Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

y

y y

b

b

b

f(y)

f(y) 0

0

0

0

0

0

0

0

a

a a

x

x x

g(x)

g(x)

(a) (b)

(c)

Figure 21.3 Superposition of boundary conditions for a metal plate.

whilstw(x, y) is the solution satisfying the boundary conditions in figure 21.3(b). It is clear
thatv(x, y) is simply given by the solution to the previous example,

v(x, y)=

∑

nodd

Bnsinh

[

nπ(a−x)

b

]

sin

(nπy

b

)

,

whereBnis given by (21.21). Moreover, by symmetry,w(x, y) must be of the same form as
v(x, y) but withxandainterchanged withyandb, respectively, and withf(y) in (21.21)
replaced byg(x). Therefore the required solution can be written down immediately without
further calculation as

u(x, y)=

∑

nodd

Bnsinh

[

nπ(a−x)

b

]

sin

(nπy

b

)

+

∑

nodd

Cnsinh

[

nπ(b−y)

a

]

sin

(nπx

a

)

,

theBnbeing given by (21.21) and theCnby

Cn=

2

asinh(nπb/a)

∫a

0

g(x)sin(nπx/a)dx.

Clearly, this method may be extended to cases in which three or four sides of the plate
have non-zero boundary conditions.

As a final example of the usefulness of the principle of superposition we now

consider a problem that illustrates how to deal with inhomogeneous boundary

conditions by a suitable change of variables.