Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

x

u

x=a

t 1

t 2

t 3

Figure 21.10 Diffusion of heat from a point source in a metal bar: the curves

show the temperatureuat positionxfor various timest 1 <t 2 <t 3 .Thearea

under the curves remains constant, since the total heat energy is conserved.

line we have used the standard result for the integral of a Gaussian, given in subsection
6.4.2. (Strictly speaking the change of variable fromktok′shifts the path of integration
off the real axis, sincek′is complex for realk, and so results in a complex integral, as will
be discussed in chapter 24. Nevertheless, in this case the path of integration can be shifted
back to the real axis without affecting the value of the integral.)
Thus the temperature in the bar at a later timetis given by

u(x, t)=

1

√

4 πκt

∫∞

−∞

exp

[

−

(x−x′)^2

4 κt

]

f(x′)dx′, (21.74)

which may be evaluated (numerically if necessary) when the form off(x)isgiven.

As we might expect from our discussion of Green’s functions in chapter 15,

we see from (21.74) that, if the initial temperature distribution isf(x)=δ(x−a),

i.e. a ‘point’ source atx=a, then the temperature distribution at later times is

simply given by

u(x, t)=G(x−a, t)=

1

√

4 πκt

exp

[

−

(x−a)^2

4 κt

]

.

The temperature at several later times is illustrated in figure 21.10, which shows

that the heat diffuses out from its initial position; the width of the Gaussian

increases as

√

t, a dependence on time which is characteristic of diffusion processes.

The reader may have noticed that in both examples using integral transforms

the solutions have been obtained in closed form – albeit in one case in the form

of an integral. This differs from the infinite series solutions usually obtained via

the separation of variables. It should be noted that this behaviour is a result of