Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSconditions in the new variables are homogeneous although the equation itself isgenerally inhomogeneous. In this section, however, we extend the use of Green’sfunctions to problems with inhomogeneous boundary conditions (and equations).This provides a more consistent and intuitive approach to the solution of suchboundary-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 for ρ(r) =0andso our discussion is equally applicable to this case.We wish to solve (21.80) in some region V bounded by a surface S, which mayconsist of several disconnected parts. As stated above, we shall allow the possibilitythat the boundary conditions on the solution u(r) may be inhomogeneous on S,although as we shall see this method reduces to those discussed above in thespecial case that the boundary conditions are in fact homogeneous.The two common types of inhomogeneous boundary condition for Poisson’sequation are (as discussed in subsection 20.6.2):(i) Dirichlet conditions, in which u(r) is specified on S, and(ii) Neumann conditions, in which ∂u/∂n is specified on S.In general, specifying both Dirichlet and Neumann conditions on S overdeterminesthe problem and leads to there being no solution.The specification of the surface S requires some further comment, since Smay have several disconnected parts. If we wish to solve Poisson’s equationinside some closed surface S then the situation is straightforward and is shownin figure 21.11(a). If, however, we wish to solve Poisson’s equation in the gapbetween two closed surfaces (for example in the gap between two concentricconducting cylinders) then the volume V is bounded by a surface S that hastwo disconnected parts S 1 and S 2 , as shown in figure 21.11(b); the direction ofthe normal to the surface is always taken as pointing out of the volume V .Asimilar situation arises when we wish to solve Poisson’s equation outside someclosed surface S 1 . In this case the volume V is infinite but is treated formallyby taking the surface S 2 as a large sphere of radius R and letting R tend toinfinity.In order to solve (21.80) subject to either Dirichlet or Neumann boundaryconditions on S, we will remind ourselves of Green’s second theorem, equation(11.20), which states that, for two scalar functions φ(r) andψ(r) defined in somevolume V bounded by a surface S,∫∫(φ∇ 2 ψ − ψ∇ 2 φ) dV = (φ∇ψ − ψ∇φ) · ˆn dS, (21.81)V754S