A partial differential equation is an equation involving an unknown function of several variables and finitely many of its partial derivatives. The study of these began with the invention of calculus by Newton and Leibniz and the historic development of the subject has paralleled the numerous applications in science and engineering. These have continually provided more equations and many questions concerning the nature and accessibility of solutions. In this course we will address many of the mathematical issues. A brief outline of topics to be covered is as follows:

Brief survey of various PDE's and associated problems

Some basic ODE theorems.

The Laplace equation: Maximum Principle, potential theory, existence, regularity, the Dirichlet problem, eigenvalues

The heat equation: Maximum Principle, fundamental solution, regularity and similarity properties

The wave equation: existence, Huygen's Principle, energy results

An introduction to 1st order equtions and the method of characteristics

Distribution and fundamental solutions of linear equations

Hilbert Space and Sobolev spaces

Second Order parabolic equations

Very brief introduction to the calculus of variations- Euler-Lagrange equations

Another useful reference, with less detail, is

R. McOwen, *Partial Differential Equations*, Methods and Applications, Prentice Hall, 1995.

Homework 35%

Hour exam 25%

Final exam 40%

There will be a homework assignment each week when there is not an hour exam scheduled. All homework is usually due each Tuesday, one week after it is assigned. The homework is not pledged. You are encouraged to discuss the homework, and to work together on the problems. However each student is responsible for the final preparation of his or her own homework papers. I will be adding hints on this homepage to problems even after they are due. If you don't get a problem the first time, check the hint and give it another try. Then hand it in for some partial credit. Latest homework.

**Midterm Exam. ** This take-home exam will be distributed Thursday Oct.30, and collected Tuesday Nov. 4. It should be taken over a continuous 3 hour period. Textbooks and notes are allowed.

It will cover through page 74, with the following adjustments:

We have skipped 2.23(b)(c)(d)(e), 2.4(d)(e), and 2.3.3 (except for Theorem 1, which we did discuss) . Also we have been using Appendix A and some of 4.1. Solutions to Midterm Exam.

**Some Handouts:**

** **A fixed point theorem for a contraction mapping

This page is maintained by Robert Hardt, hardt@rice.edu.

Last edited 10/25/08.