Fourier series and boundary value problems

James Ward Brown, Ruel V. Churchill0070082022, 9780070082021

This is an introductory treatment of Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It is designed for students who have completed a first course in ordinary differential equations and the equivalent of a term of advanced calculus. In order that the book be accessible to as many students as possible, there are footnotes referring to texts which give proofs of the more delicate results in advanced calculus that are occasionally needed. The physical applications, explained in some detail, are kept on a fairly elementary level. The first objective of the book is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. Representations of functions by Fourier series involving sine and cosine functions are given special attention. Fourier integral representations and expansions in series of Bessel functions and Legendre polynomials are also treated. The second objective is a clear presentation of the classical method of separations of variables used in solving boundary value problems with the aid of those representations. Some attention is given to the verification of solutions and to uniqueness of solutions, for the method cannot be presented properly without such considerations. Other methods are treated in the authors’ book Complex Variables and Applications, and in Professor Churchill’s book, Operational Mathematics.

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Table of contents :
Front cover……Page 1
Seies……Page 2
Joseph Fourier……Page 4
Title page……Page 5
Date-line……Page 6
About the authors……Page 7
Dedication……Page 9
Joseph Fourier……Page 11
CONTENTS……Page 13
Preface……Page 15
1 Partial Differential Equations of Physics……Page 19
2 Fourier Series……Page 57
3 The Fourier Method……Page 123
4 Boundary Value Problems……Page 147
5 Sturm-Liouville Problems and Applications……Page 186
6 Fourier Integrals and Applications……Page 235
7 Bessel Functions and Applications……Page 260
8 Legendre Polynomials and Applications……Page 311
9 Uniqueness of Solutions……Page 339
Bibliography……Page 354
Index……Page 359
Back cover……Page 367