James V. Herod*

Web page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.

**First Order and Second Order, Hyperbolic**

By:

James V. Herod

School of Mathematics

Georgia Institute of Technology

Atlanta, Georgia 30332-0160

herod@math.gatech.edu

Summer, 1994

**Preface**

More and more it is coming to be common for students to be able to manage to take an undergraduate course in partial differential equations. A student that can not manage this is unlucky, some would say.

Though science and engineering students have a hard time putting an undergraduate course in partial differential equation into their schedules, they see ordinary and partial differential equations arise repeatedly in their science and engineering studies. There is often not time for the solutions of these equations to be derived in the engineering classes. Perhaps, methods are suggested and references are given.

Because of the importance of the subject, however, student are stealing away time for undergraduate partial differential equations courses. A year of undergraduate partial differential equations is becoming less rare.

If the offerings in differential equations are sequential -- first PDE I, then PDE II, etc. -- the problem is compounded and impossible for the student who can get a slot for math in the Winter Term, only to find the prerequisite ran Fall Term.

One resolution is to construct courses to be "linearly independent" so that there might be several offerings in partial differential equations, each covering a different method for viewing and solving some types of equations. No course would depend on the student having had another and the courses might be taken in any order. Of course, maturity and experience helps in every mathematics course.

A student who takes undergraduate partial differential equations usually gets a taste of some of the methods reflected by these names: the methods of characteristics, or the construction of Green's functions, or the separation of variables. There are also numerical methods, transform methods, and Lie symmetries.

These notes make one part in a series of five. The perspective is to explore first order and second order, hyperbolic equations. In the construction of these notes three ideas were on the forefront:

1. The notes should be intended for undergraduates and require no more than sophomore differential equations.

2. Enough examples and exercises should be included so that a student will understand that there is a reasonable class of partial differential equations that can be solved by these methods, and that many cannot. Most important, the student should understand the principles of the methods.

3. Incorporating a computer algebra system into the methods can not only help the student to understand the analytic work, but also help with the tedious calculations. The computer algebra system's graphing capability can enable insight into the nature of solutions heretofore found only in the more experienced student.

MAPLE was chosen as the computer algebra system to use in this set of notes. Each section of the notes will have accompanying syntax to create the examples and figures. A worksheet for each section can be found at the back of these notes. These worksheet will illustrate and give visualization to the methods of the section. Solutions for the exercises are written in MAPLE. The syntax may not be the most efficient. It is intended to be intuitive.

The notes should remain alive -- in the sense that students, faculty, and readers will add examples and exercises. Different perspectives of the analysis and of the pedagogy will come. Also, MAPLE will change. Does anyone ever finish such a collection of ideas ? Or, is this not just a part of a conversation between the student-reader, the student-writer?

Questions, comments, or suggestions concerning the notes are welcomed at herod@math.gatech.edu and will receive a response.