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626 CHAPTER 17 The Heat Equation

17. Solve and
2
∂u ∂ u
= 4 − Au for 0 < x < 9,t > 0,
∂t ∂x 2 u(x,0) = f (x) for 0 ≤ x ≤ L.
u(0,t) = u(9,t) = 0for t ≥ 0,
Choose a value of time, and using the same set of
u(x,0) = 0for0 ≤ x ≤ 9. axes, graph the twentieth partial sum of the solution
together with the solution of the problem with F(x,t)
Graph the twentieth partial sum of the solution for
t = 0.2 with A = 1/4. Repeat this for t = 0.7and removed. Repeat this for several times. This gives some
t = 1.4 sense of the influence of F(x,t) on the temperature
distribution.
18. Solve
∂u ∂ u 19. k = 4, F(x,t) = t, f (x) = x(π − x), L = π
2
= 9 for 0 < x < L,t > 0,
∂t ∂x 2 20. k = 1, F(x,t) = x sin(t), f (x) = 1, L = 4
u(0,t) = T,u(L,t) = 0for t ≥ 0, 2
21. k = 1, F(x,t) = t cos(x), f (x) = x (5 − x), L = 5
u(x,0) = 0for0 ≤ x ≤ L. 22. k = 4, f (x) = sin(πx/2), L = 2,
In each of Problems 19 through 23, solve the problem
K for 0 ≤ x ≤ 1
2
∂u ∂ u F(x,t) =
= k + F(x,t) for 0 < x < L,t > 0, 0 for 1 < x ≤ 2
∂t ∂x 2
u(0,t) = u(L,t) = 0for t ≥ 0, 23. k = 16, F(x,t) = xt, f (x) = K, L = 3

17.3 Solutions in an Infinite Medium

We will consider problems involving the heat equation over the entire line or half-line.

17.3.1 Problems on the Real Line
For a setting in which one dimension is very much greater than the others, it is sometimes useful
to model heat conduction or a diffusion process by imagining the space variable free to vary
over the entire real line. In this case, there is no boundary condition, but we look for bounded
solutions. The problem we will solve is
2
∂u ∂ u
= k for −∞ < x < ∞,t > 0
∂t ∂x 2
and
u(x,0) = f (x) for −∞ < x < ∞.
Separation of variables yields

X + λX = 0, T + λkT = 0.

2
As with the wave equation on the line, the eigenvalues λ = ω ≥ 0 and the eigenfunctions have
the form a ω cos(ωx) + b ω sin(ωx).
2
2
The problem for T is T + kω T = 0 with solutions that are constant multiples of e −ω kt .

For ω ≥ 0, we now have functions
2
u ω (x,t) =[a ω cos(ωx) + b ω sin(ωx)]e −ω kt
that satisfy the heat equation and are bounded for all x. To satisfy the initial condition, attempt a
superposition
∞ 2

u(x,t) = [a ω cos(ωx) + b ω sin(ωx)]e −ω kt dω. (17.11)
0

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October 14, 2010 15:25 THM/NEIL Page-626 27410_17_ch17_p611-640