()0(:.C.I0),(:.E.D

uuxxfuu

Influence Function R ,x V.I.P.:

1)(.E.D1

0:.C.I

,0:.E.D 22

xpxd

dR

R

xRdx

Rd

x

x

I.V.P. )sin(),( xxR I.V.P.

x

dxfxu0

)sin()()(

Ex. D.E. : ( ), 0

I.V.P. :I.C. : (0) 1, (0) 1

u u f x xu u

, xxR sin),(

xcxcdxfxux

cossin)sin()()( 210 1)0(,1)0(:.C.I 12 cucu

x

xxdxfxu0

cossin)sin()()(

** < x u(x) x f ( )