Math 220 Homepage, Autumn 2009

Partial Differential Equations of Applied Mathematics

Instructor: András Vasy

Office: 383M

Phone: 723-2226

E-mail: andras "at" math.stanford.edu

Office hours: M3-4, T11-12, W12-1.

Office hours on Monday, December 7th, are moved to Thursday, December 10th, 2:30-4pm (one time only). There are regular office hours on Tuesday and Wednesday during finals week.

Class location: TTh 9:30-10:45am, Building 530-127.

Course assistant: Vorrapan Chandee. Office: 381F.

E-mail: vchandee "at" math.stanford.edu

Office hours: T4-6, W9-11, Th11-1.

Textbook:

Strauss' `Partial Differential Equations: An introduction' covers most topics, but the course is at a higher level, especially regarding first order PDE's, which is the first major topic covered, as well as distributions and the Fourier transform.
Evans' `Partial Differential Equations' is a more advanced text, and it covers course topics not dealt with in Strauss' book.
Both of these, as well as John's `Partial Differential Equations' will be on reserve at the Math CS library.

The running syllabus is here.

Grading policy: The grade will be based on the weekly homework (25%), on the in-class midterm exam (30%) and on the in-class final exam (45%).

The homework will be due in class or in the instructor's mailbox by 9pm on the designated day, which will usually (but not always) be Thursdays. You are allowed to discuss the homework with others in the class, but you must write up your homework solution by yourself. Thus, you should understand the solution, and be able to reproduce it yourself. This ensures that, apart from satisfying a requirement for this class, you can solve the similar problems that are likely to arise on the exams.

Lecture Notes

First order scalar semilinear equations. This has been corrected as compared to the printed version, thanks to vigilant readers!
First order scalar quasilinear equations.
Distributions and weak derivatives.
Second order constant coefficient PDE.
Properties of solutions of second order PDE.
The Fourier transform -- basic properties and the inversion formula. This has been corrected as compared to the printed version, thanks to vigilant readers!
The Fourier transform -- tempered distributions. The notation C^-\infty also stands for D'; it has been removed from this revised version of the notes.
PDEs and boundaries. There should be an overall minus sign in the formulae involving Laplace's equation compared to the printed version; this online version already corrects it.
Duhamel's principle.
Separation of variables, with a typo corrected.
Inner product spaces, symmetric operators, orthogonality.
Convergence of the Fourier series.
Solving PDEs, with some typos corrected.

The final is on Friday, December 11th, 12:15-3:15pm, in 380-380Y.

The exam is 3 hours, closed book, notes, etc. The exam should be similar to midterm, except it should be about 60% longer.

The exam will cover the whole quarter, but with an emphasis on the material since the midterm, so somewhere between half and two-thirds of the exam will use material since the midterm.

As in the midterm, most questions will be computational, but there will be some theoretical questions too. Thus, questions similar to those on distributions and energy estimates on the midterm may show up. In addition, recent theoretical topics included tempered distributions and the Fourier transform, the relation of the convergence rate of the Fourier series to smoothness, as well as inner products. Typical more computational topics covered since the midterm included separation of variables, use of the appropriate Fourier series in this method, use of the Fourier transform to solve PDE, including being ready to compute with tempered distributions, the method of reflection to solve PDE in half spaces, etc., and Duhamel's principle.

The general suggestions for preparation are similar to the midterm: go through the lecture notes as well as the problem sets first. You may want to make a photocopy of your problem set solutions to help you study. Even concepts we covered on the problem sets may show up on the exam.

Next, you should try the practice exam. You should give yourself a time slot of 3 hours to get a feel for how much time you will have in the exam. Solutions to the exam are here, with a typo fixed. Please keep in mind that a practice exam cannot cover every topic that might arise on the actual final, so you should be prepared for the full range of problems we covered in the class and on the problem sets. Also, while it has been updated to match the material we covered this quarter (including two brand new problems and two modified problems), this practice exam is based on a final given three years ago, so its emphasis is slightly different from what it would have been if it it had been written from scratch.

The final has been graded!

If you would like to know your score, please write me an e-mail asking for it explicitly.

