QUIZ 3 (385) FALL 2004
Problem 1. Find a solution to the following initial value problem:
utt = c2 uxx
<x<
u(x, 0) = f (x),
ut (x, 0) = g (x)
where f (x) = 1 for |x| 1 and f (x) = 0 for |x| > 1; g (x) = 1 |x| for |x| 1
and g (x) = 0 for |x| > 1;
Problem 2
QUIZ 2 385 FALL 2004
Problem 1. Show that the Sturm-Liouville problem
X + X = 0,
1
1
3X (log ) + 5X (log ) = 0
2
2
X (0) = 0,
has a nontrivial solution for = 1
Problem 2. Show that the problem
X + X = 0,
X (0) = 0,
2X (1) = 3
has a solution for some < 0.
OUIZ 1 385 FALL 2004
Problem 1.Find a Fourier series for the function f (x) = x2 where 0 <
x < .
Problem 2. Given a function f (x) = 1cos(x) for 0 x . Consider
x
the even extension of this function on the interval [, ]. Find all points on
the segment [, ]
EXAM 2 385 FALL 2004
Problem 1. Consider the problem
x2 X + xX + X = 0,
X (1) = 0,
X (e) = 0
(1)
A) Find all possible solutions to problem (1).
B) Is this problem a REGULAR Sturm-Liouville problem?
Problem 2. Show that the eigenvalues of the Sturm-Liouvil
QUIZ 4
Problem 1. Solve the boundary value problem
1
r r
r
v
r
+
1 2v
=0
r2 2
0 < r < c, ,
v (c, ) = cos2 ().
Problem 2.Solve the boundary value problem
1
r r
r
v
r
+
1 2v
=0
r2 2
0 < r < c, ,
vr (c, ) = sin2 ().
Problem 3.Solve the boundary value problem
Math 385 Fall 2013 Quiz 5
Show all work. Your answers must be justied to get full
credit.
(1) Let
5
u(m,t) = 8W[cos(27r(2: 20) cos(27r(:c + 2t))].
Show by directly computing the necessary quantities that u(a:, it)
satises each of the parts the follo
Math 385 Fall 2013 Quiz 3
Show all work. Your answers must be justied to get full
credit.
(1) Use the method of separation of variables to nd product solu
g tions of the following equations
2
%%=gx2ozu, 0<:I:<L,t>0,
u(0,t) = u(L,t) : 07 15> 0.
Assume
MATH 385 Spring 2015
Homework 1 - Answer
1.2.5: The derivation is similar to equations (1.2.11) (1.2.14) but with an additional term which is due
to the source. We start from the conservation law for the chemical concentration:
A
d
dt
b
b
u( u)dx,
u(x, t)