Fourier Series and Boundary Value Problems

Preface

1 Fourier Series

Piecewise Continuous Functions

Fourier Cosine Series

Examples

Fourier Sine Series

Examples

Fourier Series

Examples

Adaptations to Other Intervals

2 Convergence of Fourier Series

One-Sided Derivatives

A Property of Fourier Coefficients

Two Lemmas

A Fourier Theorem

A Related Fourier Theorem

Examples

Convergence on Other Intervals

A Lemma

Absolute and Uniform Convergence of Fourier Series

The Gibbs Phenomenon

Differentiation of Fourier Series

Integration of Fourier Series

3 Partial Differential Equations of Physics

Linear Boundary Value Problems

One-Dimensional Heat Equation

Related Equations

Laplacian in Cylindrical and Spherical Coordinates

Derivations

Boundary Conditions

Duhamel's Principle

A Vibrating String

Vibrations of Bars and Membranes

General Solution of the Wave Equation

Types of Equations and Boundary Conditions

4 The Fourier Method

Linear Operators

Principle of Superposition

Examples

Eigenvalues and Eigenfunctions

A Temperature Problem

A Vibrating String Problem

Historical Development

5 Boundary Value Problems

A Slab with Faces at Prescribed Temperatures

Related Temperature Problems

Temperatures in a Sphere

A Slab with Internally Generated Heat

Steady Temperatures in Rectangular Coordinates

Steady Temperatures in Cylindrical Coordinates

A String with Prescribed Initial Conditions

Resonance

An Elastic Bar

Double Fourier Series

Periodic Boundary Conditions

6 Fourier Integrals and Applications

The Fourier Integral Formula

Dirichlet's Integral

Two Lemmas

A Fourier Integral Theorem

The Cosine and Sine Integrals

Some Eigenvalue Problems on Undounded Intervals

More on Superposition of Solutions

Steady Temperatures in a Semi-Infinite Strip

Temperatures in a Semi-Infinite Solid

Temperatures in an Unlimited Medium

7 Orthonormal Sets

Inner Products and Orthonormal Sets

Examples

Generalized Fourier Series

Examples

Best Approximation in the Mean

Bessel's Inequality and Parseval's Equation

Applications to Fourier Series

8 Sturm-Liouville Problems and Applications

Regular Sturm-Liouville Problems

Modifications

Orthogonality of Eigenfunctions adn Real Eigenvalues

Real-Valued Eigenfunctions

Nonnegative Eigenvalues

Methods of Solution

Examples of Eigenfunction Expansions

A Temperature Problem in Rectangular Coordinates

Steady Temperatures

Other Coordinates

A Modification of the Method

Another Modification

A Vertically Hung Elastic Bar

9 Bessel Functions and Applications

The Gamma Function

Bessel Functions Jn(x)

Solutions When v = 0,1,2,...

Recurrence Relations

Bessel's Integral Form

Some Consequences of the Integral Forms

The Zeros of Jn(x)

Zeros of Related Functions

Orthogonal Sets of Bessel Functions

Proof of the Theorems

Two Lemmas

Fourier-Bessel Series

Examples

Temperatures in a Long Cylinder

A Temperature Problem in Shrunken Fittings

Internally Generated Heat

Temperatures in a Long Cylindrical Wedge

Vibration of a Circular Membrane

10 Legendre Polynomials and Applications

Solutions of Legendre's Equation

Legendre Polynomials

Rodrigues' Formula

Laplace's Integral Form

Some Consequences of the Integral Form

Orthogonality of Legendre Polynomials

Normalized Legendre Polynomials

Legendre Series

The Eigenfunctions Pn(cos ?)

Dirichlet Problems in Spherical Regions

Steady Temperatures in a Hemisphere

11 Verification of Solutions and Uniqueness

Abel's Test for Uniform Convergence

Verification of Solution of Temperature Problem

Uniqueness of Solutions of the Heat Equation

Verification of Solution of Vibrating String Problem

Uniqueness of Solutions of the Wave Equation

Appendixes

Bibliography

Some Fourier Series Expansions

Solutions of Some Regular Sturm-Liouville Problems

Some Fourier-Bessel Series Expansions

Index

Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus.

There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations.

The book is a thorough revision of the seventh edition and much care is taken to give the student fewer distractions when determining solutions of eigenvalue problems, and other topics have been presented in their own sections like Gibbs' Phenomenon and the Poisson integral formula.