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Supplementary Problems

solution of the Cauchy problem with odd data is odd. In particular, the solution
with odd data is zero for x = 0 and all t ≥ 0.
6
3
2
(b) Solve the problem with f(x) = x + x , and g(x) = sin x, and evaluate
u(1, i) for i = 1, 2, 3. Is the solution classical?

Exercise 133. (a) Solve the following initial boundary value problem for a vibrat-
ing semi-infinite string with a free boundary condition: u − u xx = 0, 0 < x < ∞,
tt
t > 0,

u (0, t) = 0, t > 0,
x

u(x, 0) = f(x), 0 ≤ x < ∞,

u (x, 0) = g(x), 0 ≤ x < ∞,

t
2
1
where f ∈ C ([0, ∞)) and g ∈ C ([0, ∞)) satisfy the compatibility conditions
0
0
f (0) = g (0) = 0.
+
+
Hint: Extend the functions f and g as even functions f and eg on the line. Solve
e
the Cauchy problem with initial data f and eg, and show that the restriction of this
e
solution to the half-plane x ≥ 0 is a solution of the problem.
3
3
6
(b) Solve the problem with f(x) = x + x , g(x) = sin x, and evaluate u(1, i)
for i = 1, 2, 3. Is the solution classical?
Exercise 134. (a) Solve the problem u − u xx = 1, −∞ < x < ∞, t > 0,
tt
(
u(x, 0) = x 2 −∞ < x < ∞,
u (x, 0) = 1 −∞ < x < ∞.
t

Exercise 135. Solve the Darboux problem: u − u xx = 0, t > max{−x, x},
tt
(
φ(t), x = t, t ≥ 0,
2
t ≥ 0, u(x, t) = where φ, ψ ∈ C ([0, ∞)) satisfies φ(0) =
ψ(t), x = −t, t ≥ 0,
ψ(0). (b) Prove that the problem is well posed.

Exercise 136. A pressure wave generated as a result of an explosion satisfies
the equation P − 16P xx = 0 in the domain {(x, t) : −∞ < x < ∞, t > 0},
tt
where P(x, t) is the pressure at the point x and time t. The initial conditions at
( (
10, |x| ≤ 1, 1, |x| ≤ 1,
the explosion time t = 0 are P(x, 0) = P (x, 0) =
t
0, |x| > 1, 0, |x| > 1.
A building is located at the point x = 10. The engineer who designed the building
0
determined that it will sustain a pressure up to P = 6. Find the time t when the
0
pressure at the building is maximal. Will the building collapse?

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