Chapter 1
Introduction to spectral theory
1.1
An ODE problem
Before dealing with partial differential equations, lets revisit ordinary differential equations, e.g.,
d xE
D AxE
dt
with x.0/
E
D xE0 :
(1.1.1)
Lets consider the simple case where A is a symme

Chapter 3
Greens Functions and Distribution theory
3.1
Introduction to the Greens function method for solving ODEs1
Consider the 1-D wave equation for a stretched string with harmonic forcing
't t
c 2 'xx D F .x/e
i !t
Lets look for harmonic solutions by

Chapter 6
Regular and singular boundary value problems
Now that weve learned how to use the Greens function to derive eigenfunction expansions and
transforms associated with boundary value problems, its worth revisiting some of the basic theory.
One of th

Chapter 2
The basic differential equations
Before delving into solution methods, its worth motivating things a bit. We do this by considering
some partial differential equations that come up in various applications.
2.1
Electrostatics
Maxwells electromagn

Chapter 4
Inner Products and Adjoints
4.1
Eigenfunction Expansions
The next thing we need to do is better understand the expansions of the type,
f .x/ D
1
X
fn sin.nx/ ;
nD1
fn D
2
f ./ sin.n/ d ;
0
and how they are connected with boundary value problems.

Chapter 5
Spectral Theory of 2nd-order Differential Operators1
5.1
The Greens function method
Recall that one way of looking at an eigenvalue problem for an operator L is to consider the forced
problem
.L C /u D f
with u confined to some domain D by the b

411-1 Final Exam Solutions
1. Consider the half-plane problem
r 2 g D .x x0 / .y y0 / for
y; y0 > 0 and
1 < x; x0 < 1 ;
with gy D 0 on y D 0. In how many ways can this problem be solved using transforms or eigenfunction expansions? (Explain.) Pick one and

411-1 Homework 3
1. What is .x 2
2. What is
a2 / equal to in the sense of distributions?
d
H.x 2
dx
3. Solve x 2
a2 / equal to in the sense of distributions?
du
D 0 in the sense of distributions.
dx
[Hint: first show that a test function f is of the form

411-1 Homework 2
1. The goal of this problem is to develop a general way to determine the specific form of the
Laplacian in different coordinate systems. [Note: you only have to do the parts (a)-(g); the
notes in between these is information given to you.

411-1 Homework 4
1. By modifying the method used in class (or using some other way, such as the one outlined
on page 119 of Stakgold), show that
Z 1
sin kx
lim
f .x/ dx D f .0/:
k!1
x
1
Solution: Heres one way to do this, I think, when f .x/ is a testing

411-1 Homework 1
1. If z D x C iy, D C i and 2 D z, determine and in terms of x and y. (Hint: first eliminate
to find a quadratic equation for 2 in terms of x and y.) This gives a way to calculate complex
square roots without using the polar form. Use it

Fall 2015
ESAM 441-1 PS2
1. The goal of this problem is to develop a general way to determine the specific form of the
Laplacian in different coordinate systems. [Note: you only have to do the parts (a)-(g);
the notes in between these is information given

411-1 Homework 5
1. Use two different spectral representations to find the solution of Laplaces equation for
'.x; y/ in the rectangular region 0 < x < 1, 0 < y < b, with the boundary conditions '.x; 0/ D 0, '.x; b/ D 0 and '.0; y/ D 1. Show from either of

Fall 2015
ESAM 441-1 PS3
1. What is .x 2
a2 / equal to in the sense of distributions?
Solution: First, we can assume a > 0 without loss of generality. The problem is that
x 2 a2 is not monotonic, so the inverse is multi-valued. So we split up the integral

411-1 Homework 4
1. By modifying the method used in class (or using some other way, such as the one outlined
on page 119 of Stakgold), show that
Z 1
sin kx
lim
f .x/ dx D f .0/:
k!1
x
1
2. Determine the spectral representation for
Lu D uxx
for
0<x<1
with

411-1 Exam #1 Solutions
1.
(a) Verify the result x 0 .x/ D .x/ (in the sense of distributions).
(10 pts)
(a) Solution. We have
h x 0 .x/; f .x/i D h 0 .x/; xf .x/i D h.x/; .xf .x/0 i
D h.x/; f .x/ C xf 0 .x/i D f .0/ D h.x/; f .x/i
for any testing functio

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