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Supplementary Problems

(a) Using the separation of variables method find a (formal) solution of the fol-
lowing periodic heat problem:
u − ku xx = 0 0 < x < 2π, t > 0,
t
u(0, t) = u(2π, t), u (0, t) = u (2π, t) t ≥ 0,
x
x
u(x, 0) = f(x) 0 ≤ x ≤ 2π,
where f is a smooth periodic function. This system describes the heat evolu-
tion on a circular insulated wire of length 2π.

(b) Find lim t→∞ u(x, t) for all 0 < x < 2π, and explain the physical interpretation
of your result.
(c) Show that if v is an arbitrary partial derivative of the solution u, then v(0, t) =
v(2π, t) for all t ≥ 0.

Exercise 192. Solve the following heat problem:
u − ku xx = A cos αt, 0 < x < 1, t > 0,
t
u (0, t) = u (1, t) = 0 t > 0,
x
x
2
u(x, 0) = 1 + cos πx 0 ≤ x ≤ 1, t > 0.
Exercise 193. Consider the problem
−t
u − u xx = e sin 3x, 0 < x < π, t > 0,
t
u(0, t) = u(π, t) = 0 t ≥ 0,
u(x, 0) = f(x) 0 ≤ x ≤ π.
(a) Solve the problem using the method of eigenfunction expansion.
(b) Find u(x, t) for f(x) = x sin x.
(c) Show that the solution u(x, t) is indeed a solution of the equation u − u xx =
t
−t
e sin 3x, 0 < x < π, t > 0.

Exercise 194. Consider the problem
u − u xx − hu = 0, 0 < x < π, t > 0,
t
u(0, t) = u(π, t) = 0 t ≥ 0, where h is a real constant.

u(x, 0) = x(π − x) 0 ≤ x ≤ π,
(a) Solve the problem using the method of eigenfunction expansion.
(b) Does lim u(x, t) exist for all 0 < x < π?
t→∞
Hint: Distinguish between the following cases: (a) h < 1, (b) h = 1, (c) h > 1.

Exercise 195. Consider the problem
u = u xx + αu 0 < x < 1, t > 0,
t
u(0, t) = u(1, t) = 0 t ≥ 0,

u(x, 0) = f(x) 0 ≤ x ≤ 1, f ∈ C([0, 1]).
(a) Assume that α = −1 and f(x) = x and solve the problem.
(b) Prove that for all α ≤ 0 and all f, the solution u satisfies lim u(x, t) = 0.
t→∞
2
2
(c) Assume now that π < α < 4π . Does lim u(x, t) exist for all f? If your
t→∞
answer is no, find a necessary and sufficient condition on f which ensures the
existence of this limit.

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