Math 453.01 Spring 2013: Quiz # 9 Solutions
Instructor: Dr. Herzog
April 11th, 2013
1. Find the eigenvalues and eigenfunctions of the heat equation on the
rectangle G = cfw_(x, y ) : 0 < x < a, 0 < y < b with boundary conditions
u(0, y, t) = u(a, y, t) =
Math 453.01 Spring 2013: Quiz # 8 Solutions
Instructor: Dr. Herzog
April 4th, 2013
1. Solve the inhomogenous equation
utt c2 uxx = sin(2x/L),
u(x, 0) = 0,
ut (x, 0) = 0
with boundary conditions u(0, t) = u(L, t) = 0. Does the solution oscillate
or damp ou
Math 453.01 Spring 2013: Quiz # 7 Solutions
Instructor: Dr. Herzog
March 26th, 2013
1. If a solution to the wave equation is in the form
u(x, t) = F (x ct) + G(x + ct), show that the energy
1
e(t) = 2 R u2 (x, t) + c2 u2 (x, t) dx satises
t
x
e(t) = e(0)
Math 453.01 Spring 2013: Quiz # 5 Solutions
Instructor: Dr. Herzog
March 19th, 2013
1. Let n (x) = 1 L exp(inx/L) be the normalized eigenfunctions, i.e.,
2
n = 1 for all n. Let
L
cn = f , n =
f (x)n (x) dx.
L
Use Parsevals equality to show that
f
2
|cn |2
Math 453.01 Spring 2013: Quiz # 5 Solutions
Instructor: Dr. Herzog
February 26th, 2013
1. Let u solve ut = kuxx with boundary conditions ux (0, t) = u(L, t) = 0
and u(x, 0) = f (x). Show that
k
L
ux (L, t) dt =
f (x) dx.
0
0
Hint: Use the fact that
(ut ku
Math 453.01 Spring 2013: Quiz # 4 Solutions
Instructor: Dr. Herzog
February 19th, 2013
1. Compute the coecients An of the eigenfunction expansion for the
function f (x) = x on [0, L]. How fast do the coecients tend to zero as
n ? Keeping in mind that the
Math 453.1 Spring 2013: Quiz # 3 Solutions
Instructor: Dr. Herzog
February 7th, 2013
1. Let u(x, t) be a solution to the following IBVP:
ut = kuxx ,
x, t > 0,
u(x, 0) = f (x),
x > 0,
u(0, t) = 0,
t0
and suppose that u, ux , uxx vanish rapidly as x .
(a) C
Math 453.1 Spring 2013: Quiz # 2 Solutions
Instructor: Dr. Herzog
January 31st, 2013
1. Let u(x, t) and v (x, t) both be twice continuously dierentiable solutions
of the equation
ut kuxx = q.
Suppose that u(x, t) v (x, t) for |x| L, t = 0, and for
x = L,
Math 453.1 Spring 2013: Quiz # 1 Solutions
Instructor: Dr. Herzog
January 23rd, 2013
1. Solve:
ut + cux + u = 0, u(x, 0) = f (x).
Solution. The characteristics are
dx
= c,
dt
implying x(t) = ct + x0 . Setting v (t) = u(x(t), t) we see that
v (t) = v (t)
g
Math 453.01 Spring 2013: Exam # 1
Instructor: Dr. Herzog
March 5th, 2013
1. (a) Solve the IVP
ut + x2 ux = 0, u(x, 0) = f (x)
using the fact that the general solution x(t) to x (t) = x(t)2 is
x(t) =
x0
.
1 x0 t
(b) Over what region in the x, t plane does