Math 286 Homework 5
Due Tuesday March 24
1. Find all radial
p solutions to the two-dimensional Laplace’s equation. That is, find all solutions of the form u = u(r),
where r = x2 + y 2 . Find the steady state temperature distribution in the annular domain 1 ≤ r ≤ 2 if the inner
circle r = 1 is held at 0 degrees and the outer circle r = 2 is held at 10 degrees.
2. Find the solution of the PDE
2uxx − 4uxt + ux = 0
in terms of two arbitrary functions.
Hint: Make a transformation.
3. If |φ(x)| ≤ M for all x, where M is a positive constant, show that the solution u to the Cauchy problem (2.1) − (2.2)
satisfies |u(x, t)| ≤ M for all x and t > 0.
Hint: Use
R the Rcalculus fact that the absolute value of an integral is less than or equal to the integral of the absolute
value: | f | ≤ |f |.
4. Show that if u(x, t) and v(x, t) are any two solutions to the heat equation (2.1), then w(x, y, t) = u(x, t)v(y, t) solves the
two-dimensional heat equation wt = k(wxx + wyy ). Can you guess the solution to the two-dimensional Cauchy problem
wt = k(wxx + wyy ), (x, y) ∈ R2 , t > 0
w(x, y, 0) = ψ(x, y),
(x, y) ∈ R2 ?
5. Solve the outgoing signal problem
utt − c2 uxx = 0, −∞ < t < ∞
ux (0, t) = s(t), −∞ < t < ∞
where s(t) is a known signal.
Hint: Look for a right-traveling wave solution.
6. The three-dimensional wave equation is
utt − c2 ∆u = 0,
wherepu = u(x, y, z, t) and ∆ is the Laplacian operator. For waves with spherical symmetry, u = u(ρ, t), where
ρ = x2 + y 2 + z 2 . In this special case the Laplacian is given by (Section 1.8) ∆u = uρρ + ρ2 uρ . By introducing a
change of dependent variable U = ρu, show that the general solution for the spherically symmetric wave equation
2
utt = c2 (uρρ + uρ )
ρ
is
1
(F (ρ − ct) + G(ρ + ct)).
ρ
Why do you think an outward-moving wave u = F (ρ − ct)/ρ decays in amplitude?
u=
Note: The waves modeled in this problem are truly 3-dimensional - think of a perfectly spherical light bulb - the light
waves that propagate from the bulb would be modeled by this problem. By the way, that should help you understand
why the waves must decay....)
7. Show that the Cauchy problem for the backward diffusion equation,
ut + uxx = 0, x ∈ R, t > 0,
u(x, 0) = f (x), x ∈ R,
is unstable by considering the solutions
u(x, t) = 1 +
for large n.
1
1 n2 t
e sin(nx)
n
8. Find a d’Alembert type of solution for the problem
utt = c2 uxx , −∞ < x < ∞, t > 0,
u(x, 0) = f (x), ut (x, 0) = g(x),
−∞ < x < ∞
Verify that your solution satisfies the differential equation and initial conditions.
9. Suppose you are given the solution u(x, y) which satisfies the wave equation
utt = c2 uxx
u(x, 0) = f (x), ut (x, 0) = g(x),
u(a, t) = u(b, t) = 0
a < x < b, t > 0
a<x<b
t>0
Use the energy
Z
b
E(t) =
a
1 2 1 2 2
u + c ux dx
2 t 2
to show the solution is unique.
Hint: Start by showing that E 0 (t) ≤ 0. You might recall we did something similar in 1.3.2
2