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MAT 1060H:
Introduction to Partial Differential Equations

__ Professor__: Mary Pugh

__ Contact information__: mpugh@math. utoronto.ca

__ 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.