Description
1. Green’s function of the 1-D heat equation in a semi-infinite domain, , is
defined by:
subject to zero initial condition:
The boundary condition is either (a): or
(b):
The solution in a semi-infinite domain can be constructed from the solution in the infinite
domain by adding or subtracting another source located at , so that the
contributions cancel at for (a), or the contributions are symmetric about
Find the Green’s function defined above for boundary condition (a). Then repeat the
problem for boundary condition (b).
2. Find the Greens function for the wave equation in two-dimensions governed by
The solution is
(a) Derive this solution using Fourier transform in and .
Hint: In the inverse transform, use polar coordinates to get
(b) Derive this solution using Laplace transform in .
Hint: First show that the Laplace transform of G is
Gxt (,; , ) x t
2
2 ( ) ( ) ( ), 0 , , 0, 0. DGx t x t
t x
d xd t x t ¶ ¶ – = – – < < ¥ > >
¶ ¶
G t = = 0 at 0.
Gx x = = 0 at 0 and , ® ¥
Gx x 0 at 0 and .
x
¶ = = ® ¥
¶
x = -x
x = 0 x = 0.
2 22
2 22
222
( ) ( ) ( ) ( ).
0 as , where .
0 for 0.
G G txy
t xy
G r rxy
G t
dd d ¶ ¶¶ – + =
¶ ¶¶
® ®¥ = +
º <
2 2
1( ) , where is the Heaviside function. 2
Ht r G H
p t r
– = –
x y
0 0
1 ( )sin . Then use integral tables. 2
G J kr ktdk
p
¥
= ò
t
G! = 1
2π
K0 (sr), where K is modified Bessel function of the second kind. Then use Laplace transform tables.