Page 638 - Advanced_Engineering_Mathematics o'neil

618 CHAPTER 17 The Heat Equation

17.2.4 Transformation of Problems
As with the wave equation, it may be impossible to separate variables in a diffusion problem,
depending on the partial differential equation and/or the boundary conditions. Sometimes we
can transform such a problem into one to which separation of variables applies.

EXAMPLE 17.3
A thin, homogeneous bar extends from x = 0to x = L. The left end is maintained at constant
temperature T 1 and the right end at constant temperature T 2 . The initial temperature in the cross
section at x is f (x).
The initial-boundary value problem modeling this setting is
2
∂u ∂ u
= k for 0 < x < L,t > 0,
∂t ∂x 2
u(0,t) = T 1 ,u(L,t) = T 2 for t > 0,
and

u(x,0) = f (x) for 0 ≤ x ≤ L.
If we set u(x,t)= X(x)T (t),wemusthave X(0)T (t)= T 1 .If T 1 =0, this is satisfied by requiring
that X(0) = 0. If T 1 = 0, then X(0) = 0, and we must have T (t) = T 1 /X(0) = constant, which is
not acceptable.
Perturb the temperature distribution function by setting

u(x,t) = U(x,t) + ψ(x).
The idea is to choose ψ to obtain a problem we know how to solve. Substitute U into the wave
equation to get
2
∂U ∂ U

= k + ψ (x) .
∂t ∂x 2

This is the standard heat equation (17.1) if ψ (x) = 0, hence, if ψ(x) = cx + d.Now
u(0,t) = T 1 = U(0,t) + ψ(0),
and this is the more friendly condition U(0,t) = 0if ψ(0) = T 1 . Thus, choose d = T 1 to have
ψ(x) = cx + T 1 . Next we need
u(L,t) = T 2 = U(L,t) + ψ(L),

and this is just U(L,t) = 0if ψ(L) = T 2 . We need cL + T 1 = T 2 ,so c = (T 2 − T 1 )/L, and
1
ψ(x) = (T 2 − T 1 )x + T 1 .
L
Finally,
u(x,0) = f (x) = U(x,0) + ψ(x),

so
1
U(x,0) = f (x) − ψ(x) = f (x) − (T 2 − T 1 )x − T 1 .
L

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:25 THM/NEIL Page-618 27410_17_ch17_p611-640