Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.5 Inhomogeneous problems – Green’s functions

the infinite range inx, rather than of the transform method itself. In fact the

method of separation of variables would yield the same solutions, since in the

infinite-range case the separation constant is not restricted to take on an infinite

set of discrete values but may have any real value, with the result that the sum

overλbecomes an integral, as mentioned at the end of section 21.2.

An infinite metal bar has an initial temperature distributionf(x)along its length. Find

the temperature distribution at a later timetusing the method of separation of variables.

This is the same problem as in the previous example, but we now seek a solution by
separating variables. From (21.12) a separated solution for the one-dimensional diffusion
equation is given by

u(x, t)=[Aexp(iλx)+Bexp(−iλx)] exp(−κλ^2 t),

where−λ^2 is the separation constant. Since the bar is infinite we do not require the
solution to take a given form at any finite value ofx(for instance atx=0)andsothere
is no restriction onλother than its being real. Therefore instead of the superposition of
such solutions in the form of a sum over allowed values ofλwe have an integral over
allλ,

u(x, t)=

1

√

2 π

∫∞

−∞

A(λ)exp(−κλ^2 t)exp(iλx)dλ, (21.75)

where in takingλfrom−∞to∞we need include only one of the complex exponentials;
we have taken a factor 1/

√

2 πout ofA(λ) for convenience. We can see from (21.75)
that the expression foru(x, t) has the form of an inverse Fourier transform (whereλis
the transform variable). Therefore, Fourier-transforming both sides and using the Fourier
inversion theorem, we find
̃u(λ, t)=A(λ)exp(−κλ^2 t).

Now, the initial boundary condition requires

u(x,0) =

1

√

2 π

∫∞

−∞

A(λ)exp(iλx)dλ=f(x),

from which, using the Fourier inversion theorem once more, we see thatA(λ)= ̃f(λ).
Therefore we have

̃u(λ, t)= ̃f(λ)exp(−κλ^2 t),

which is identical to (21.73) inthe previous example (but withkreplaced byλ), and hence
leads to the same result.

21.5 Inhomogeneous problems – Green’s functions

In chapters 15 and 17 we encountered Green’s functions and found them a useful

tool for solving inhomogeneous linear ODEs. We now discuss their usefulness in

solving inhomogeneous linear PDEs.

For the sake of brevity we shall again denote a linear PDE by

Lu(r)=ρ(r), (21.76)

whereLis a linear partial differential operator. For example, in Laplace’s equation