Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

This gives a particular solution of the original PDE (21.3)

u(x, y, z, t)=exp(ilx) exp(imy) exp(inz) exp(−icμt)

= exp[i(lx+my+nz−cμt)],

which is a special case of the solution (20.33) obtained in the previous chapter

and represents a plane wave of unit amplitude propagating in a direction given

by the vector with componentsl, m, nin a Cartesian coordinate system. In the

conventional notation of wave theory,l,mandnare the components of the

wave-number vectork, whose magnitude is given byk=2π/λ,whereλis the

wavelength of the wave;cμis the angular frequencyωof the wave. This gives

the equation in the form

u(x, y, z, t)=exp[i(kxx+kyy+kzz−ωt)]

= exp[i(k·r−ωt)],

and makes the exponent dimensionless.

The method of separation of variables can be applied to many commonly

occurring PDEs encountered in physical applications.

Use the method of separation of variablesto obtain for the one-dimensional diffusion

equation

κ

∂^2 u

∂x^2

=

∂u

∂t

, (21.9)

a solution that tends to zero ast→∞for allx.

Here we have only two independent variablesxandtand we therefore assume a solution
of the form

u(x, t)=X(x)T(t).

Substituting this expression into (21.9) and dividing through byu=XT(and also byκ)
we obtain
X′′
X

=

T′

κT

.

Now, arguing exactly as above that the LHS is a function ofxonlyandtheRHSisa
function oftonly, we conclude that each side must equal a constant, which, anticipating
the result and noting the imposed boundary condition, we will take as−λ^2. This gives us
two ordinary equations,

X′′+λ^2 X=0, (21.10)

T′+λ^2 κT=0, (21.11)

which have the solutions

X(x)=Acosλx+Bsinλx,

T(t)=Cexp(−λ^2 κt).

Combining these to give the assumed solutionu=XTyields (absorbing the constantC
intoAandB)

u(x, t)=(Acosλx+Bsinλx)exp(−λ^2 κt). (21.12)