MATH 435- SPRING 2011-COURSE LOG
W 1/12 syllabus and course policies/ examples of PDE: transport
equation, wave equation, diffusion equation/
Laplace
operator
HW (1.4): 1,
4, 6 (due 1/21) (For 4(a), recall the eqn for s-s temperatures is
u_{xx}=f(x).)
F 1/14 concept of well-posed problem: examples
(transport eqn, Laplace eqn in upper-half plane)
Stability
estimate for the transport eqn. Examples: 1.4.
7 (gas dynamics) 1.4.3 (steady-state temperatures; start)
HW (1.5) 2, 4
(due 1/21)
W 1/19 discussion of problem 1.4.3 (steady-state
temperature). 1D wave eqn: solution of the Cauchy problem.
Remarks: (1)
smoothness needed for the data (2) stability estimate for u(., t)
in the supremum norm
(3) linear
solution (in t) from bounded data (example). Example: 2.1.9
(start)
HW (2.1): 1, 7,
9 (For 7, consider also the case when the initial data are even
functions of x) DUE 1/21
F 1/21 1-dWE topics: causality(2.2), energy (2.2),
source terms (3.4)
HW 2.2.4,
2.2.5, 3.4.1 (due 1/28)
M 1/24 1-dWE topics: solution on half-line (3.1),
spherical waves (2.1.8), discussion of HW1. Problem: 3.2.5 (due 1/28)
Addendum to HW policy: having
turned in a HW set, students will have a chance to re-do
incorrect/incomplete solutions,
with the
second attempt weighted at 70%. For the 1st HW set only, problems may still be turned
in on Monday 1/24 (with the grade
corrected by a factor of 0.8) . (Some students added the
course late, or only got a hold of the text recently).
W 1/26 Cauchy problem for the heat eqn on the line--the heat
kernel (2.4) Problems: 2.4.4, 2.4.9 (due 1/28)
F 1/28 Diffusion equation (heat equation): the
maximum principle (2.3) Problems: 2.3.3, 2.3.7, 2.4.15, 2.4.16, 2.4.18,
2.5.1, 3.3.2, 3.5.1, 3.5.2 (due 2/4)
M 1/31--discussion of HW problems- contrast between HE and WE (read
section 2.5)
W 2/2 discussion of HW problems/ max principle for subsolutions/
eigenfunctions, separation of variables
Problems: (4.1) 4, 5
(4.2) 3,4 (Due 2/11) Remark: section 4.3 won't be discussed
F 2/4 Fourier series, ch. 5 (start) Problems: (5.1)
5,9 (5.2) 10, 11, 12 (Due 2/11)
M 2/7 Complex Fourier series, even/odd functions
W 2/9 Notions of convergence (5.3) 2,3 (practice problems)
F 2/11 Discussion of HW/ L2 convergence (5.4) 1, 4, 6,7, 13,16
(practice)
M 2/14 Discussion of problems (work
on the practice problems listed above for preparation)
W 2/16 Exam 1 (Material covered from sections 1 through 5.)
Exam
1
(Note: of the previous 3 times I taught
this course, Fall 2000 is the closest to the current one.)
F 2/18 Pointwise convergence (proof of Dirichlet's
theorem-start)/ discussion of Exam 1
M 2/21 pointwise and uniform convergence of Fourier series (proofs)
HW (5.4) 6, 7, 10, 11 (5.5)
2, 14 (due 2/25)
W 2/23 Solution of the IVP on bounded intervals: wave eqn (pp.
144-5), heat eqn (5.5.6), non-hom problems (5.6)
Wirtinger's inequality
(5.5.12), application to the isoperimetric problem (not in text!)
F 2/25 invariance of the Laplacian (coordinate-free
approach). Problems: (6.1): 2,4,5,7,10,11 (DUE 3/7)
Invariance
of the Laplacian
M 2/28 Laplacian of radial functions/ examples of rotationally
symmetric boundary-value problems
W 3/2 Discussion of homework problems/ smoothness and decay
of Fourier coefficients
Second test: Friday, 3/4. Sections
included: 2.3, 2.4, 2.5, 4.1, 4.2, 5.4, 5.5 (emphasis on the problems
in each section). Only the highest of
the grades on tests 1 and 2 will count towards the course average.
F 3/4 Exam
2
solutions
M 3/7 Poisson's formula for the disk in R^2 (6.3)
W 3/9 Problems from chapter 6
Done in class: 6.1.11, 6.2.3,
6.2.7, 6.3.1, 6.3.2, 6.4.11
Homework (due 3/21): 6.2.4,
6.2.6, 6.3.3, 6.4.1, 6.4.2, 6.4.6
F 3/11 Green's first identity (7.1) Mean-value
property, maximum principle, Dirichlet's principle/
Green's second identity
and representation formula (7.2)
M 3/14 to F 3/18: SPRING BREAK
M 3/21 Green's functions for a domain; the general Poisson
formula.
Problems for chapter 7 (due 3/25) (7.1) 3,5 (7.2) 1 (7.3) 2
W 3/23 Green's fn and Poisson formula for the upper half-space;
problems from 7.4
F 3/25 Problems from Ch. 7 HW (7.4): 6, 7, 8, 11, 12, 13 (due 4/4)
M 3/28 Wave eqn in R^3, R^2 (9.2)
W 3/30 Wave eqn in R^3, R^2: problems: data supported in a disk, time
decay
Time
decay estimates for the wave equation (elementary treatment)
F 4/1 Heat eqn in R^n: heat kernel, solution of the Cauchy
problem/ discussion of HW problems from Ch.7
Problems from Ch. 9 (due 4/11):
9.1.1, 9.1.6(a), 9.2.9, 9.2.16, 9.2.17, 9.2.18
M 4/4 Eigenvalues/eigenfunctions of bounded domains. Examples: square
and disk in R^2/Bessel functions
Problems (practice): 10.1.1,
10.1.5, 10.2.2, 10.2.4, 10.2.5
W 4/6 Eigenvalues/eigenfunctions of a ball in R^3
Problems (practice): 10.3:
4,5,8,9,10
F 4/8: Problems from 7.4, 10.2
M 4/11 Problems from 10.1, 9.1, 9.2, 10.3
W 4/13 Spherical harmonics; problems from 10.3
F 4/15 Minimizing properties of eigenfunctions; Rayleigh
quotient
Problems (due 4/25) 11.1: 1, 3, 4
M 4/18 Third Exam. Material:
chapters 6, 7,9,10 (sections
covered in class)- OPEN BOOK exam
Exam
3
Solutions
W 4/20 Rayleigh-Ritz approximation (11.2)
Problems (due 4/25): 1,
3, 7
Reading for pleasure:
Can One Hear the Shape of a
Drum? (By Mark Kac)
F 4/22: Spring recess (no classes)
M 4/25 Completeness of eigenfunctions; problems 11.3.1, 11.3.2.
Practice: 11.3.3
W 4/27 Fredholm alternative (Thm. 1, 11.5)/ Asymptotics of
eigenvalues (11.6, Thms1- Examples 2 and 5/
Monotonicity of
the eigenvalues (Thm 4). Example: 11.6.3 Practice: 11.5.1, 11.5.2
F 4/29 Problem session--Chapter 11 (last day)
FINAL EXAM: Thursday, May 5, 12:30--2:30 PM
Open book exam (but use of
class notes will not be allowed; textbook only). The exam will consist
of six problems:
four from exams 1,2 or 3 (possibly with slight changes) and two from
Chapter eleven (material discussed in class only).