Orthogonal Series and Boundary Value Problems

Evans M. Harrell II*

version of 7 April 1996

This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. The subject could be described as "linear methods for solving differential equations," especially Fourier series, other orthogonal series, and integral operators (Green's functions), but these techniques are useful for many other problems as well, such as signal processing, filtering, and numerical approximation. An equally good description of the subject would be "introduction to linear operators and linear methods."

The text is intended for a first course on the subject, to be taken by students who have had two years of calculus and an introduction to ordinary differential equations and vector spaces. Actually, there are two variants of the course which are interwoven in the text; they correspond to two different 10-week undergraduate courses at Georgia Tech, in which the text has been used. The course emphasizing the use of Fourier series and orthogonal series, Mathematics 4582, follows the "orthogonal track" with the plan listed below, while the course emphasizing integral operators and the method of Green's functions, Mathematics 4348, follows the Green track. The materials in the orthogonal track originated as class notes by Evans Harrell, while the ones in the Green track originated as class notes by James Herod.

Most of our students are engineers, with a few physicists and mathematics majors. If the students are adequately prepared, they can cover all the material in either track in ten weeks or so. On the other hand, if a substantial amount of review is necessary for linear algebra or ordinary differential equations, then they will need more time, and the course might take an entire semester. A semester-long course for well prepared students could also be made by doing all of either track and selected parts of the other track.

The book is evolving and will probably always evolve - this is one of the features of Web publishing. It has been fully useful for several months, but is still in some respects a beta version. Later this year, when we are ready to market it widely, some access control may be put in place.

Because of our short terms, we at Georgia Tech have divided the syllabus on partial differential equations into several different courses organized by method - separation of variables, characteristics, Green functions, transforms, and numerical analysis. A student can currently take some or all of these courses in any order. We have made many efforts to incorporate technology into these courses and to write text and other materials for them. This text is the first to evolve into a book residing on the Web. If you are interested in using this book as a text in a class at another university, we would be quite interested in such an experiment, and we could discuss the mechanics of doing this.

The course following the "Green track" through this book is about linear functions L and solving linear equations L(u) = f where f has been previously specified. Students who come into this class know a lot about such equations already. For example, if L is an nxn matrix and f is an n-dimensional vector, then an n-dimensional vector u can be found such that L(u) = f if the determinant of L is not zero. If the determinant of L is zero, there are some f's for which there is a solution and some for which there is none. This situation is typical even when the equation is not a matrix equation and there is no determinant defined on the linear function L. The existence of such L's and f's provokes a reasonable pair of questions for linear equations:

(1) How can one determine whether there are solutions to L(u) = f for all given f's?

(2) In case there are not solutions for all f's, how can one characterize those f's for which there is a solution.

We will answer these two questions for classes of integral equations, classes
of ordinary differential equations, and classes of partial differential
equations. When possible, the inverse of L, denoted in the usual manner by
L^{-1}, will be created. This defines the Green function!

Students planning to take this course often ask at least these three questions: What do I need to know to understand the course? What will be required of me? and, What will be the pace through the lecture notes? Here are some answers.

One should recall a little elementary geometry of R^{n}: that
perpendicularity is determined by the dot product and not by visual inspection,
and that the matrix A and its adjoint A* are also related through this dot
product: < Ax, y > = < x, A*y >. In addition to knowing about
solutions for A(u) = f, the student should know about solutions of A(u) = 0,
especially in case the determinant of A is zero. Be reminded that all such u
is called the null space of A.

In the section on integral equations, one must recall the notion of a collection of linearly independent vectors or functions. A recurring idea from the calculus is that if

where f is differentiable and g is continuous, then

A review of the differentiation of integrals with variable limits of integration might be advisable.

From an introductory course in ordinary differential equations, one should be
able to solve low order, constant coefficient differential equations. For
example, one would be expected to choose the pair cos(3x) and sin(3x) as
solutions of y'' + 9 y = 0, instead of the pair
e^{3ix} and e^{-3ix}. While both are technically correct,
there are many advantages to using the first pair in the course.

In the last section of these notes, the multi-dimensional calculus is used.
One should feel comfortable computing gradients and Laplacians of smooth
functions and computing normals to simple surfaces in R^{3}. The
divergence theorem in R^{2} and R^{3} is fundamental in
computing the adjoints of some to the linear functions of that section.

There are computer projects in these notes. They should be done in the course for they show the utility of the methods developed. Use any machine and any language. Maple syntax and output will be used for computation and visualization in these notes.

The pace through the notes will be to use about one third time for each of the chapters- chapters on integral equations, ordinary differential equations, and partial differential equations. It is hoped that the future editions of these notes will be improved because of your suggestions. Thanks are expressed to previous classes who have provided corrections to previous editions and answers to exercises.

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