Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

superposing solutions corresponding to different allowed values of the separation

constants. To take a two-variable example: if

uλ 1 (x, y)=Xλ 1 (x)Yλ 1 (y)

is a solution of a linear PDE obtained by giving the separation constant the value

λ 1 , then the superposition

u(x, y)=a 1 Xλ 1 (x)Yλ 1 (y)+a 2 Xλ 2 (x)Yλ 2 (y)+···=

∑

i

aiXλi(x)Yλi(y)

(21.16)

is also a solution for any constantsai, provided that theλiare the allowed values

of the separation constantλgiven the imposed boundary conditions. Note that

if the boundary conditions allow any of the separation constants to be zero then

the form of the general solution is normally different and must be deduced by

returning to the separated ordinary differential equations. We will encounter this

behaviour in section 21.3.

The value of the superposition approach is that a boundary condition, say that

u(x, y) takes a particular formf(x)wheny= 0, might be met by choosing the

constantsaisuch that

f(x)=

∑

i

aiXλi(x)Yλi(0).

In general, this will be possible provided that the functionsXλi(x) form a complete

set – as do the sinusoidal functions of Fourier series or the spherical harmonics

discussed in subsection 18.3.

A semi-infinite rectangular metal plate occupies the region 0 ≤x≤∞and 0 ≤y≤bin

thexy-plane. The temperature at the far end of theplate and along its two long sides is

fixed at 0 ◦C. If the temperature of the plate atx=0is also fixed and is given byf(y),find

the steady-state temperature distribution u(x,y) of the plate. Hence find the temperature

distribution iff(y)=u 0 ,whereu 0 is a constant.

The physical situation is illustrated in figure 21.1. With the notation we have used several
times before, the two-dimensional heat diffusion equation satisfied bythe temperature
u(x, y, t)is

κ

(

∂^2 u

∂x^2

+

∂^2 u

∂y^2

)

=

∂u

∂t

,

withκ=k/(sρ). In this case, however, we are asked to find the steady-state temperature,
which corresponds to∂u/∂t= 0, and so we are led to consider the (two-dimensional)
Laplace equation

∂^2 u

∂x^2

+

∂^2 u

∂y^2

=0.

We saw that assuming a separable solution of the formu(x, y)=X(x)Y(y)ledto
solutions such as (21.14) or (21.15), or equivalent forms withxandyinterchanged. In
the current problem we have to satisfy the boundary conditionsu(x,0)=0=u(x, b)and
so a solution that is sinusoidal inyseems appropriate. Furthermore, since we require
u(∞,y) = 0 it is best to write thex-dependence of the solution explicitly in terms of