Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PDES: SEPARATION OF VARIABLES AND OTHER METHODS


we haveL=∇^2 , whereas for Helmholtz’s equationL=∇^2 +k^2. Note that we have


not specified the dimensionality of the problem, and (21.76) may, for example,


represent Poisson’s equation in two or three (or more) dimensions. The reader


will also notice that for the sake of simplicity we have not included any time


dependence in (21.76). Nevertheless, the following discussion can be generalised


to include it.


As we discussed in subsection 20.3.2, a problem is inhomogeneous if the fact

thatu(r) is a solution doesnotimply that any constant multipleλu(r)isalsoa


solution. This inhomogeneity may derive from either the PDE itself or from the


boundary conditions imposed on the solution.


In our discussion of Green’s function solutions of inhomogeneous ODEs (see

subsection 15.2.5) we dealt with inhomogeneous boundary conditions by making a


suitable change of variable such that in the new variable the boundary conditions


were homogeneous. In an analogous way, as illustrated in the final example


of section 21.2, it is usually possible to make a change of variables in PDEs to


transform between inhomogeneity of the boundary conditions and inhomogeneity


of the equation. Therefore let us assume for the moment that the boundary


conditions imposed on the solutionu(r) of (21.76) are homogeneous. This most


commonly means that if we seek a solution to (21.76) in some regionVthen


on the surfaceSthat boundsVthe solution obeys the conditionsu(r)=0or


∂u/∂n=0,where∂u/∂nis the normal derivative ofuat the surfaceS.


We shall discuss the extension of the Green’s function method to the direct so-

lution of problems with inhomogeneous boundary conditions in subsection 21.5.2,


but we first highlight how the Green’s function approach to solving ODEs can


be simply extended to PDEs for homogeneous boundary conditions.


21.5.1 Similarities to Green’s functions for ODEs

As in the discussion of ODEs in chapter 15, we may consider the Green’s


function for a system described by a PDE as the response of the system to a ‘unit


impulse’ or ‘point source’. Thus if we seek a solution to (21.76) that satisfies some


homogeneous boundary conditions onu(r) then the Green’s functionG(r,r 0 )for


the problem is a solution of


LG(r,r 0 )=δ(r−r 0 ), (21.77)

wherer 0 lies inV. The Green’s functionG(r,r 0 ) must also satisfy the imposed


(homogeneous) boundary conditions.


It is understood that in (21.77) theLoperator expresses differentiation with

respect toras opposed tor 0. Also,δ(r−r 0 ) is the Dirac delta function (see


chapter 13) of dimension appropriate to the problem; it may be thought of as


representing a unit-strength point source atr=r 0.


Following an analogous argument to that given in subsection 15.2.5 for ODEs,
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