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
Answers and comments- Ch. 2 homework 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).
Answer
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
Answers and comments- Ch.4 homework problems
Answers and comments- Ch. 9 homework problems
First test: Tuesday, Oct 9 (Ch. 1, 2, 4, 5, material on WE from ch.9)
Th 10/4 Discussion of homework problems
Answers and comments- Ch. 5 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
Answers and comments- Ch.7 homework 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
Answers
Course grades: A=5, B=3, C+=1, C=1 (10 students took the final)