MATH 435-SPRING 2014-COURSE LOG

W   1/8     Overview of PDE. For Friday: read section 1.1,  bring (in writing) a PDE problem from a
class in another department you have taken, are taking or will likely take in the future.

F   1/10    1-dim wave equation: simplified derivation. d'Alembert's solution of the Cauchy problem. Estimate for the
norm of the solution in terms of the norm of the data
Problems: p.17, 3 [answer in the form u(x,t)=f(t)g(x)], 7 (Dirichlet BC) [no need to include (x,t) diagram]

M  1/13  the 1DWE on a bounded interval: boundary-value problems/ sets of influence, sets of dependence
Problems: p.17:9,  p.24: 5, 7 (Neumann BC), 10

W 1/15  Homogeneous 1DWE on the half-line and bounded intervals: Dirichlet and Neumann problems, solution by extension of the
initial data as even/odd functions, domain of dependence at (x,t)
Problems: p.17 4, 5 (Dirichlet BC)
p. 24 1(mixed BC), 2 (Neumann BC) [answer in the form u(x,t)=f(t)g(x)], 3 (mixed BC)

HW1 (due Friday 1/17) p. 17: 3, 4, 7 p. 24 1, 2, 7 (follow the instructions given above in italics for some of the problems)
HW1solutions

F 1/17 Non-homogeneous 1DWE/ example: x-independent force. Problems: p. 27: 1--6 (all) [HW 2, due 1/24]

M 1/20 MLK Holiday (no classes)

W 1/22  Periodicity in time for BVP, non-homog. problems

F  1/24 Characteristic parallelograms, non-hom Dirichlet BC on half-line

M 1/27  Pulses, bump functions and support: interpretation of the solution/ conservation of energy

W 1/29  Stability, uniqueness and existence for the 1DWE/ HW 2 collected
HW2 solutions

F 1/31  The energy-momentum vector field and energy balance/ solutions by eigenfunction expansion (start)
Notes on the one-dimensional wave equation
(preliminary version, 20 pages. Includes Exercises 1-7 (in boldface)=HW 3, due Friday 2/7

M 2/3 Solutions by eigenfunction expansion

W 2/5 Discussion of HW3/ eigenfunction expansions

F  2/7 Introduction to Fourier series; Fourier coefficients
HW3solutions

M 2/10: Heat equation: derivation, basic properties.

W 2/12: Heat equation: derivation of the heat kernel, properties

F 2/14: Heat equation: solution of the Cauchy problem on the real line (proof)
One-dimensional heat equation (notes)
version date: 2/22, 9PM
(Preliminary version--includes 6 problems on p.8=HW 4, due Friday 2/21)

M 2/17: Maximum principle and the energy method for the heat equation

W 2/19: norms in function spaces, stability estimates, BVPs for the heat equation on the half-line

F 2/21: BVPs for the heat equation on bounded intervals via eigenfunction expansions

M 2/24 Discussion of hw problems (heat equation)
HW4 solutions
(includes also solutions to problems on p.22 of heat equation notes.)

W 2/26 Review

F 2/28 First test (in-class)
Exam 1 (with solutions)

M 3/3: Fourier series: notions of convergence, convergence theorems.

W 3/5: Fourier series: decay of Fourier coeficients/Bessel's ineq/Parseval's equality
Convergence of Fourier series
(includes 6 problems, due as HW 5 on Friday, 3/14)

F 3/7: Fourier series: convergence theorems, applications, examples
HW5 solutions

M 3/10: Fourier synthesis: Functions defined by series, Fourier synthesis, applications to HE

W 3/12: Two applications: the Weierstrass density theorem, the isoperimetric inequality (via Wirtinger's inequality)

F 3/14: HE in higher dimensions: derivation, product solutions
Heat equation in higher dimensions

3/17--3/21: SPRING BREAK

M 3/24: HE in higher dimensions: heat kernel, Cauchy problem

W 3/26: WE in higher dimensions: Laplacian in polar coordinates

F 3/28: WE in higher dimensions/spherical means HW6 due (from handout)
HW6 solutions

M 3/31: WE in R^3: d'Alembert's formula, Huygens' principle

W 4/2: WE in R2; examples
Wave Equation in 2d and 3d: examples
(includes 6 problems, due Monday 4/6 as HW set 7)

F 4/4: Review: Fourier series

M 4/7: Review: heat and wave equations in 2d and 3d
HW7 solutions

W 4/9: Second test: lectures from 3/3 to 4/2 (Fourier series, higher-dim wave and heat equations)
Test 2(with solutions)

F 4/11: Time-independent problems: Dirichlet problem on the disk

M 4/14: Poisson formula, properties of harmonic functions

W 4/16: Green's functions: theory, examples (half-space, ball)-HW 8 posted
Harmonic functions, Green's functions, potentials
(includes 6 HW problems, =HW8)

F 4/18: SPRING RECESS (no classes)

M 4/21: Green's functions-examples-HW 8 due, Test 3 given (take-home)
HW8 solutions

W 4/23: Spectrum of the disk in R2 (Bessel functions)-test 3 collected
Test 3
Test 3-solutions

F 4/25 (last day) Spectrum of the ball in R3; relation with harmonic polynomials
spectrum of bounded domains (includes 8 problems)
solutions

Review session: Monday, 4/28, 12:30-2:30 (in Ayres 121, the usual classroom)

FINAL:
Wednesday, 4/30, 12:30--2:30
1) Comprehensive--emphasis on test and homework questions (and possibly a problem or two from the last handout)
2) Open book: you may bring the texts by Weinberger and Strauss and copies of the online handouts (but not your class notes)
Final Exam (with solutions)