MATHEMATICS 435-PARTIAL DIFFERENTIAL EQUATIONS-SPRING 2002

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