Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS

y

b

0 x

u=0

u=0

u=f(y) u→ 0

Figure 21.1 A semi-infinite metal plate whose edges are kept at fixed tem-

peratures.

exponentials rather than of hyperbolic functions. We therefore write the separable solution
in the form (21.15) as

u(x, y)=[Aexpλx+Bexp(−λx)](Ccosλy+Dsinλy).

Applying the boundary conditions, we see firstly thatu(∞,y) = 0 impliesA=0ifwe
takeλ>0. Secondly, sinceu(x,0) = 0 we may setC= 0, which, if we absorb the constant
DintoB, leaves us with

u(x, y)=Bexp(−λx)sinλy.

But, using the conditionu(x, b) = 0, we require sinλb=0andsoλmust be equal tonπ/b,
wherenis any positive integer.
Using the principle of superposition (21.16), the general solution satisfying the given
boundary conditions can therefore be written

u(x, y)=

∑∞

n=1

Bnexp(−nπx/b) sin(nπy/b), (21.17)

for some constantsBn. Notice that in the sum in (21.17) we have omitted negative values of
nsince they would lead to exponential terms that diverge asx→∞.Then= 0 term is also
omitted since it is identically zero. Using the remaining boundary conditionu(0,y)=f(y)
we see that the constantsBnmust satisfy

f(y)=

∑∞

n=1

Bnsin(nπy/b). (21.18)

This is clearly a Fourier sine series expansion off(y) (see chapter 12). For (21.18) to
hold, however, the continuation off(y) outside the region 0≤y≤bmust be an odd
periodic function with period 2b(see figure 21.2). We also see from figure 21.2 that if
the original functionf(y) does not equal zero at either ofy=0andy=bthen its
continuation has a discontinuity at the corresponding point(s); nevertheless, as discussed
in chapter 12, the Fourier series will converge to the mid-points of these jumps and hence
tend to zero in this case. If, however, the top and bottom edges of the plate were held not
at 0◦C but at some other non-zero temperature, then, in general, the final solution would
possess discontinuities at the cornersx=0,y=0andx=0,y=b.
Bearing in mind these technicalities, the coefficientsBnin (21.18) are given by

Bn=

2

b

∫b

0

f(y)sin

(nπy

b

)

dy. (21.19)