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# Fourier Series and Boundary Value Problems

## byJames Ward Brown, Ruel V Churchill

### Hardcover | February 8, 2011

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### about

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.

### Details & Specs

The following ISBNs are associated with this title:

ISBN - 10:007803597X

ISBN - 13:9780078035975

### Customer Reviews of Fourier Series and Boundary Value Problems

### Extra Content

Table of Contents

### 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 *J*_{n}(*x*)

_{n}

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

#### Recurrence Relations

#### Bessel's Integral Form

#### Some Consequences of the Integral Forms

#### The Zeros of *J*_{n}(*x*)

_{n}

#### 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 *P*_{n}(cos *?*)

_{n}