MATH 435, FALL 2007- COURSE LOG

Errata for text

Th,  Aug 23   Course policies/  Derivation of the heat equation/ well-posed problems//
(sect 1.4),  Initial and boundary conditions (1.5)/ Poisson's equation, electrostatic potential

Tu, Aug 28     Simple examples of PDE solutions (1.6)
1-dim wave equation
discussion of problems in Ch.1:  1.3, 1.7(a)(b)
Exercise (1.4) will be discussed 8/30; other problems (inc. 1.7(c)) fair game for HW- due 9/4.
Rk. For problems like 1.1 or 1.2 or 1.5, with several similar (independent) items, only ONE
item will count towards HW credit (if correct). Of course, you can always do the others for practice.

Th, Aug 30      Radial solutions of the minimal surface equation (problem 1.4)
Method of characteristics (Ch 2, 2.1 to 2.5)- start

Tu, Sept 4     Method of characteristics- examples (2.1 to 2.5)
HW problems (ch. 2) : 2, 3, 4, 8, 10, 12, 16, 17, 24, 26 (due 9/13)
In problems 8 and 12, the word `ray' should really be `half-plane'. All solutions of PDE
should include the domain of definition.
Hints: 8. Consider the equation satisfied by v=u^2/2
10. Treat y as a parameter (only x and tau are involved in the PDE)
(The odd-numbered problems have answers in the book, but I expect a complete solution)

Th, Sept 6      Method of characteristics- conclusion: proof of the local E/U theorem
The E/U theorem for vector fields and the inverse mapping theorem

Tu, Sept 11     Discussion of HW problems - generalized solutions for u_t+uu_x=0 (2.7, p.41 to 46)
The one-dimensional wave equation (ch. 4)- d'Alembert's solution of the Cauchy problem

Th, Sept 13     1D wave eqn- well-posedness, domains of dependence/influence, propagation of singularity
Non-homog. wave eqn.
Problems for ch. 4 (due 9/25):  2, 6, 9, 12, 14, 16, 17(a)

Tu, Sept 18       Well-posedness for 1DWE;  (p. 234-241)
Discussion of HW problems

Th 9/20           WE in 3 dim.  Additional HW for 9/25: p.279, 9.5. Also:
Problem: let u(x,t) be a solution of the homogeneous wave equation in R^3 (c=1).
Assume the initial data is nonzero exactly on the set r_0<|x|<r_1. Compute the
support K(t) of u at time t>0 (that is, the set where u is not zero at time t).
Hint: three cases to consider (i) t<r_0 (K(t) is an annular region); (ii)r_0<t<r_1 (K(t) is a ball)
(iii) t>r_1 (K(t) is again an annular region).

Tu  9/25            Representation formulas -WE in 2 dim
Representation formulas, time decay and stability for the wave equation
Separation of variables for the 1 dim heat eqn (start)

Th   9/27       Eigenfunctions on an interval- orthogonality, Fourier sine series
Convergence of series of functions: pointwise, uniform, L2; Weierstrass M-test
Convergence and smoothness of formal solutions of the heat eqn
Differentiability and decay of Fourier coefficients
HW from Ch. 5 (due  10/4):  4, 5, 8, 9, 10, 12, 16
Remark: when doing the hw, assume k=1 (heat eqn) and c=1 (wave eqn) and L=pi
for problems in an interval. These parameters are a mere nuisance, with no mathematical
significance. (In part. for problem 10(c) the condition is 1<alpha <4).

Tu 10/2         Separation of variables for the wave equation; non-homogeneous problems
Uniqueness via the energy method

First test:  Tuesday, Oct 9 (Ch. 1, 2, 4, 5, material on WE from ch.9)

Th 10/4           Discussion of homework problems

Tu 10/9          Exam 1

Th  10/11       Fall Break (no classes)

Tu   10/16       Basic facts on Fourier series: Riemann-integrable functions, the space
L^2[-pi,pi], minimizing property of the coefficients, Bessel's inequality
and Parseval's equality. (Notes and HW problems- to be posted soon)/
Brief discussion of  Exam 1 (returned.)

Th 10/18         Convergence of Fourier series
Preliminary version of the handout, inluding material presented up to 10/18.
There are 6 homework problems, due 10/25. Problems 1 and 5 are worth 3
HW pts each, problem 6 is worth 2 pts, Problems 2,3,4: 1pt.

Tu  10/23      Proof  of Dirichlet's theorem; the Gibbs phenomenon/
Heat eqn: uniform convergence to the initil data (Dirichlet BCs)

Th  10/25       The isoperimetric inequality/ Eigenvalues of domains in higher dimensions/
Variational properties of Dirichlet eigenvalues.

Tu 10/28        Variational properties- first Neumann eigenvalue and higher eigenvalues.
Discussion of homework problems (Fourier series)

Th  11/1         Connection between eigenvalues of the Laplacian and Poincare'-Sobolev inequalities
(for functions in bounded domains)/ Application to L2 convergence of generalized Fourier series/
Examples: eigenvalues of rectangles, degeneracy for squares. (Sect. 9.5 in text)
Homework problems- minimizing properties of eigenvalues (due date: 11/8)
Solutions

Tu  11/6       Eigenfunctions of a disk and Bessel functions (9.5)/ eigenfunctions of a ball in R^3/
connection between spherical harmonics (=eigenvalues of the spherical Laplacian)
and homogeneous harmonic polynomials in 3 variables

Th   11/8     Finding a basis for harmonic polynomials in R^3 of given degree/ harmonic functions:
weak maximum principle.
PROBLEMS from Ch. 7: 2 (hint: Green's identities), 3, 8, 10 (hint:same), 12, 15, 17, 19, 21
(those turned in by Tu 11/20 before the test will count twd the HW grade; some will
be solved in class on 11/15)

notes on spherical harmonics

Tu   11/13     mean value property/strong maximum principle/ maximum principle for the heat equation

Th    11/15    separation of vbles for Laplace's eqn, heat and wave eqn: examples, hw problems

Tu    11/20    EXAM 2
solutions

Exam 2: Tu, 11/20,  topics:  lectures from 10/16 to 11/15 (class notes, online handouts,
sect 9.5 to 9.7 and 7.1 to 7.7.1 in text)

Th    11/22     Thanksgiving holiday

Tu     11/27    Poisson's formula for the disk; representation of harmonic functions in terms of boundary values
Brief discussion of Exam 2 (returned.)

Th      11/29    Green's functions for domains in the plane and space; method of images. Application to
Poisson-type formulas
Poisson formulas and Green's functions-problems
(this is the last homework set; it will be discussed on Tuesday 12/4)

Tu     12/4       Discussion of homework problems
The basic eigenfunctions, Green's functions, Poisson kernelsFIFIN

FINAL EXAM: Thursday, December 6, 7:15-9:15 PM (in the usual classroom)

STUDY GUIDE:
The final will consist of 6 or 7 problems.  The material included consists of the lectures from 9/27
to 11/29 (this means the method of characteristics and the whole-space wave equation are not included). The problems will
be based on Exam 1 (excluding problems 1,2 and 3), on the homework problems from the online handout "Convergence
of Fourier series" (10/18- for which written solutions were given), on Exam 2 and on the last homework set (Poisson
formulas and Green's functions), to be discussed on 12/4.  Read the handout from 10/18, but don't worry about the `notes on
spherical harmonics'. If you have time, review also the solutions to the homework sets on this material, and the
corresponding sections in the text (in addition to the four sources listed above.)

FINAL