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 factthatu(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 (seesubsection 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 ODEsAs 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 withrespect 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,