Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

conditions in the new variables are homogeneous although the equation itself is

generally inhomogeneous. In this section, however, we extend the use of Green’s

functions to problems with inhomogeneous boundary conditions (and equations).

This provides a more consistent and intuitive approach to the solution of such

boundary-value problems.

For definiteness we shall consider Poisson’s equation

∇^2 u(r)=ρ(r), (21.80)

but the material of this section may be extended to other linear PDEs of the form

(21.76). Clearly, Poisson’s equation reduces to Laplace’s equation forρ(r)=0and

so our discussion is equally applicable to this case.

We wish to solve (21.80) in some regionVbounded by a surfaceS, which may

consist of several disconnected parts. As stated above, we shall allow the possibility

that the boundary conditions on the solutionu(r) may be inhomogeneous onS,

although as we shall see this method reduces to those discussed above in the

special case that the boundary conditions are in fact homogeneous.

The two common types of inhomogeneous boundary condition for Poisson’s

equation are (as discussed in subsection 20.6.2):

(i) Dirichlet conditions, in whichu(r) is specified onS,and

(ii) Neumann conditions, in which∂u/∂nis specified onS.

In general, specifyingbothDirichletandNeumann conditions onSoverdetermines

the problem and leads to there being no solution.

The specification of the surfaceSrequires some further comment, sinceS

may have several disconnected parts. If we wish to solve Poisson’s equation

inside some closed surfaceSthen the situation is straightforward and is shown

in figure 21.11(a). If, however, we wish to solve Poisson’s equation in the gap

between two closed surfaces (for example in the gap between two concentric

conducting cylinders) then the volumeVis bounded by a surfaceS that has

two disconnected partsS 1 andS 2 , as shown in figure 21.11(b); the direction of

the normal to the surface is always taken as pointingoutof the volumeV.A

similar situation arises when we wish to solve Poisson’s equationoutsidesome

closed surfaceS 1. In this case the volumeVis infinite but is treated formally

by taking the surfaceS 2 as a large sphere of radiusRand lettingRtend to

infinity.

In order to solve (21.80) subject to either Dirichlet or Neumann boundary

conditions onS, we will remind ourselves of Green’s second theorem, equation

(11.20), which states that, for two scalar functionsφ(r)andψ(r) defined in some

volumeVbounded by a surfaceS,
∫

V

(φ∇^2 ψ−ψ∇^2 φ)dV=

∫

S

(φ∇ψ−ψ∇φ)·nˆdS , (21.81)