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)