Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

f(y)

−b 0 b y

Figure 21.2 The continuation off(y) for a Fourier sine series.

Therefore, iff(y)=u 0 (i.e. the temperature of the side atx= 0 is constant along its
length), (21.19) becomes

Bn=

2

b

∫b

0

u 0 sin

(nπy

b

)

dy

=

[

−

2 u 0

b

b

nπ

cos

(nπy

b

)]b

0

=−

2 u 0

nπ

[(−1)n−1] =

{

4 u 0 /nπ fornodd,

0forneven.

Therefore the required solution is

u(x, y)=

∑

nodd

4 u 0

nπ

exp

(

−

nπx

b

)

sin

(nπy

b

)

.

In the above example the boundary conditions meant that one term in each

part of the separable solution could be immediately discarded, making the prob-

lem much easier to solve. Sometimes, however, a little ingenuity is required in

writing the separable solution in such a way that certain parts can be neglected

immediately.

Suppose that the semi-infinite rectangular metalplate in the previous example is replaced

by one that in thex-direction has finite lengtha. The temperature of the right-hand edge

is fixed at 0 ◦Cand all other boundary conditions remain as before. Find the steady-state

temperature in the plate.

As in the previous example, the boundary conditionsu(x,0) = 0 =u(x, b) suggest a solution
that is sinusoidal iny. In this case, however, we requireu=0onx=a(rather than at
infinity) and so a solution in which thex-dependence is written in terms of hyperbolic
functions, such as (21.14), rather than exponentials is more appropriate. Moreover, since
the constants in front of the hyperbolic functions are, at this stage, arbitrary, we may
write the separable solution in the most convenient way that ensures that the condition
u(a, y) = 0 is straightforwardly satisfied. We therefore write

u(x, y)=[Acoshλ(a−x)+Bsinhλ(a−x)](Ccosλy+Dsinλy).

Now the conditionu(a, y) = 0 is easily satisfied by settingA=0.Asbeforethe
conditionsu(x,0)=0=u(x, b)implyC=0andλ=nπ/bfor integern. Superposing the