Feb 5, 2001

Notes for Lecture Notes for Lecture #8

Orthogonal function expansions and Green's functions

Suppose we have a ``complete'' set of orthogonal functions {un(x)} defined in the interval x1 £ x £ x2 such that

ó
õ 		x2

x1 
 un(x) um(x)   dx = dnm.
(1)
We can show that the completeness of this functions implies that
¥
å
 n = 1 
 un(x) un(x¢) = d(x-x¢).
(2)

This relation allows us to use these functions to represent a Green's function for our system. For the 1-dimensional Poisson equation, the Green's function satisfies

2
x2
 G(x,x¢) = -4pd(x-x¢).
(3)
Therefore, if
d2
dx2
 un(x) = -an un(x),
(4)
where {un(x)} also satisfy the appropriate boundary conditions, then we can write the Greens functions as
G(x,x¢) = 4 p
å
 n 
 un(x) un(x¢)
an
 .
(5)
For example, if un(x) = Ö[(2/a)] sin(npx/a), then
G(x,x¢) = 8 p
a
å
 n 
 sin(npx/a)sin(np x¢/a)
æ
ç
è 		np
a
 ö
÷
ø 		2

 
.
(6)
These ideas can easily be extended to two and three dimensions. For example if {un(x)}, {vn(x)}, and {wn(x)} denote the complete functions in the x, y, and z directions respectively, then the three dimensional Green's function can be written:
G(x,x¢,y,y¢,z,z¢) = 4 p
å
 lmn 
 ul(x)ul(x¢)vm(y)vm(y¢)wn(z)wn(z¢)
al + bm + gn
 ,
(7)
where
d2
dx2
 ul(x) = -al ul(x),     d2
dy2
 vm(x) = -bm vm(y),    and     d2
dz2
 wn(z) = -gn wn(z).
(8)
See Eq. 3.167 in Jackson for an example.

An alternative method of finding Green's functions for second order ordinary differential equations is based on a product of two independent solutions of the homogeneous equation, u1(x) and u2(x), which satisfy the boundary conditions at x1 and x1, respectively:

G(x,x¢) = K u1(x < ) u2(x > ),     where    K º 4p
u1 du2
dx
 - du1
dx
 u2
 ,
(9)
with x < meaning the smaller of x and x¢ and x > meaning the larger of x and x¢. For example, we have previously discussed the example of the one dimensional Poisson equation with the boundary condition F(0) = 0 and [(dF(a))/ dx] = 0 to have the form:
G(x,x¢) = -4px < .
(10)
For the two and three dimensional cases, we can use this technique in one of the dimensions in order to reduce the number of summation terms. These ideas are discussed in Section 3.11 of Jackson. For the two dimensional case, for example, we can assume that the Green's function can be written in the form:
G(x,x¢,y,y¢) =
å
 n 
 un(x) un(x¢) gn(y,y¢).
(11)
If the functions {un(x)} satisfy Eq. 4, then we must require that G satisfy the equation:
Ñ2 G =
å
 n 
 un(x) un(x¢) é
ê
ë 		-an + 2
y2
 ù
ú
û 		gn(y,y¢) = - 4 pd(x-x¢)d(y-y¢).
(12)
The y-dependence of this equation will have the required behavior, if we choose:
é
ê
ë 		-an + 2
y2
 ù
ú
û 		gn(y,y¢) = -4 pd(y-y¢),
(13)
which in turn can be expressed in terms of the two independent solutions vn1(y) and vn2(y) of the homogeneous equation:
d2
d y2
 vni(y) = an vni(y),
(14)
and a constant related to the Wronskian:
Kn º 4p
vn1 dvn2
dx
 - dvn1
dx
 vn2
 .
(15)
If these functions also satisfy the appropriate boundary conditions, we can then construct the 2-dimensional Green's function from
G(x,x¢,y,y¢) =
å
 n 
 un(x) un(x¢) Kn vn1(y < ) vn2(y > ),
(16)
where the constant Kn is defined in a similar way to the one-dimensional case. For example, a Green's function for a two-dimensional with 0 £ x £ a and 0 £ y £ b, which vanishes on each of the boundaries can be expanded:
G(x,x¢,y,y¢) = 8 ¥
å
 n = 1 
sin æ
ç
è 		npx
a
 ö
÷
ø 		sin æ
ç
è 		npx¢
a
 ö
÷
ø 		sinh æ
ç
è 		npy <
a
 ö
÷
ø 		sinh æ
ç
è 		np
a
 (b-y > ) ö
÷
ø
n sinh æ
ç
è 		npb
a
 ö
÷
ø
.
(17)

This example can be used to solve the 2-dimensional Laplace equation in the square lattice discussed in Lecture notes#5. In this case,

F(r)
= 1
4pe0
 ó
õ

V 
 d3r¢ r(r¢) G(r,r¢) + 1
4p
 ó
õ

S 
 é
ê
ë 		G(r,r¢) F
n¢
 - F(r¢) G(r,r¢)
n¢
 ù
ú
û 		da¢
(18)
=  -  1
4p
 ó
õ

S 
 F(r¢) G(r,r¢)
n¢
 da¢.
For this example, a = b = 1, and the integral over the ``surface'' da¢ º dx¢ is a line integral 0 £ x¢ £ 1 for y¢ = 1. It can be shown that the result takes the form:
F(x,y) = ¥
å
 n = 0 
     4 V0       
sin[(2n+1)px/a] sinh[(2n+1)py/a]
 (2n+1) psinh[(2n+1)p]
(19)

Refer to the beginning of the maple file lecture8.mws to see plots of F(x,y) for 2 different summations of the series.

File translated from TEX by TTH, version 2.20.
On 5 Feb 2001, 08:48.