
74 Applied calculus of variations for engineers
The coefficient of the solution is also a straightforward generalization as
c
mnr
=
8
abc
c
0
b
0
a
0
f(x, y, z)sin
mπx
a
sin
nπy
b
sin
rπz
c
dxdydz.
These last two solutions were the analytic solutions to the two- and three-
dimensional heat conduction problems. The computational solution of the
two-dimensional problem will be further addressed in Chapter 12.
Let us now solve a problem of two spatial variables again, but with a tem-
poral variable whose second derivative is present:
h
2
(
∂
2
u
∂x
2
+
∂
2
u
∂y
2
)=
∂
2
u
∂t
2
.
We assume constant boundary conditions:
u(0,y,t)=u(a, y, t)=0,
and
u(x, 0,t)=u(x, b, t)=0.
The a, b are the dimensions ...