# Fourier Series and Boundary Value Problems

8^{th}Edition

Copyright: 2012

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ISBN10: 007803597X | ISBN13: 9780078035975

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The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

The estimated amount of time this product will be on the market is based on a number of factors, including faculty input to instructional design and the prior revision cycle and updates to academic research-which typically results in a revision cycle ranging from every two to four years for this product. Pricing subject to change at any time.

### Program Details

### 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 *θ*)

_{n}

#### 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

### 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 *θ*)

_{n}

#### 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

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