*Alex Freire- Ayres 207A, 974-4313, freire@math.utk.edu*
*Office hours (spring '02): Th+F , 11-12 (or by appointment)*

Goal: first course in PDE for mathematics, science and
engineering majors.

Emphasis on the application of techniques of mathematical analysis to

the rigorous construction of solutions to the standard linear PDE of

mathematical physics (by eigenfunction expansions and potential theory
methods)

Prerequisites- vector calculus, first course in differential equations

Text: Partial Differential Equations of Mathematical Physics
and Integral Equations

by Ronal B. Guenther and John W. Lee (Dover)

Grading: Based on homework (20%), two in-class exams (25%
each) and a

comprehensive final exam (30%)

HOMEWORK PROBLEMS: see the course
log Problems from sections covered

on Tuesday-Thursday of week **n **are due at the beginning
of the Thursday class on

week **n+1.**

*COURSE OUTLINE*

Part I- Fourier series, linear problems in bounded intervals

3-1 Convergence theorems for Fourier series

3-2 L^2 convergence and uniform convergence

3-3 Proof of Dirichlet's theorem

Fourier
series-summary (PDF file)

4-2 , 5-1 Homogeneous IBVP on bounded intervals

5-2 Maximum principle for the heat equation

4-3 Non-homogeneous problems

5-3 Representation formulas, heat eqn. for continuous
IC

EXAM 1: March 5 (topics from Part I)

Part II-Linear problems in unbounded regions

3-4 Fourier transforms

3-5 Convergence of Fourier transforms

5-4 Heat equation on the line and half-line

4-1 WE on the line: d'Alembert's solution

4-5 WE on the half-line

EXAM 2: TAKE-HOME

INSTRUCTIONS

PROBLEMS

This is a PDF file; there are two pages. Draft posted 3/15/02,8:00p.m.

Corrected final version posted 3/16/02, 3:30 p.m.- problems 2 and

4 expanded, with hints.

PROBLEMS-Postscript
file

(may be better for printing)

8-5 Harmonic functions and Poisson equation in R^n

8-3 Green's functions for bounded domains

Harmonic
functions and potentials (in preparation)

8-4 Max prple., mean-value property

Poisson kernels and Liouville theorems

9-1 Heat kernel in R^2,R^3-Cauchy problem,

maximum principle

EXAM 3: April 18 (in class, open
book/notes- 4 sections listed above)

Problem session: Tuesday, April
16, 4:30-5:30, room 309B

Part III-Wave equation, problems on bounded domains

8-2 Problems with symmetry: eigenvalues of disks,
rectangles, balls

9-2,9-3,10-1,10-2: heat and wave equation in bounded
domains

10-4 Solution of the Cauchy problem for the WE in R^n

FINAL: May 4 (Saturday, 10:15-12:15)-comprehensive,
open book

Review session:Monday, April
29, 5:00, Ayres 309B