Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS

if the boundary conditions onu(r) are homogeneous then a solution to (21.76)

that satisfies the imposed boundary conditions is given by

u(r)=

∫

G(r,r 0 )ρ(r 0 )dV(r 0 ), (21.78)

where the integral onr 0 is over some appropriate ‘volume’. In two or more

dimensions, however, the task of finding directly a solution to (21.77) that satisfies

the imposed boundary conditions onScan be a difficult one, and we return to

this in the next subsection.

An alternative approach is to follow a similar argument to that presented in

chapter 17 for ODEs and so to construct the Green’s function for (21.76) as a

superposition of eigenfunctions of the operatorL, providedLis Hermitian. By

analogy with an ordinary differential operator, a partial differential operator is

Hermitian if it satisfies
∫

V

v∗(r)Lw(r)dV=

[∫

V

w∗(r)Lv(r)dV

]∗

,

where the asterisk denotes complex conjugation andvandware arbitrary func-

tions obeying the imposed (homogeneous) boundary condition on the solution of

Lu(r)=0.

The eigenfunctionsun(r),n=0, 1 , 2 ,...,ofLsatisfy

Lun(r)=λnun(r),

whereλnare the corresponding eigenvalues, which are all real for an Hermitian

operatorL. Furthermore, each eigenfunction must obey any imposed (homo-

geneous) boundary conditions. Using an argument analogous to that given in

chapter 17, the Green’s function for the problem is given by

G(r,r 0 )=

∑∞

n=0

un(r)u∗n(r 0 )

λn

. (21.79)

From (21.79) we see immediately that the Green’s function (irrespective of how

it is found) enjoys the property

G(r,r 0 )=G∗(r 0 ,r).

Thus, if the Green’s function is real then it is symmetric in its two arguments.

Once the Green’s function has been obtained, the solution to (21.76) is again

given by (21.78). For PDEs this approach can become very cumbersome, however,

and so we shall not pursue it further here.

21.5.2 General boundary-value problems

As mentioned above, often inhomogeneous boundary conditions can be dealt

with by making an appropriate change of variables, such that the boundary