MATH 5163
Homework 9
Due Fri, 4/18/14, 5 p.m.
Problem 1. [Spaces of sequences]
Let p , , and s stand for linear spaces of sequences x = (xj )jN = (x1 , x2 , . . .) endowed with
the norms
x
p
1/p
p
= ( xj )
for p [1, ) ,
j=1
x
= sup xj ,
jN
x
s
1 xj
.
j
j=
MATH 5163
Homework 7
Due Fri, 3/6/14, 5 p.m.
Problem 1. [Green function for Dirichlet BVP for Poisson equation in half-ball]
In this problem you will use the methods of electrostatic images to construct the Green
function for the Dirichlet boundary value
MATH 5163
Homework 3
Due Thu, 2/6/14
Problem 1. [Fourier transform applied to the advection-diusion-decay equation]
Let u(x, t) be the concentration of a substance in an innitely long narrow channel through
which water ows in positive x-direction with con
MATH 5163
Homework 2
Due Thu, 1/30/14
Problem 1. [Dierentiation of an integral depending on a parameter]
The solution of the initial value problem for the wave equation
1 2u 2u
=
,
c2 t2 x2
xR ,
t0 ,
u(x, 0) = 0 ,
ut (x, 0) = h(x)
(where c is a positive c
MATH 5163
Homework 6
Due Fri, 2/28/14, 5 p.m.
Problem 1. [Derivatives of distributions]
Find the rst and the second derivatives of the following distributions from D (R):
(a) H(a x ) (where a > 0);
(b) the oor function x, where x is the smallest integer n
MATH 5163
Homework 8
Due Wed, 4/9/14, 5 p.m.
Problem 1. [The 1-dimensional wave equation]
In this problem you will rederive the expressions we obtained for the solution of the wave
equation in one spatial dimension, using Fourier transform and Duhamels pr
MATH 5163
Homework 5
Due Fri, 2/21/14, 4 p.m.
Problem 1. [Fundamental solution of the heat equation on R1+1 ]
In this problem you will derive the so-called fundamental solution of the heat equation on
R1+1 , i.e., in one spatial dimension (x) and one temp
MATH 5163
Homework 10
Due Fri, 5/2/14, 5 p.m.
Problem 1. [Parallelogram identity]
(a) Let H be an inner product linear space. Prove the parallelogram identity,
u+v
2
+ uv
2
= 2( u
2
+ v 2)
for any u, v H .
(b) Consider the functions f (x) = x and g(x) = 1
MATH 5163
Homework 4
Due Thu, 2/13/14
Problem 1. [Test functions]
Let D(R). Consider the following sequences of functions:
(a) k (x) =
1
(x) ;
k
(b) k (x) =
1
(kx) ;
k
(c) k (x) =
1
x
( ) .
k
k
For each sequence explain if it converges in D(R) as k if it
MATH 5163
Homework 1
Due Thu, 1/23/14
Problem 1. Consider the rst-order PDE
xux + yuuy = u
(1)
for the unknown function u(x, y), in the rst quadrant, i.e., in the domain
U := cfw_(x, y) R2 : x > 0, y > 0 .
(a) Prove that, for any smooth function of one va