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CHAPTER 5

Solutions Using Fourier Series

and Integrals

We have already demonstrated solution of partial differential equations for some simple cases
in rectangular Cartesian coordinates in Chapter 2. We now consider some slightly more
complicated problems as well as solutions in spherical and cylindrical coordinate systems to
further demonstrate the Fourier method of separation of variables.

5.1 CONDUCTION (OR DIFFUSION) PROBLEMS
Example 5.1 (Double Fourier series in conduction). We now consider transient heat con-
duction in two dimensions. The problem is stated as follows:

u t = α(u xx + u yy )

u(t, 0, y) = u(t, a, y) = u(t, x, 0) = u(t, x, b) = u 0
u(0, x, y) = f (x, y) (5.1)

That is, the sides of a rectangular area with initial temperature f (x, y) are kept at a constant
temperature u 0 . We first attempt to scale and nondimensionalize the equation and boundary
conditions. Note that there are two length scales, a and b. We can choose either, but there will
remain an extra parameter, either a/b or b/a in the equation. If we take ξ = x/a and η = y/b
then (5.1) can be written as
a 2 a 2
u t = u ξξ + u ηη (5.2)
α b 2
2
2
The time scale is now chosen as a /α and the dimensionless time is τ = αt/a . We also choose
a new dependent variable U(τ, ξ, η) = (u − u 0 )/( f max − u 0 ). The now nondimensionalized
system is
2
U τ = U ξξ + r U ηη (5.3)
U(τ, 0,η) = U(τ, 1,η) = U(τ, ξ, 0) = U(τ, ξ, 1) = 0
U(0,ξ,η) = ( f − u 0 )/( f max − u 0 ) = g(ξ, η)