Orthogonal Series and Boundary Value Problems

Evans M. Harrell, II, and James V. Herod

Copyright 1994,1995, 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 integral operators and the method of Green's functions, Mathematics 4348, follows the "Green track" with the plan listed below, while the course emphasizing Fourier series and other orthogonal series, Mathematics 4582, follows the orthogonal 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. A good 15-week course could be made from all of one track and selected pieces of the other.

You are welcome to browse, but if you make more than casual use, such as downloading files or using them as study materials, certain restrictions and fees apply. Before proceeding, please read this copyright notice.

Syllabus: An outline of the content and objectives of a course at Georgia Tech

A diagnostic homework assignment, which you should take before embarking on this course.

Other possibly useful links are to the usenet newsgroup devoted to mathematics and mail to other professors at Georgia Tech who teach these courses, Profs. Chafee, Loss, Neff, and Swiech,

Contents for the Green Track

(Switch here to the shorter index)
(Switch here to the Orthogonal Track)
(Switch here to Prof. Herod's lecture notes on partial differential equations and the method of characteristics. They are in Web format but not systematically marked as hypertext.)

PREFACE

INTRODUCTION

BACKGROUND MATERIAL

CHAPTER I: INTEGRAL EQUATIONS

Section 1.1 Geometry and integral operators
Section 1.2 The Fredholm Alternative Theorems
Section 1.3 Solving Y = KY + f where K has a separable kernel
Section 1.4 Solving Y = KY + f where K is small
Section 1.5 An alternate meaning for small
The Resolvent
Section 1.6 K does not have a separable kernel and is not small
A COMPENDIUM OF PROBLEMS

A test at this stage.

CHAPTER II: BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

Section 2.1 More about function space
Section 2.2 Differential Operators and Their Adjoints
Section 2.3 The Fredholm Alternative Theorems
Section 2.4 A Method