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

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).