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Chapter 7

The Diffusion Equation

7.1 Solved Problems

7.2 Supplementary Problems

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Exercise 142. Consider the solution 1 − x − 2kt of the diffusion equation. Find
the locations of its maximum and its minimum in the closed rectangle (0 ≤ x ≤
1, 0 ≤ t ≤ T).

Exercise 143. Solve the diffusion equation with the initial condition φ(x) = 1
for Ix| < l and φ(x) = 0 for Ixl > l. Write your answer in terms of Erf(x).

Exercise 144. Do the same for φ(x) = 1 for x > 0 and φ(x) = 3 for x < 0.

3x
Exercise 145. Use (??) to solve the diffusion equation if φ(x) = e .

Exercise 146. Solve the diffusion equation if φ(x) = e −x for x > 0 and φ(x) = 0

for x < 0.

Exercise 147. Solve the diffusion equation u = ku xx with the initial condition
t
2
u(x, 0) = x by the following special method. First show that u xxx satisfies the
diffusion equation with zero initial condition. Therefore, by uniqueness, u xxx = 0.
2
Integrating this result thrice, obtain u(x, t) = A(t)x + B(t)x + C(t). Finally, it’s
easy to solve for A, B, and C by plugging into the original problem.

Exercise 148. Solve the diffusion equation with constant dissipation: u −ku +
xx
t
bu = 0 for −∞ < x < ∞ with u(x, 0) = φ(x), whereb > 0 is a constant. (Hint:
Make the change of variables u(x, t) = e −bt v(x, t).)

Exercise 149. Solve the diffusion equation with variable dissipation: u −ku +
t
xx
2
bt u = 0 for −∞ < x < ∞ with u(x, 0) = φ(x), where b > 0 is a constant. (Hint:
3
2
The solutions of the ODE w + bt w = 0 are Ce −bt /3 . So make the change of
t
3
variables u(x, t) = e −Bt /3 v(x, t) and derive an equation for v.)

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