Lectures Notes - Winter 2013

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Lecture Notes, Winter 2013

WARNING: downloading the lecture notes and putting them under your pillow at night will not help you learn the material.

Preparation for the course : This graduate class relies heavily on undergraduate mathematics including

Calculus of several variables, including partial differentiation, vector calculus and spherical/cylindrical coordinate systems
Concepts of linear algebra
Ordinary differential equations
Fourier series, Laplace transforms

If you do not master these concepts you will struggle with this course. To prepare for it adequately, make sure you practice by reading and doing examples of Chapter 5 (in particular 5.1, 5.2, 5.3, 5.4, 5.5, 5.6), 10 (in particular 10.7 and 10.9), and 14 (in particular 14.1 and 14.2) of the Riley, Hobson & Bence textbook (see Syllabus page).

Preparation for each class : You must come to class prepared - just "showing up" is not sufficient. Graduate studies involve progressively more independent working practices, and this class will slowly get you used to this. Prior to each class, you must:

Read the material from the last lecture, and come prepared with questions on what you do not understand. You must be ready to be able to summarize the previous lecture in front of the class.
Read the assigned material, and work through it. Come prepared with questions on material you did not understand. The class will assume that you have read the assigned chapters, and you must be prepared to answer questions on them.

Preparing for class should take you about 2 hours per class. This does not include the homework preparation time.

Week 1:
Preparation for class:

Preparation for Tuesday class: Read Chapter 1 of textbook.
Preparation for Thursday class: Read Chapter 2.1-2.4 of textbook.

Lecture notes:

Lecture 0: Reviews (not lectured, but please read if you think you need it).
Lecture 1: Introduction.
Lecture 1: PPT Introduction.
Lecture 2: Method of separation of variables - example of the diffusion equation.

Week 2:
Preparation for class:

Material for Tuesday class: Read Chapter 3.1-3.5, and Chapter 2.5
Material for Thursday class: Prepare for Quiz 1 (see Homework). Read Chapter 4.

Lecture notes:

Lecture 3: Fourier Series. The diffusion equation and Laplace's equation.
Lecture 4: The wave equation

Week 3:
Preparation for class:

Preparation for Tuesday class: Read Chapter 5.1-5.4 of textbook.
Preparation for Thursday class: Read Chapter 5.5-5.9 of textbook. Prepare for Quiz 2 (cf. Hw2)

Lecture notes:

Lecture 5: Sturm-Liouville theory (part 1).
Lecture 6: Sturm-Liouville theory (part 2).

Week 4:
Preparation for class:

Preparation for Tuesday. As for last Thursday.
Preparation for Thursday class: Read Chapter 7.1-7.6

Lecture notes:

Lecture 7: Applications of SL theory: the circular drum. Higher dimensional problems (part 1)
Examples of Maple files, to help you with the syntax

Week 5:
Preparation for class:

Preparation for Tuesday class: Read Chapter 7.7-7.10
Preparation for Thursday class: Read Chapter 8.1-8.3

Lecture notes:

Lecture 8: Higher dimensional problems (part 2). Examples of heat diffusion in a rectangular plate.
Lecture on stellar oscillations.
Lecture 9: Forced problems (part 1)

Week 6:
Preparation for class:

Preparation for Tuesday class: For this whole week: Read end of Chapter 8. We will not cover all of it in class, but you will be asked to do some HW problems on the parts we do not cover.

Lecture notes:

Lecture 10: Forced problems (part 2)
Lecture 11: Introduction to Green's functions (part 1)

Week 7:
Preparation for class:

Preparation for Tuesday class: Chapters 9.1-9.3. Chapter 9.4 treats the case when there is a 0 eigenvalue. Read if you are interested.
Preparation for Thursday class: Read Chapter 9.5

Lecture notes: Note that I have scanned everything I have on the subject for completeness. We will definitely not cover everything in class.

Lecture 12: Introduction to Green's functions (part 2).

Week 8:
Lecture notes: Note that the subject is not particularly well-covered in the textbook. See the Pinchover and Rubinstein textbook for a slightly better introduction to the topic.

Lecture 13 and 14: Method of characteristics for semilinear equations and associated figures

Week 9:
Preparation for class:

Preparation for Thursday class: Read Chapter 12.6

Lecture notes:

Lecture 15: Method of characteristics for quasilinear equations. Note that we will not cover "existence and uniqueness" this year.
Lecture 16: Shocks and Traffic flow
Lecture 17: Expansion Shocks (if we have time)

Week 10:
Preparation for class:

Preparation for Tuesday class: Review change of coordinate systems (RHB Chapter 5.5 and 5.6)

Lecture notes:

Lecture 19: Canonical Forms, part 1.
Lecture 20: Canonical Forms, part 2.