Math 560 Homework 1
Due: Sept 18, 2012
1. Dene the (ordered) set T := cfw_1, x, x2 in L2 [1, 1] with inner
product
1
|x| f (x)g (x) dx
< f, g >=
(1)
1
Use the Gram Schmidt procedure to nd an othogonal basis
for span(T ).
2. Let S = IR3 and f = (4, 1, 1).
Math 560-561: Homework 6.
Due: Tuesday, February 26, 2013.
1. [10] Dene the functional
1
1
y 2 + y 2 dx
L(y, y ) dx =
J (y ) =
0
0
on A where
1
A=
y C 2 [0, 1] : y (0) = 0, y (1) = 1,
y 2 dx = 5
0
The modied Lagrangian for the problem is
M (y, y , ) L(y,
Math 560-561: Homework 7.
Due: Thursday, April 4, 2013.
1. [25] Find two term expansions1 for all regular solutions of:
x5 + sin( x) 1
=
x3 y
y x2
0
(y 1)ex + y
x3
=
y y
=
y + x2 y
y2
,
= ( )y
y (0) = 1 , y (1) = e
,
y (0) = y ( ) = 0
1
y y
y (x)3 dx
=
,
Math 560-561: Homework 5.
Due: Thursday, Jan 24, 2013.
1. [10] Consider the functional
b
L(y, y ) dx
J (y ) =
a
where
L(y, y ) = y 2 (1 y )2
a) Use a rst integral to nd the general solution of the Euler Lagrange
equations associated with extremizing J (y
Math 560: Homework 4.
Due: Dec 11 noon (no later).
1. [10] Use the eigenfunctions of Lu u with domain
D(L) = cfw_u C 2 [0, ] : u (0) = u ( ) = 0
to nd a series representation of the Greens function g (x, t)
associated with the problem
(L I )u = f
u D(L)
,
Math 560: Assignment 2
1. [15] Dene
T = cfw_n cfw_sin nx
n=1
T is a complete orthogonal set for L2 [0, ]. Next, dene the
kernel
n+1 (x)n (y )
k (x, y ) =
(1)
n2
n=1
and corresponding integral operator
k (x, y )u(y )dy
Ku =
(2)
0
i) Prove K is a bounded o
Math 560: Assignment 3
Due: Thursday Nov 1, 2012.
1. [5] Consider the distribution whose action on a test function
D is:
1
1
< , >= x + L x L
2
2
This could represent the electrostatic potential for two charges
of opposite sign (charge 1 ) a distance L a
Math 560-561: Homework 8.
Due: Thursday, April 25, 2013.
1. [10] Find leading-order composite asymptotic approximation for the following single boundary layer problem using Prandtl matching.
y + 2y + y 3 = 0 ,
y (0) = 0
,
y (1) = 1/2
Sketch the solution.