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Supplementary Problems

u = iu xx on (0, l) with Dirichlet conditions at the ends. Separate variables and
t
find its representation as a series.

Exercise 187. Solve the diffusion problem u = ku xx in 0 < x < l, with the
t
mixed boundary conditions u(0, t) = u (l, t) = 0.
x
2
1 2 2
Solution: The eigenvalues are (n + ) π /l and the eigenfuncfions are sin[(n +
2
1 )πx/l] for n = 0, 1, 2, . . .
2
Exercise 188. Consider diffusion inside an enclosed circular tube. Let its length
(circum-ference) be 2l. Let x denote the arc length parameter where −l ≤ x ≤ l.

Then the concentration of the diffusing substance satisfies u = ku xx for −l ≤ x ≤
t
l, u(−l, t) = u(l, t) and u (−l, t) = u (l, t). These are called periodic boundary
x
x
conditions.
2
(a) Show that the eigenvalues are λ = (πn/l) for n = 0, 1, 2, 3, . . .
(b) Show that the concentration is

∞
a 0 X |pinx |pinx 2 2 2
u(x, t)= + (a cos + b sin )e −n π kt/l .
n
n
2 l l
n=1
Exercise 189. Solve the equation u = 17u , 0 < x < π, t > 0, with the
t
xx
boundary conditions u(0, t) = u(π, t) = 0 t ≥ 0, and the initial conditions
(
0 0 ≤ x ≤ π/2,
u(x, 0) =
2, π/2 < x ≤ π.

Exercise 190.
(a) Using the method of separation of variables, find a (formal) solution of the

problem
u − ku xx = 0 0 < x < L, t > 0,
t
u (0, t) = u (L, t) = 0 t ≥ 0,
x
x
u(x, 0) = f(x) 0 ≤ x ≤ L,
describing the heat evolution of an insulated one-dimensional rod (Neumann
problem).

(b) Solve the heat equation u = 12u xx in 0 < x < π, t > 0 subject to the
t
following boundary and initial conditions:

u (0, t) = u (π, t) = 0, t ≥ 0,
x
x
3
u(x, 0) = 1 + sin x, 0 ≤ x ≤ π.
(c) Find lim t→∞ u(x, t) for all 0 < x < π, and explain the physical interpretation

of your result.

Exercise 191.

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