Feb 2, 2000
Notes for Lecture
Notes for Lecture #7
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 showed 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)
|
. |
| (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¢) = |
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
|
, |
| (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:
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) |
We 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) |
This form 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) |
If vn1(y) and vn2(y) are solutions to the homogeneous
equation
|
|
é
ê
ë
|
-an + |
d2
d y2
|
ù
ú
û
|
vni(y) = 0, |
| (14) |
satisfying 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 > ), |
| (15) |
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, with
the potential vanishing 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
|
|
|
. |
| (16) |
File translated from TEX by TTH, version 2.20.
On 2 Feb 2000, 12:04.