Lectures Notes - Winter 2011

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.

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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.

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Week 2:
Preparation for class:

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

Lecture notes:

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

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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.

Lecture notes:

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

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Week 4:
Preparation for class:

• Preparation for Tuesday class: Read Chapter 7.1-7.6
• Preparation for Thursday class: Read Chapter 7.7-7.10

Lecture notes:

• Lecture 7: Applications of SL theory: the circular drum. Higher dimensional problems (part 1)
• Lecture 8: Higher dimensional problems (part 2). Examples of heat diffusion in a rectangular plate, and of stellar oscillations.

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Week 5:
Preparation for class:

• Preparation for Tuesday class:
• Preparation for Thursday class: For this whole week: Read 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 9: Forced problems (part 1)
• Lecture 10: Forced problems (part 2)

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Week 6:
Preparation for class:

• Preparation for Tuesday class: Review Laplace Transforms (see Riley, Hobson & Bence, Chapter 13.2).
• Preparation for Thursday class: Midterm!

Lecture notes:

• Lecture 11: Forced problems (part 3). Formal solution of the generalized non-homogeneous diffusion equation
• Lecture 12: Forced problems (part 4). Formal solution of the generalized non-homogeneous wave equation. Introduction to Green's functions

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Week 7:
Preparation for class:

• Preparation for Tuesday class: Chapters 9.1-9.3 cover material we did this last lecture. Please read for complement. Chapter 9.4 treats the case when there is a 0 eigenvalue. Read if you are interested. In preparation for Tuesday, read Chapter 9.5.
• Preparation for Thursday class: Read Chapter 12.1-5 (Note: I do not like much the way the topic of the method of characteristics is introduced in this textbook. The lectures will proceed somewhat differently. )

Lecture notes:

• Lecture 13: Introduction to Green's functions (part 2)
• Lecture 14: Introduction to Green's functions (part 3). Introduction to the method of characteristics for quasilinear equations.

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Week 8:
Preparation for class:

• Preparation for Tuesday class: Enjoy the long weekend, and review 2nd order PDE material for questions if you have any.
• Preparation for Thursday class:

Lecture notes:

• Lecture 15: Method of characteristics for semilinear equations and Handouts
• Lecture 16: Method of characteristics for quasilinear equations. Existence and uniqueness of solutions

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Week 9:
Preparation for class:

• Preparation for Tuesday class:
• Preparation for Thursday class:

Lecture notes:

• Lecture 17: Shocks and Traffic flow
• Lecture 18: Expansion Shocks

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Week 10:
Preparation for class:

• Preparation for Tuesday class: Read the following lecture notes on the use of 1st order ODES in stochastic modeling. Review change of coordinate systems (RHB Chapter 5.5 and 5.6)
• Preparation for Thursday class:

Lecture notes:

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