Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

We may treat the diffusion equation

κ∇^2 u=

∂u

∂t

in a similar way. Separating the spatial and time dependences by assuming a

solution of the formu=F(r)T(t), and taking the separation constant ask^2 ,we

find

∇^2 F+k^2 F=0,

dT

dt

+k^2 κT=0.

Just as in the case of the wave equation, the spatial part of the solution satisfies

Helmholtz’s equation. It only remains to consider the time dependence, which

has the simple solution

T(t)=Aexp(−k^2 κt).

Helmholtz’s equation is clearly of central importance in the solutions of the

wave and diffusion equations. It can be solved in polar coordinates in much the

same way as Laplace’s equation, and indeed reduces to Laplace’s equation when

k= 0. Therefore, we will merely sketch the method of its solution in each of the

three polar coordinate systems.

Helmholtz’s equation in plane polars

In two-dimensional plane polar coordinates, Helmholtz’s equation takes the form

1

ρ

∂

∂ρ

(

ρ

∂F

∂ρ

)

+

1

ρ^2

∂^2 F

∂φ^2

+k^2 F=0.

If we try a separated solution of the formF(r)=P(ρ)Φ(φ), and take the

separation constant asm^2 , we find

d^2 Φ

dφ^2

+m^2 φ=0,

d^2 P

dρ^2

+

1

ρ

dP

dρ

+

(

k^2 −

m^2

ρ^2

)

P=0.

As for Laplace’s equation, the angular part has the familiar solution (ifm=0)

Φ(φ)=Acosmφ+Bsinmφ,

or an equivalent form in terms of complex exponentials. The radial equation

differs from that found in the solution of Laplace’s equation, but by making the

substitutionμ=kρit is easily transformed into Bessel’s equation of orderm

(discussed in chapter 16), and has the solution

P(ρ)=CJm(kρ)+DYm(kρ),

whereYmis a Bessel function of the second kind, which is infinite at the origin