Fourier Series and Boundary Value Problems

by Brown, James Ward; Churchill, Ruel Vance

• ISBN13: 9780072325706
• ISBN10: 0072325704
• Edition: 6th
• Format: Hardcover
• Copyright: 2000-08-02
• Publisher: McGraw-Hill Science/Engineering/Math

Summary

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.

Author Biography

James Ward Brown is Professor of Mathematics at The University of Michigan--Dearborn Ruel V. Churchill was, at the time of his death in 1987, Professor Emeritus of Mathematics at The University of Michigan, where he began teaching in 1922

Table of Contents

Preface

1 Partial Differential Equations of Physics

Linear Boundary Value Problems

Conduction of Heat

Higher Dimensions

Cylindrical Coordinates

Spherical Coordinates

Boundary Conditions

A Vibrating String

Vibrations of Bars and Membranes

Types of Equations and Boundary Conditions

Methods of Solution

2 The Fourier Method

Linear Operators

Principle of Superposition

A Generalization

A Temperature Problem

The Nonhomogeneous Condition

A Vibrating String Problem

The Nonhomogeneous Condition

Historical Development

3 Orthonormal Sets and Fourier Series

Piecewise Continuous Functions

Inner Products and Orthonormal Sets

Examples

Generalized Fourier Series

Fourier Cosine Series

Fourier Sine Series

Fourier Series

Examples

Best Approximation in the Mean

Bessel's Inequality and a Property of Fourier Constants

4 Convergence of Fourier Series

One-Sided Derivatives

Two Lemmas

A Fourier Theorem

Discussion of the Theorem and Its Corollary

Fourier Series on Other Intervals

A Lemma

Uniform Convergence of Fourier Series

Differentiation of Fourier Series

Integration of Fourier Series

Convergence in the Mean

5 Boundary Value Problems

A Slab with Faces at Prescribed Temperatures

Related Problems

A Slab with Internally Generated Heat

A Dirichlet Problem

Cylindrical Coordinates

A String with Prescribed Initial Velocity

Resonance

An Elastic Bar

Double Fourier Series

Periodic Boundary Conditions

6 Sturm-Liouville Problems and Applications

Regular Sturm-Liouville Problems

Modifications

Orthogonality of Eigenfunctions

Real-Valued Eigenfunctions and Nonnegative Eigenvalues

Methods of Solution

Examples of Eigenfunction Expansions

Surface Heat Transfer

A Dirichlet Problem

Modifications of the Method

A Vertically Hung Elastic Bar

7 Fourier Integrals and Applications

The Fourier Integral Formula

Dirichlet's Integral

Two Lemmas

A Fourier Integral Theorem

The Cosine and Sine Integrals

More on Superposition of Solutions

Temperatures in a Semi-Infinite Solid

Temperatures in an Unlimited Medium

8 Bessel Functions and Applications

Bessel Functions Jn

General Solutions of Bessel's Equation

Recurrence Relations

Bessel's Integral Form

The Zeros of J0(x)

Zeros of Related Functions

Orthogonal Sets of Bessel Functions

Proof of Theorem

The Orthonormal Functions

Fourier-Bessel Series

Temperatures in a Long Cylinder

Internally Generated Heat

Vibration of a Circular Membrane

9 Legendre Polynomials and Applications

Solutions of Legendre's Equation

Legendre Polynomials

Orthogonality of Legendre Polynomials

Rodrigues' Formula and Norms

Legendre Series

The Eigenfunctions Pn(cos theta)

Dirichlet Problems in Spherical Regions

Steady Temperatures in a Hemisphere

10 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

On Laplace's and Poisson's Equations

An Example

Appendixes

1 Bibliography

2 Some Fourier Series Expansions

3 Solutions of Some Regular Sturm-Liouville Problems