MAT 1060H: Introduction to Partial Differential Equations

Professor: Mary Pugh
Contact information: mpugh@math.
Office hours: by appointment
Office location: room 6268, Bahen Building

Meeting time and place: The class meets in BA 6183: Tuesdays 1:10-2:00 and Thursdays 1:10-3:00. The first lecture will be on Tuesday September 10 and the last on Tuesday December 3.

What it is: This is the first semester of the two-semester core course in partial differential equations. It is followed by Mat1061 in the Winter semester, taught by Michael Sigal.

Prerequisites: It's nice if you've already taken a course in ordinary differential equations (ODEs) and PDEs but it's not required. If you haven't already taken a course on measure and integration, you should be willing to suspend disbelief when presented with things like the Cauchy-Scwartz inequality and the Dominated Convergence Theorm.

Assessment: There will be a fifty-minute, open-book midterm on a Tuesday (date to be chosen) and two-hour open-book final exam. The midterm will be worth 25\% of the course mark and the final worth 45\% of the course mark. In addition, there will be seven problem sets, worth 30\% of the course mark. They'll be due Sept 26, Oct 8, Oct 17, Oct 29, Nov 7, Nov 19, and Nov 28. I encourage you to work with your classmates on the homework, and to hand in a single joint homework as a team.

Textbook: The textbook is "Partial Differential Equations: Methods and Applications" by Robert C. McOwen. It's in stock at the campus bookstore. In addition, I've asked the Math library to put it on reserve along with the supplementary text: "Partial differential equations" by Lawrence C. Evans. If you're having problems with the text's being too advanced then I'd suggest having a look at "Partial differential equations: an introduction" by Walter Strauss. Strauss' book is at the undergraduate level and is more accessible.

What We'll Cover:
Chapter 1: First-Order Equations
1.1 The Cauchy Problem for Quasilinear Equations
1.2 Weak Solutions for Quasilinear Equations
1.3 General Nonlinear Equations

Chapter 2: Principles for Higher-Order Equations
2.1 The Cauchy Problem
2.2 Second-Order Equations in Two Variables
2.3 Linear Equations and Generalized Solutions

Chapter 3: The Wave Equation
3.1 The One-Dimensional Wave Equation
3.2 Higher Dimensions
3.3 Energy Methods
3.4 Lower-Order Terms
3.5 Applications to Light and Sound

Chapter 4: The Laplace Equation
4.1 Introduction to the Laplace Equation
4.2 Potential Theory and Green's Functions
4.3 Existence Theory
4.4 Eigenvalues of the Laplacian
4.5 Applications to Vector Fields

Chapter 5: The Heat Equation
5.1 The Heat Equation in a Bounded Domain
5.2 The Pure Initial Value Problem
5.3.Regularity and Similarity
5.4 Application to Fluid Dynamics

The above information, ready to be printed

First homework set due on Thursday September 26
Second homework set due on Tuesday October 8
Third homework set due on Thursday October 17. LaTeX file for third homework set in case you need it.

Sample exam: this was a three-hour in-class final that I wrote and gave in 2011.
The in-class midterm exam given on October 22.
Solutions to the midterm.

Fourth homework set due on Thursday November 7.
Fifth homework set due on Tuesday November 19.

Here is a very nice explanation of the Perron method by Xinwei Yu of the University of Alberta. Note that there's a typo in the top half of the first page: the integrand in the integrals should be v(y) and v(w)

Sixth homework set due on Thursday November 28.