Page 33 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 23
Recall that

V = W(ξ, 1 − η) and U = V + W

2.1.18 Lessons
If there are two nonhomogeneous boundary conditions break the problem into two problems
that can be added (since the equations are linear) to give the complete solution. If you are
unsure of the sign of the separation constant just assume a sign and move on. Listen to what the
mathematics is telling you. It will always tell you if you choose wrong.

Example 2.4. A Non-homogeneous Heat Conduction Problem

Consider now the arrangement above, but with a heat source, and with both boundaries held
at the initial temperature u 0 . The heat source is initially zero and is turned on at t = 0 . The
+
exercise illustrates the method of solving the problem when the single nonhomogeneous condition is in
the partial differential equation rather than one of the boundary conditions.

ρcu t = ku xx + q (2.75)

u(0, x) = u 0
u(t, 0) = u 0 (2.76)
u(t, L) = u 0

2.1.19 Scales and Dimensionless Variables
Observe that the length scale is still L, so we define ξ = x/L. Recall that k/ρc = α is the
diffusivity. How shall we nondimensionalize temperature? We want as many ones and zeros
in coefficients in the partial differential equation and the boundary conditions as possible.
Define U = (u − u 0 )/S, where S stands for “something with dimensions of temperature” that
we must find. Dividing both sides of the partial differential equation by q and substituting
for x

2
L SρcU t kSU ξξ
= + 1 (2.77)
q q

Letting S = q/k leads to one as the coefficient of the first term on the right-hand side.
2
Choosing the same dimensionless time as before, τ = αt/L results in one as the coefficient of