Name
Spring 2008
Score
Math 451 Final Exam
Due Tuesday, May 6 by 5pm
Please provide enough details of your work so that I can follow your reasoning. Without
details, I cannot assign partial credit.
1. (20 points) The initial boundary value problem for the damped wave equation governs
the displacement of a string immersed in a fluid. The string has unit length and is fixed
at its ends. Its initial displacement is given by f (x), and it has zero initial velocity.
The constant k > 0 is the damping constant, and you may assume that k < 2πc. Use
separation of variables to find the solution.
utt + kut = c2 uxx ,
0 < x < 1, t > 0
u(0, t) = 0, u(1, t) = 0,
u(x, 0) = f (x), ut (x, 0) = 0,
t>0
0<x<1
2. (a) (5 points) Assume that the function u ∈ S and a ∈ lR. Verify the following
property of the Fourier Transform.
F(u(x + a)) = e−iaξ û(ξ)
(b) (15 points) Use the Fourier Transform to find a formula for the solution to the
initial value problem for the convection-diffusion equation
ut − cux − uxx = 0,
u(x, 0) = f (x),
x ∈ lR, t > 0
x ∈ lR.
You may assume that f ∈ S, and if you do this problem correctly, part (a) should
be useful to you.
3. (20 points) Let c > 0 denote a constant, and let g denote the gravitational constant.
Solve the following pde using Laplace Transforms. The solution shows what
happens to a falling cable (or an elastic string) that is lying on a table (or otherwise
supported underneath) when the table is suddenly removed. See Problem 3 on page 396
of the Logan textbook. Also assume that the deflection u(x, t) is bounded. Generate
a time series plot of your solution using the parameter values c = 2.0, g = 32.2 at the
time instants t = 1.0, 2.0, 3.0, 5.0.
utt − c2 uxx = g,
u(0, t) = 0,
x > 0, t > 0
t>0
u(x, 0) = 0, ut (x, 0) = 0,
x > 0.
4. (10 points) Problem 2 on page 429 of Logan textbook.
5. (10 points) Let c > 0 be constant. Given any continuously differentiable function
f : lR → lR, show that
u(x, t) = f (x − ct),
and
v(x, t) = f (x + ct)
both solve the second order linear PDE
utt − c2 uxx = 0.
6. (10 points) Problem 3a on page 429 of Logan textbook. In addition, find a formula for
the solution to the pde.
7. (10 points) Problem 1a on page 429 of Logan textbook.
2