Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

in whichκis a constant with the dimensions length^2 ×time−^1. The physical

constants that go to make upκin a particular case depend upon the nature of

the process (e.g. solute diffusion, heat flow, etc.) and the material being described.

With (20.34) we cannot hope to repeat successfully the method of subsection

20.3.3, since nowu(x, t) is differentiated a different number of times on the two

sides of the equation; any attempted solution in the formu(x, t)=f(p) with

p=ax+btwill lead only to an equation in which the form offcannot be

cancelled out. Clearly we must try other methods.

Solutions may be obtained by using the standard method of separation of

variables discussed in the next chapter. Alternatively, a simple solution is also

given if both sides of (20.34), as it stands, are separately set equal to a constant

α(say), so that

∂^2 u

∂x^2

=

α

κ

,

∂u

∂t

=α.

These equations have the general solutions

u(x, t)=

α

2 κ

x^2 +xg(t)+h(t)andu(x, t)=αt+m(x)

respectively and may be made compatible with each other ifg(t) is taken as

constant,g(t)=g(wheregcould be zero),h(t)=αtandm(x)=(α/ 2 κ)x^2 +gx.

An acceptable solution is thus

u(x, t)=

α

2 κ

x^2 +gx+αt+ constant. (20.35)

Let us now return to seeking solutions of equations by combining the inde-

pendent variables in particular ways. Having seen that a linear combination of

xandtwill be of no value, we must search for other possible combinations. It

has been noted already thatκhas the dimensions length^2 ×time−^1 and so the

combination of variables

η=

x^2

κt

will be dimensionless. Let us see if we can satisfy (20.34) with a solution of the

formu(x, t)=f(η). Evaluating the necessary derivatives we have

∂u

∂x

=

df(η)

dη

∂η

∂x

=

2 x

κt

df(η)

dη

,

∂^2 u

∂x^2

=

2

κt

df(η)

dη

+

(

2 x

κt

) 2

d^2 f(η)

dη^2

,

∂u

∂t

=−

x^2

κt^2

df(η)

dη

.

Substituting these expressions into (20.34) we find that the new equation can be