Homework Set 2: Math 716, Due: Friday, January 26th
1. Consider the wave equation
utt = c2 uxx , u(x, 0) = (x), ut (x, 0) = (x)
for x R with c = 0. Assume C2 and C1 have compact support (i.e. there is
Solution to Homework Set 1, Math 716
1. Calculate and plot the characteristic curves of
yux + xuy = 0
where (x, y ) R2 . Afterwards, derive the general solution u(x, y ) under the additional
assumptio
Week 10 Lectures, Math 716, Tanveer
1
Examples of Fourier-Transform of Distributions
Example: Using the same argument, as for F [1], except with k replaced by k , we nd:
F [exp[i x] (k) = (2 )m/2 (k )
Week 9 Lectures, Math 716, Tanveer
1
1.1
Greens function as a distribution
Laplace Operator
For the Poisson-Problem with homogeneous boundary condition:
u = f for x , u = 0 on
(1)
we know that
u(x0 )
Week 8 Lectures, Math 716, Tanveer
In the next few weeks lecture, we will consider Greens function for dierent linear dierential
equations. As we shall see, this is a very valuable constructive techni
Week 7 Lectures, Math 716, Tanveer
1
Introduction
Recall we discussed completeness of Fourier Series. This is relevant for constant coecient partial
dierential equations in simple rectangular geometri
Week 6 Lectures, Math 716, Tanveer
1
Fourier Series
In the context of separation of variable to nd solutions of PDEs, we encountered
f (x) =
nx
for x (0, l)
l
bn sin
n=1
or
a0
nx
f (x) =
+
an cos
for
Solution to homework Set 6: Math 716
1. Determine the Greens function G(x, x0 ) for Dirichlet condition for Laplaces equation in
3-D dimensions in the hemispherical domain:
= cfw_x : |x| < 1, x3 > 0
Solution to Set 5: Math 716
1. Show that the partial dierential operator A dened by:
Au =
(p u) + qu
in a bounded Rn for p(x) > 0 is symmetric, with respect to the usual L2 inner-product.
What condi
Solution to Homework Set 4: Math 716
1. Using the two approaches listed below as (a) and (b), nd two dierent representation of
solution to the following Initial-Boundary value problem for the heat equ
Homework Set 3: Math 716, Due: Wednesday, February 7th
1. Use energy method to prove uniqueness of classical solution to the initial value problem
for the damped wave equation ( > 0):
utt +ut = u for
Homework Set 2: Math 716, Due: Monday, January 30th
1. Consider the wave equation
utt = c2 uxx , u(x, 0) = (x), ut (x, 0) = (x)
for x R with constant c = 0. Assume C2 and C1 have compact support (i.e.
Finals, Math 716, Tanveer, Due: March 15th, 12 noon
Instructions: No collaboration or any discussion of any kind relating to nals.
Do not consult material outside of class notes, text and homework sol