The mean was 141, the median 145 (out of a maximum score of 200). There is no grade for the final -- the course grade, as described above, has a number of components, and it is the actual scores that count. To give you an idea what your score corresponds to, i.e. what your course grade might be if you did similarly on the midterm and on the homeworks (similarly does not mean that you have a similar percentage of the maximum score, e.g. on the homeworks the expectations are much higher since you have a lot more time to do them and since you can talk to others), here are some rough ranges:

A+: high 190s and above,
A: 170ish to mid-190s,
A-: mid 160s,
B+: low 150s-160ish,
B: mid-120s-high 140s,
B-: high 100s-low 120s,
C+: mid 90s-low 100s,
C: 80ish-low 90s,
C-: mid-70s,
D: 70ish and below.

The midterm is on Thursday, October 29, in class!

The exam is 75 minutes, closed book, notes, etc. The exam should be similar to homeworks, but usually be less computationally intensive due to the lack of time.

The exam covers the material through next Tuesday's (Oct 27) lecture, i.e. including the Fourier transform.

The best preparation for the midterm is to go through the lecture notes as well as the problem sets first. You may want to make a photocopy of your problem set solutions to help you study. Even concepts we covered on the problem sets may show up on the exam.

Next, you should try the practice exam. You should give yourself a time slot of 75 minutes to get a feel for how much time you will have in the exam. Solutions to the exam are here. Please keep in mind that a practice exam cannot cover every topic that might arise on the actual midterm, so you should be prepared for the full range of problems we covered in the class and on the problem sets.

Here is the actual midterm and the solutions.

The midterm has been graded!

The mean was 72, the median 75. There is no grade for the midterm -- the course grade, as described above, has a number of components, and it is the actual scores that count. To give you an idea what your score corresponds to, i.e. what your course grade might be if you do similarly on the final and on the homeworks (similarly does not mean that you have a similar percentage of the maximum score, e.g. on the homeworks the expectations are much higher since you have a lot more time to do them and since you can talk to others), here are some rough ranges:

A+: high 90s and above,
A: mid-80s to mid-90s,
A-: 80ish,
B+: mid-70s,
B: 60ish to low 70s,
B-: low-mid-50s,
C+: mid-high-40s,
C: high 30s-low 40s,
D: 30ish-mid-30s,
F: mid-20s and below.

Problem Sets

Problem Set 1, due Thursday, October 1. Solutions.
Problem Set 2, due Thursday, October 8. Solutions, thanks to our CA!
Problem Set 3, due Thursday, October 15. Solutions, thanks to our CA!
Problem Set 4, due Thursday, October 22. Solutions, thanks to our CA!
Problem Set 5, due Tuesday, October 27. Solutions, thanks to our CA!
Problem Set 6, due Thursday, November 5. Solutions, thanks to our CA!
Problem Set 7, due Thursday, November 12. There's a typo on the second line of p.2 of the printed version, corrected here, namely f (not u) should be C^infty on the support of phi -- this is what makes sense due to the first paragraph of the problem. Solutions, thanks to our CA!
Problem Set 8, due Thursday, November 19. Solutions, thanks to our CA! There's an error in the solution of Problem 5 (i). The general solution is not exactly of the stated form. Namely, when you separate variables and for each function (in this case T and X) you get a 2-dimensional solution space, it is not sufficient to take the product of these two solutions, you need to take arbitrary linear combinations. Thus, the general solution is of the form A'_n cos(n pi c t/l) cos(n pi x/l)+ B'_n cos(n pi c t/l) sin (n pi x/l)+ C'_n sin (n pi c t/l) cos (n pi x/l) +D'_n sin (n pi c t/l) sin (n pi x/l). Note that everything in the posted solution is of this form, but the converse is not true. To see this, note that with the notation of the solution A'_n=C_n A_n, B'_n= C_n B_n, C'_n=D_n A_n, D'_n=D_n B_n. So indeed, everything in the posted solution is of this form. On the other hand, given arbitrary A'_n,B'_n,C'_n,D'_n one might not me able to find A_n,B_n,C_n,D_n such that these equations hold. For if these equations hold, then A'_n D'_n=A_n B_n C_n D_n=B'_n C'_n, so if you pick any A'_n,B'_n,C'_n, D'_n such that A'_n D'_n is not equal to B'_n C'_n, it is impossible to find A_n, B_n, C_n and D_n as in the posted solution.
Problem Set 9, due Thursday, December 3. Solutions, thanks to our CA!