MATH 435- SPRING 2002- COURSE LOG

1/10   Fourier coefficients of periodic functions
uniform convergence and L^2 convergence
statement of convergence theorems
complex form of Fourier series
Problems from 3-1
due 1/22: 4,9,10,11

1/15  Fourier series as best L^2 approximation
Bessel's ineq., Parseval's equality
Weierstrass's theorem (statement)
Proof of L^2 convergence for cts. functions
Problems from 3-2
due 1/24: 2(a),5,12

1/17  No class (talk in Washington,DC)
Make-up TBA

1/22  Uniformly convergent series/improper integrals
Weierstrass M-test
Uniform convergence of Fourier series
Piecewise C^k functions-decay of Fourier coeffs.
Problems: 3-3 no. 4,  3-4 no. 2 (due 1/31)

1/24   Riemann-Lebesgue lemma, decay of coefficients
Cesaro convergence: Fejer's theorem[not in text!]
Proof of Weierstrass' theorem
Problems from 3-2: 10, 16(d) (prove the statement made,
assuming 16(c)) (due 1/31) solutions

1/29  Pointwise convergence; Dirichlet's kernel
Fourier transforms of L^1 functions
Problems: 3-4: 5,6;  3-3 1,2 (due 2/12)

1/31  discussion of homework problems
Fourier transform: derivatives, Gaussians
Convolution theorem [not in text!]

2/5   Eigenfunctions of b.v.p. on an interval
Formal solution of 1DWE with Dirichlet BCs
Conditions on the data for convergence
Problems: 4-2 1,3,9,10 (due 2/14) solutions
5-1 1(a)(b)(c); 3,4 (all due 2/19)

2/7   discussion of HW: Lipschitz and Holder
conditions, solving ODEs w/ Fourier transforms
1DWE with Dirichlet BCs: existence theorem

2/12   WE on an interval: uniqueness, stability
Heat eqn on an interval, Dirichlet BCs: existence

2/14   Maximum principle for the heat equation
Applications: uniqueness, stability
Heat equation with source term (formal solution)
5-2:2,7 4-3:5,6    5-3:5 solutions

2/19  Heat equation with source term-conditions for convergence
Duhamel's principle for second order ODEs [not in text!]
Wave equation with forcing term (Dirichlet): formal sln.

2/21   WE with forcing term: convergence
D'Alembert's representation formula for the WE;
solution of the WE for C^2 data
Solution of the heat eqn for continuous data

2/26   Heat kernel in an interval (Dirichlet)
Representation of the solution
HW problems: non-hom heat equation

2/28    Discussion of homework problems

3/5       EXAM 1 (open book)problems
solutions

3/7       Heat kernel on the real line:
derivation using Laplace transform, properties,
examples of solutions of the IVP

3/12    Solution of the IVP for the heat equation: proof
of  convergence, examples. Uniqueness and
maximum principle.

3/14   Comparison of the heat and wave equation.
Non-homogeneous problems via Duhamel's principle
HE and WE on the half-line (Dirichlet)

3/17- 3/22 SPRING BREAK

3/26   Laplacian in R^2,R^3 (polar coords.); Green's identities
Uniqueness for BVPs in bounded regions
Harmonic polynomials in R^2,R^3

3/28   Green's functions for R^2,R^3
Representation formula in R^3; solution of Poisson's eqn.
Poisson's kernel for a disk in R^2

4/2    Poisson kernel in R^3
Applications: solution of Dirichlet's problem for a ball (theorem);
mean-value property, maximum principle
homework problems (due 4/11)-PDF

4/4    Harnack's inequality and Liouville's theorem
Stokes representation formula on bounded domains
Green's functions and Poisson kernel for domains
Poisson formula for  the upper-half space

4/9       Heat kernel in R^n; solution of Cauchy problem
Uniqueness, non-hom h.e. in R^n
max/min principle (bounded domains)
homework problems (due 4/16)-PDF

4/11    Examples: heat  eqn on a 3D strip and in upper-half plane
Eigenvalues of symmetric operators: spectral theorem, positivity
of eigenvalues of the Laplacian

4/16   Eigenfunction expansions
Eigenvalues of a square, solution of IVPs
Eigenvalues of a disk, Bessel functions

4/16, afternoon: problem session (homework problems)

4/18  EXAM 3 problems
solutions

4/23      Example: Neumann eigenvalues of a cylinder, heat eqn.
Spectrum of a ball in R^3

4/23, afternoon: problem session (exams 2 and 3)

4/25      Cauchy problem for the WE in R^3: Kirchoff's formula
Huygens'  principle
Solution of Cauchy's problem in R^2, failure of Huygens'  prple

4/29, 5:00- problem session
review problems (PDF file)
solutions (problem 4)