318 Physics Electromagnetism Greens Function for the Poisson Equation 1. The nonhomogeneous problem for the Poisson Equation
2
R.G. Palmer 4/13/09
(x) = (x)/0
in a domain V
Closed boundary conditions on (x) at the surface S 2. The Greens function G(x, x )
2
G(x, x ) = 4 (x x )
3. The magic rule (x) = 1 40 G(x , x)(x ) d3x
V
1 4
(x )
S
G (x , x) G(x , x) (x ) da n n
4. Homogeneous boundary conditions for G(x, x ) We can choose homogeneous boundary conditions for G(x, x ) on S . Then G(x, x ) = G(x , x) Often we use the Dirichlet BCs (G(x, x ) = 0 for x on S ), giving (x) = 1 40 GD (x, x )(x ) d3x
V
1 4
(x )
S
GD (x, x )da n (x )da n
(Dirichlet Greens Fn)
or the Neumann BCs (G(x, x )/n = for 4/S x on S ), giving (x) =
S
+
1 40
GN (x, x )(x ) d3x +
V
1 4
GN (x, x )
S
(Neumann Greens Fn)
5. Techniques for constructing Greens function a. Method of images. [J2.6] b. Fundamental solution (1/|x x |) + solutions for Laplaces equation. [J1.10] c. Eigenfunction expansions for two dimensions and using the matching method for the third. [J3.9, J3.11] d. Eigenfunction expansions for three dimensions (see below). ...