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 ( )