Page 63 - 6637

Chapter 9

The Heat Equation

9.1 Solved Problems

Exercise 177. Is the solution of the Cauchy problem for the heat equation unique?

u − κu xx = q(x, t), −∞ < x < ∞, t > 0
t
u(x, 0) = f(x)

Solution: Let u and v both be solutions of the Cauchy problem for the heat equation.
Let w be the difference of these solutions. w satisfies the problem

w − κw xx = 0, −∞ < x < ∞, t > 0,
t
w(x, 0) = 0.

We can solve this problem with the Fourier transform.

2
ˆ w + κω ˆw = 0, ˆ w(ω, 0) = 0
t
ˆ w = 0
w = 0

Since u − v = 0, we conclude that the solution of the Cauchy problem for the heat
equation is unique.

Exercise 178. Consider the heat equation with a time-independent source term
and inhomogeneous boundary conditions.

u = κu xx + q(x)
t
u(0, t) = a, u(h, t) = b, u(x, 0) = f(x)

Solution: Let µ(x) be the equilibrium temperature. It satisfies an ordinary differ-
ential equation boundary value problem.

q(x)
00
µ = − , µ(0) = a, µ(h) = b
κ

To solve this boundary value problem we find a particular solution µ that satisfies
p
homogeneous boundary conditions and then add on a homogeneous solution µ h

59