Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

Thus, writingu(ρ, φ)=P(ρ)Φ(φ) and using the expression (21.23), Laplace’s

equation (21.26) becomes

Φ

ρ

∂

∂ρ

(

ρ

∂P

∂ρ

)

+

P

ρ^2

∂^2 Φ

∂φ^2

=0.

Now, employing the same device as previously, that of dividing through by

u=PΦ and multiplying through byρ^2 , results in the separated equation

ρ

P

∂

∂ρ

(

ρ

∂P

∂ρ

)

+

1

Φ

∂^2 Φ

∂φ^2

=0.

Following our earlier argument, since the first term on the RHS is a function of

ρonly, whilst the second term depends only onφ, we obtain the twoordinary

equations

ρ

P

d

dρ

(

ρ

dP

dρ

)

=n^2 , (21.27)

1

Φ

d^2 Φ

dφ^2

=−n^2 , (21.28)

where we have taken the separation constant to have the formn^2 for later

convenience; for the present,nis a general (complex) number.

Let us first consider the case in whichn= 0. The second equation, (21.28), then

has the general solution

Φ(φ)=Aexp(inφ)+Bexp(−inφ). (21.29)

Equation (21.27), on the other hand, is the homogeneous equation

ρ^2 P′′+ρP′−n^2 P=0,

which must be solved either by trying a power solution inρor by making the

substitutionρ=exptas described in subsection 15.2.1 and so reducing it to an

equation with constant coefficients. Carrying out this procedure we find

P(ρ)=Cρn+Dρ−n. (21.30)

Returning to the solution (21.29) of the azimuthal equation (21.28), we can

see that if Φ, and henceu, is to be single-valued and so not change whenφ

increases by 2πthennmust be an integer. Mathematically, other values ofnare

permissible, but for the description of real physical situations it is clear that this

limitation must be imposed. Having thus restricted the possible values ofnin

one part of the solution, the same limitations must be carried over into the radial

part, (21.30). Thus we may write a particular solution of the two-dimensional

Laplace equation as

u(ρ, φ)=(Acosnφ+Bsinnφ)(Cρn+Dρ−n),