642:527 Fall 2008 Home

642:527

METHODS OF APPLIED MATHEMATICS

FALL 2008

Course overview, prerequisites, placement quiz

For general information on these topics please see the main web page for Math 527.

Basic information for Fall 2008

Instructor: Professor E. Speer, Hill Center 520, 732–445–1313, speer at math dot rutgers dot edu
Office hours:
- Monday 1:40–3:00 PM, Hill 520
- Tuesday 3:20–4:40 PM, Hill 520
- Wednesday 10:20–11:40 AM, Hill 520
- Or by appointment or chance in Hill 520
Detailed course information and policies: As web page and as a PDF file.
Tentative syllabus: As web page and as a PDF file.
Homework assignments and solutions: Click here for assignments and solutions.

Final exam

The final exam is scheduled for Friday, December 19, 8:00–11:00 AM, in SEC 203.

The exam is cumulative, but will probably emphasize Chapters 17, 18, and 19, and in particular material that we have covered since the last exam.
Here is the formula sheet for the final exam. It consists of the formula sheets for exams 1 and 2, plus formulas for the Fourier transform and for d'Alembert's solution. You will also be given the tables in the appendices of our text for the Laplace and Fourier transforms.
We will hold a problem session Thursday, December 11, 2:00–4:00 PM, in SEC 212.
I will hold office hours Thursday, December 18, 10:30–12:00 AM and 3:00–4:30 PM.
Here is the 2006 final exam for your amusement. However, I did not construct this exam and it may not completely reflect the nature of our exam.

Announcements

Here are the lecture notes from Thursday, December 4. My thanks to Vinesh Nishawala, who provided these.
Here are notes on the heat equation in a circular disk.

We are treating material in somewhat different order from that of the book. Below is a guide (current as of October 31) to how the lectures and book relate.

We have covered parts of sections 9.6, 9.9, 9.10, and 17.6; you should look over these, but the essential points are summarized in these lecture notes .

We have covered 17.3.

We talked a bit about 17.5, but not much, and will do no more with that material.

On Thursday 10/30 we did the heat equation with homogeneous Dirichlet and Neumann boundary conditions (covered, with other topics, in 18.3). This provided motivation for the study of half-range sine and cosine series, from 17.4.

On Tuesday 11/04 we will discuss quarter-range sine and cosine series (the balance of 17.4) then move on to Sturm-Liouville problems (17.7 and 17.8). As we discuss the Sturm-Liouville theory we will continue to consider problems from separation of variables for PDEs as motivation.

Next we will finish some remaining material from 18.3, then move on to the Fourier transform, sections17.9, 17.10, and 18.4.

I regret that I must cancel my office hours for Tuesday, November 4.

Here are some lecture notes on Expansions in Orthogonal Bases .

Here are the phase plane portraits for predator-prey models that were shown in class on Thursday, October 16.
Here are the phase plane portraits for competing species models that were shown in class on Tuesday, October 14.
Here are some notes on the geometry of the orbits for centers and foci. Your comments on these notes, and on any errors they contain, are welcome.
Here is a file with some plots of the phase plane of the pendulum
My office hours on Monday, September 29, are rescheduled from 10:20 to 11:40 AM.
9/12/2008: Here are the lecture notes from Thursday, September 11. My thanks to Vinesh Nishawala, who provided these.
9/8/2008: Here are the solutions to the placement quiz in PDF form.
9/7/2008: Here is a summary of the Frobenius method in PDF form. This is basically a restatement of Theorem 4.3.1 of the text, and you can use it as a replacement for that theorem in working problem 1 of section 4.3.
I was told by some of the students in the course that not everyone has been able to purchase a text. For this reason, I am posting here copies of the exercises from Sections 4.2 and 4.3 of Greenberg which will be needed to do Assignment 1.
The students who spoke to me planned to buy books on line. However, if you do not have a book and would like the math department to urge the bookstore to get more copies, let me know.
I cannot hold my office hour on Wednesday September 3; it is canceled. I will instead hold an office hour Friday September 5, 10:20–11:40 AM, Hill 520.

Exam 1

The first exam will be held Thursday, October 9. It will cover all our work on power series solutions of differential equations, Bessel's equation, and Laplace transforms. There is a homework "Assignment" for the week of October 6, but it is only for your use in studying; it will not be collected.

No calculators are to be used on the exam. I will try to make sure that all computations are simple.
Here is formula sheet that will be distributed with the exam. You will also be given a copy of the table of Laplace transforms from Appendix C of our text.
Here is Exam 1 from Fall 2007.
Here are the solutions to Exam 1.

Exam 2

The second exam will be held Thursday, November 13. It will cover all our work through Thursday, November 6, on trajectories in the phase plane, orthogonal expansions, Fourier series, Sturm-Liouville problems, and the one-dimensional heat equation with homogeneous Dirichlet and/or Neumann boundary conditions. See the guide below for the relation between lectures, text, and notes. There is a homework "Assignment" for the week of November 10 (Assignment 10), but it is only for your use in studying; it will not be collected. Solutions to this assignment are posted.

No calculators are to be used on the exam. I will try to make sure that all computations are simple.
Here is formula sheet that will be distributed with the exam.
Here is Exam 2 from Fall 2007. This exam does not cover any Sturm-Liouville problems; for that you should work on Assignment 10.
Here are the solutions to Exam 2.