Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

Comparing this expression forr=z,θ= 0 with thez<aexpansion of (21.60), which is
valid for anyz, establishesA 2 +1=0,A 0 =−GM/aand

A 2 =−

GM

a^2 +1

(−1)(2−1)!!

2 !

,

so that the final expression valid, and convergent, forr<ais thus

u(r, θ, φ)=−

GM

a

[

1+

∑∞

=1

(−1)(2−1)!!

2 !

(r

a

) 2 

P 2 (cosθ)

]

.

It is easy to check that the solution obtained has the expected physical value for larger
and forr= 0 and is continuous atr=a.

21.3.4 Separation of variables for inhomogeneous equations

So far our discussion of the method of separation of variables has been limited

to the solution of homogeneous equations such as the Laplace equation and the

wave equation. The solutions of inhomogeneous PDEs are usually obtained using

the Green’s function methods to be discussed below in section 21.5. However, as a

final illustration of the usefulness of the separation of variables, we now consider

its application to the solution of inhomogeneous equations.

Because of the added complexity in dealing with inhomogeneous equations, we

shall restrict our discussion to the solution of Poisson’s equation,

∇^2 u=ρ(r), (21.63)

in spherical polar coordinates, although the general method can accommodate

other coordinate systems and equations. In physical problems the RHS of (21.63)

usually contains some multiplicative constant(s). Ifuis the electrostatic potential

in some region of space in whichρis the density of electric charge then∇^2 u=

−ρ(r)/ 0. Alternatively,umight represent the gravitational potential in some

region where the matter density is given byρ,sothat∇^2 u=4πGρ(r).

We will simplify our discussion by assuming that the required solutionuis

finite on the polar axis and also that the system possesses axial symmetry about

that axis – in which caseρdoes not depend on the azimuthal angleφ.Thekey

to the method is then to assume a separated form for both the solutionuandthe

density termρ.

From the discussion of Laplace’s equation, for systems with axial symmetry

onlym= 0 terms appear, and so the angular part of the solution can be

expressed in terms of Legendre polynomialsP(cosθ). Since these functions form

an orthogonal set let us expand bothuandρin terms of them:

u=

∑∞

=0

R(r)P(cosθ), (21.64)

ρ=

∑∞

=0

F(r)P(cosθ), (21.65)