Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSwe have L = ∇ 2 , whereas for Helmholtz’s equation L = ∇ 2 +k 2 . Note that we havenot specified the dimensionality of the problem, and (21.76) may, for example,represent Poisson’s equation in two or three (or more) dimensions. The readerwill also notice that for the sake of simplicity we have not included any timedependence in (21.76). Nevertheless, the following discussion can be generalisedto include it.As we discussed in subsection 20.3.2, a problem is inhomogeneous if the factthat u(r) is a solution does not imply that any constant multiple λu(r) isalsoasolution. This inhomogeneity may derive from either the PDE itself or from theboundary conditions imposed on the solution.In our discussion of Green’s function solutions of inhomogeneous ODEs (seesubsection 15.2.5) we dealt with inhomogeneous boundary conditions by making asuitable change of variable such that in the new variable the boundary conditionswere homogeneous. In an analogous way, as illustrated in the final exampleof section 21.2, it is usually possible to make a change of variables in PDEs totransform between inhomogeneity of the boundary conditions and inhomogeneityof the equation. Therefore let us assume for the moment that the boundaryconditions imposed on the solution u(r) of (21.76) are homogeneous. This mostcommonly means that if we seek a solution to (21.76) in some region V thenon the surface S that bounds V the solution obeys the conditions u(r) =0or∂u/∂n =0,where∂u/∂n is the normal derivative of u at the surface S.We shall discuss the extension of the Green’s function method to the direct solutionof problems with inhomogeneous boundary conditions in subsection 21.5.2,but we first highlight how the Green’s function approach to solving ODEs canbe simply extended to PDEs for homogeneous boundary conditions.21.5.1 Similarities to Green’s functions for ODEsAs in the discussion of ODEs in chapter 15, we may consider the Green’sfunction for a system described by a PDE as the response of the system to a ‘unitimpulse’ or ‘point source’. Thus if we seek a solution to (21.76) that satisfies somehomogeneous boundary conditions on u(r) then the Green’s function G(r, r 0 )forthe problem is a solution ofLG(r, r 0 )=δ(r − r 0 ), (21.77)where r 0 lies in V . The Green’s function G(r, r 0 ) must also satisfy the imposed(homogeneous) boundary conditions.It is understood that in (21.77) the L operator expresses differentiation withrespect to r as opposed to r 0 . Also, δ(r − r 0 ) is the Dirac delta function (seechapter 13) of dimension appropriate to the problem; it may be thought of asrepresenting a unit-strength point source at r = r 0 .Following an analogous argument to that given in subsection 15.2.5 for ODEs,752