The College of the Holy Cross

Mathematics 174 -- Applied Mathematics 2

Syllabus, Spring 2000

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment.
Course Homepage: http://math.holycross.edu/~little/Applied9900/AM200.html

Course Description

This is a continuing course from MATH 173 (Applied Mathematics 1). For the first 3/4 of this semester, we will continue our study of partial differential equations and analytic techniques for their solution, focusing on problems of heat conduction, mechanical vibrations of strings and membranes, etc. In this part of the course, the goals will be to first to unify and then to extend what we did with Fourier series solutions of boundary value problems in the first semester. We will begin with a general theory of eigenfunctions and eigenvalues of boundary value problems for 2nd order ODE, developed first by the mathematicians Sturm and Liouville in the 19th century. The Sturm-Liouville theory includes all the special cases of eigenfunctions/eigenvalues that we studied last term, and explains, in a sense, why it is always possible to find a collection of eigenfunctions satisfying orthogonality relations for suitable boundary value problems.

Next, we will consider two applications of the Sturm-Liouville theory -- the circular drumhead, where the normal modes of vibration involve a new family of eigenfunctions you have not seen before (called the Bessel functions) and non-homogenous problems where eigenfunction expansions are useful to find solutions and understand their properties.

We next turn our attention to PDE on infinite domains. Here we will introduce an integral representation of solutions called the Fourier transform. The integral can be seen as the limit, in a certain sense, of a Fourier series, which can be thought of as a sort of Riemann sum approximation of the solution. We will also run into what are known as Green's functions for certain types of problems. We will see that solutions of PDE can be derived from initial conditions and the Green's function by another integral operation called convolution of functions. (Interestingly enough, we already ran into one instance of this operation last semester when we considered the formula for the kth partial sum of a Fourier series of a function given by integration of the function against the kth Dirichlet kernel function D_k(u) !)

The final section of the course will be devoted to a topic of your choice. Some of the possibilities are:

1. Applications of Fourier transforms, convolution, etc. to signal processing. It turns out that much of the mathematics behind the Green's function approach to solving PDE is also a very useful framework for thinking about frequency-domain analysis and manipulation of signals. This is the basis for most modern communication systems.
2. Numerical techniques for solution of PDE -- many of the PDE encountered in "real-world" problems are considerably more complicated than the ones we have studied. In many cases, analytic solutions are not available. Here we would study some computational techniques used to approximate solutions by computer.
3. The method of characteristics for 1st order PDE, applications to 1st order wave equations, shock waves, etc.

The rough schedule for the semester is as follows:

• Unit I: Sturm-Liouville eigenvalue problems, about 6 days
• Unit II: Applications of Sturm-Liouville theory, about 10 days
• Unit III: Infinite domain problems and Fourier transforms, about 6 days
• Unit IV: Final topic of your choice, the remaining 5 days.

Text

The text for the course is again Elementary Applied Partial Differential Equations by Richard Haberman. We will cover material from chapters 5, 7, 8, 9, 10 (and possibly 6) this semester.

Course Format

To get you more directly involved in the subject matter of this course, several times during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be present and available for questions and other help during these periods. At the conclusion of some of these discussions, at times the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments. \bigskip Some of the other meetings of the class will be structured as lectures or computer laboratory days when that seems appropriate.

Computer Work

We will be using Maple on the departmental Sun workstation network quite extensively throughout the course to implement the techniques we discuss and to generate solutions to problems. Several class meetings will take place in the SW 219 computer lab and some of the individual problem sets will include problems for which you will need to use Maple.

Grading

The assignments for the course will consist of:

1. One take-home midterm exam, worth 20% of course grade given out Thursday, February 24 and due Thursday, March 2.
2. Final exam worth 30% of the course grade. The final exam will also be given as a take-home problem set.
3. Individual problem sets and lab reports, worth 25% of the course grade.
4. Group reports from discussion days, worth 25% of the course grade.

If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.