Introduction to Partial Differential Equations

These notes were used in an introduction to linear partial differential equations. We begin with an overview the qualitative aspects of the classical theory to provide a foundation for the remainder of the course which focuses on the meaning of and methods for constructing weak solutions. This includes an introduction to Hilbert space methods as well as an introductory treatment of distribution theory and its applications to solving problems in partial differential equations. We do not require a specific course prerequisite as the results will be developed as we go along. However, a reasonable amount of mathematical maturity is necessary.

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Laplace's Equation (updated Sep 4)

Extra Problems I (added 2 problems on Sep 9)

The Heat Equation (updated Sep 15)

The Wave Equation (updated Sep 22)

Introduction to Function Spaces (last updated on Oct 13)

Supplement: Additional techniques

Solving PDE's with the Fourier Transform (problems)

Solving PDE's by Eigenfunction Expansion (problems)

The course follows parts of the text “Partial Differential Equations” by L. C. Evans but this text is too advanced for this course. Some less advanced texts, all of which are available in inexpensive Dover editions, are the following:

**Partial Differential Equations
of Mathematical Physics** by Guenther and Lee

**Applied Partial Differential
Equations** by DuChateau and Zachmann

**A First Course in Partial
Differential Equations** by Weinberger

**Introduction to Partial
Differential Equations** by Zachmonoglou and Thoe

**Equations of Mathematical Physics** by
Tikhonov and Samarskii