Page 76 - 6637

Solved Problems

Solution: The Green function problem for a sphere of radius R centered at the
point ξ is

∆G − λG = δ(x − ξ), G = 0. (10.2)
|x−ξ|=R
We will solve the modified Helmholtz equation,

∆u − λu = 0,

where the value of u is known on the boundary of the sphere of radius R in terms

of this Green function.
in terms of this Green function.
Let L[u] = ∆u − λu.

Z Z
(uL[G] − GL[u]) dξ = uδ(x − ξ) dξ = u(x)
S S

Z Z
(uL[G] − GL[u]) dξ = (u∆G − G∆u) dξ
S S

∂G ∂u
Z
= u − G dA ξ
∂S ∂n ∂n
∂G
Z
= u dA ξ
∂S ∂n

Z
∂G
u(x) = u dA ξ
∂S ∂n
We are interested in the value of u at the center of the sphere. Let ρ = |p − ξ|

Z
∂G
u(p) = u(ξ) (p|ξ) dA ξ
∂S ∂ρ

We do not need to compute the general solution of Equation 10.2. We only need
the Green function at the point x = p. We know that the Green function there is
a linear combination of the fundamental solutions,

−1 √ −1 √
G(p|ξ) = c 1 e λ|p−ξ| +c 2 e − λ|p−ξ| ,
4π|p − ξ| 4π|p − ξ|

such that c + c = 1. The Green function is symmetric with respect to x and ξ.
2
1
We add the constraint that the Green function vanishes at ρ = R. This gives us
two equations for c and c .
1
2
c 1 √ c 2 √
c + c = 1, − e λR − e − λR = 0
1
2
4πR 4πR
√
1 e 2 λR
c = − √ , c = √
1
2
e 2 λR −1 e 2 λR −1
72