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Supplementary Problems

Exercise 137. (a) Solve the problem


u − u xx = 0 0 < x < ∞, 0 < t,
tt

u(0, t) = t 0 ≤ t,
1+t

u(x, 0) = u (x, 0) = 0 0 ≤ x < ∞.

t
(b) Show that the limit lim u(cx, x): = φ(c) exists for all c > 0. What is the
x→∞
limit?

Exercise 138. Consider the Cauchy problem

u − 4u xx = F(x, t), −∞ < x < ∞, t > 0,
tt

u(x, 0) = f(x), u (x, 0) = g(x), −∞ < x < ∞,
t

x, 0 < x < 1,


 (

1, 1 < x < 2, 1 − x , |x| < 1,
 2
where f(x) = g(x) = and F(x, t) =
3 − x, 2 < x < 3, 0, |x| > 1




0, x > 3, x < 0,

x
−4e on t > 0, −∞ < x < ∞.
(a) Is the d’Alembert solution of the problem a classical solution? If your answer
is negative, find all the points where the solution is singular.
(b) Evaluate the solution at (1, 1).
Exercise 139. Solve the problem

x
u − 4u xx = e + sin t, −∞ < x < ∞, t > 0,
tt
u(x, 0) = 0 − ∞ < x < ∞,

1
u (x, 0) = − ∞ < x < ∞.
t
1 + x 2

Exercise 140. Find the general solution of the problem u −u xxx = 0, u (x, 0) =
x
ttx
0, u (x, 0) = sin x, in the domain {(x, t)| − ∞ < x < ∞, t > 0}.
xt
Exercise 141. Solve the problem

u − u xx = xt, −∞ < x < ∞, t > 0,
tt

u(x, 0) = 0 − ∞ < x < ∞,

x
u (x, 0) = e − ∞ < x < ∞.
t

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