Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

The first term depends only onrand the second and third terms (taken together)

depend only onθandφ. Thus (21.39) is equivalent to the two equations

1

R

d

dr

(

r^2

dR

dr

)

=λ, (21.40)

1

Θsinθ

d

dθ

(

sinθ

dΘ

dθ

)

+

1

Φsin^2 θ

d^2 Φ

dφ^2

=−λ. (21.41)

Equation (21.40) is a homogeneous equation,

r^2

d^2 R

dr^2

+2r

dR

dr

−λR=0,

which can be reduced, by the substitutionr=expt(and writingR(r)=S(t)), to

d^2 S

dt^2

+

dS

dt

−λS=0.

This has the straightforward solution

S(t)=Aexpλ 1 t+Bexpλ 2 t,

and so the solution to the radial equation is

R(r)=Arλ^1 +Brλ^2 ,

whereλ 1 +λ 2 =−1andλ 1 λ 2 =−λ. We can thus takeλ 1 andλ 2 as given by

and−(+1);λthen has the form(+ 1). (It should be noted that at this stage

nothing has been either assumed or proved about whetheris an integer.)

Hence we have obtained some information about the first factor in the

separated-variable solution, which will now have the form

u(r, θ, φ)=

[

Ar+Br−(+1)

]

Θ(θ)Φ(φ), (21.42)

where Θ and Φ must satisfy (21.41) withλ=(+1).

The next step is to take (21.41) further. Multiplying through by sin^2 θand

substituting forλ, it too takes a separated form:

[

sinθ

Θ

d

dθ

(

sinθ

dΘ

dθ

)

+(+1)sin^2 θ

]

+

1

Φ

d^2 Φ

dφ^2

=0. (21.43)

Taking the separation constant asm^2 , the equation in the azimuthal angleφ

has the same solution as in cylindrical polars, namely

Φ(φ)=Ccosmφ+Dsinmφ.

As before, single-valuedness ofurequires thatmis an integer; form= 0 we again

have Φ(φ)=Cφ+D.