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Week
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Date
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Material
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Reading Assignment
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Homework
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1
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19-Jan
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Part I: Perturbation theory and asymptotic
approximations:
Perturbation theory for algebraic equations
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Read 1-17 in Fulling's
notes. It
may be helpful for you to read Simmonds and Mann: page 3-17 as well
although it's not required.
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Homework
#1
due Jan 26.
Solutions.
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1
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21-Jan
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2
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26-Jan
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Regular perturbation theory (power series)
and its shortcomings
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Read 18-31 in Fulling's
notes and these supplementary notes
on regular perturbation theory.
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Homework
#2
due Feb 2.
Solutions.
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2
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28-Jan
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|
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3
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2-Feb
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Asymptotics and uniformity and distorted time methods.
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Read 32-40 in Fulling's
notes.
Lecture
on pointwise vs uniform convergence and the distorted time method.
Maple worksheet.
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Homework
#3
due Feb 9.
Solutions.
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3
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4-Feb
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4
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9-Feb
|
Stretched-time and two-time methods
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If you've already read up
to page 40 in Fulling's notes then you have an introduction to the
two-time methods that we'll be discussing. It might be helpful to read
ahead (page 41-50 in Fulling) although we probably won't get to the WKB
approximation until next week. It may be helpful to read pages 61-71 in
Simmond and Mann on the two-scale method.
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Homework
#4
due Feb 16.
Solutions.
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4
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11-Feb
|
|
Here are some of Prof.
Fullings' old exams: 09, 06, 05, 98.
Our exam 1 will be similar except in two respects: WKB will not be on
the exam, two-scale methods will not be on the exam.
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5
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16-Feb
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Review for Exam 1.
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Homework
#5
due Feb 25.
Solutions.
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5
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18-Feb
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Exam 1 (covers first 4 weeks)
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Solutions
to Exam 1.
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6
|
23-Feb
|
WKB (phase-integral, LiouvilleGreen) approximation |
We will use Simmonds and
Mann (pages 71-77) as the primary text for the WKB approximation. It is
also helpful to read about it in Fulling's notes (pages 41-50).
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Homework
#6
due Mar 4.
Solutions.
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6
|
25-Feb
|
Boundary-layer problems |
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7
|
2-Mar
|
Boundary layer problems
Introduction to partial differential
equations (PDEs)
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Read 50-56 in Fulling's
notes (boundary layer theory) and 57-73 (this is part II). The main
topics of part II that we will cover are: introduction to PDEs, linear
operators, equations and problems, separation of variables (to solve
the heat equation).
Alternatively, these topics are covered in Constanda's book: linear
operators (pg 6-7), and separation of variables for the heat equation
(69-81). Although it isn't part of this course, you might want to read
about how the heat equation is derived from physical principles (pg
49-57).
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Homework
#7
due Mar 11. Solutions. |
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7
|
4-Mar
|
Linearity, homogeneity, separation of variables.
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|
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8
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9-Mar
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Fourier series, solutions to the heat equation.
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We will briefly review
linear operators, linear equations and how to find fundamental
solutions to the heat equation. From there, we'll discuss Fourier
series. This is covered in Constanda chapter 2 and Fulling's notes
pages 74-84. My lecture will be modeled on Constanda's chapter 2.
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Homework
#8
due April 1.
Solutions.
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8
|
11-Mar
|
|
It's possible that we will
start on orthogonal functions and Sturm-Liouville problems March 11.
This topic is covered in Fulling's notes beginning with page 118. It is
chapter 3 of Constanda's book. My lecture will be closer to Fulling's
notes.
Here's a Maple
worksheet on Fourier series. Here's an pdf
version.
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9
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16-Mar
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Spring Break (no class)
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9
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18-Mar
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Spring Break (no class)
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10
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23-Mar
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Review for exam 2.
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Here is a practice
exam.
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|
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10
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25-Mar
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Exam 2
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Exam 2 covers the two-time
method, the WKB approximation, boundary layer problems, linear
operators and linear equations.
Exam
2 solutions.
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|
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11
|
30-Mar
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Sturm-Liouville problems.
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Sturm-Liouville problems
are covered in Chapter 3 of Constanda and 118-124 in Fulling's notes.
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Homework
#9
due April 8.
Solutions.
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11
|
1-April
|
|
We'll
discuss using separation of variables to solve linear second-order
PDEs. This material is covered in chapter 5 of Constanda. We will
probably skip chapters 6 and 7 (these chapters cover separation of
variables problems for non-homogeneous PDEs).
If there's time, we'll go over the D'Alembert's solution to the wave
equation which is discussed in Fulling's notes pages 122-140.
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12
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6-April
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Last day to drop without penalty.
More separation of variables problems & the Fourier transform.
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We will do a few more
separation of variables problems for the sake of practice. We'll also
go over D'Alembert's solution to the wave
equation which is discussed in Fulling's notes pages 122-140. If
there's time, we'll start on the Fourier transform.
To begin studying the Fourier transform, I recommend that you read item
(5) of page 81 in Fulling's notes. This discusses how to put a Fourier
series into complex number notation. Then skip ahead to page 92.
Ultimately we will cover the material in 92-105.
The same material is presented in Constanda chapter 8. However, the
notation is different, the definition of the Fourier transform is
slightly different and he uses a different method for solving PDEs via
Fourier transform. My lecture will follow Fulling's notes more than
Constanda's book.
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Homework
#10
due April 15.
Solutions.
Here's a Maple demonstration
of the wave equation problems.
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12
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8-April
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|
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13
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13-April
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The Fourier transform
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Homework
#11
Solutions.
due April 22 |
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13
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15-April
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Delta functions and Green's functions. |
The delta function is
discussed in Constanda pages 157-158. Green's functions are the subject
of Chapter 10. Fulling's discusses these topics in pages
106-117. My lecture will borrow from Fulling's notes mostly.
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14
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20-April
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Types of PDEs (parabolic, hyperbolic, elliptic);
well-posed problems
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14
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22-April
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Review for Exam III
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Here is a practice
exam.
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15
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27-April
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Exam III
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Exam
Solutions |
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15
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29-April
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Review. This is the last day of class.
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Here is a practice
sheet for the final.
I will have office hours Wed and Fri 10-3pm next week and I'll be
available online most of the time and I'm happy to answer your
questions.
Some solutions
to the practice sheet.
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Wed
12-May |
Final: 1-3pm. (The final
is comprehensive)
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I will be out of town
after May 7th. A different professor will be proctoring the exam. If
you need to talk with me about your grade or anything, please contact
me before May 7. |
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