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9780073051932: Fourier Series and Boundary Value Problems
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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.

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L'autore:
RUEL V. CHURCHILL (University of Michigan)
Contenuti:

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

Discussion of the Theorem and Its Corollary

Convergence on Other Intervals

A Lemma

Absolute and Uniform Convergence of Fourier Series

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

A Vibrating String

Vibrations of Bars and Membranes

General Solution of the Wave Equation

Types of Equations and Boundary Equations

4 The Fourier Method

Linear Operators

Principle of Superposition

A Temperature Problem

A Vibrating String Problem

Historical Development

5 Boundary Value Problems

A Slab with Faces at Prescribed Temperatures

Related Problems

A Slab with Internally Generated Heat

Steady Temperatures in a Rectangular Plate

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

More on Superposition of Solutions

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

Real-Valued Eigenfunctions and Nonnegative Eigenvalues

Methods of Solution

Examples of Eigenfunction Expansions

A Temperature Problem in Rectangular Coordinates

Another Problem

Other Coordinates

A Modification of the Method

Another Modification

A Vertically Hung Elastic Bar

9 Bessel Functions and Applications

Bessel Functions Jn(x)

General Solutions of Bessel's Equation

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

The Orthonormal Functions

Fourier-Bessel Series

Examples

Temperatures in a Long Cylinder

Internally Generated Heat

Vibration of a Circular Membrane

10 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 ¿)

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

Index

Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.

  • EditoreMcGraw-Hill Education
  • Data di pubblicazione2006
  • ISBN 10 0073051934
  • ISBN 13 9780073051932
  • RilegaturaCopertina rigida
  • Numero di pagine366
  • Valutazione libreria

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Altre edizioni note dello stesso titolo

9780070636606: Fourier Series and Boundary Value Problems, 7th Edition (Churchill-Brown Series)

Edizione in evidenza

ISBN 10:  ISBN 13:  9780070636606
Casa editrice: McGraw Hill India, 2006
Brossura

  • 9780071259170: Fourier Series and Boundary Value Problems

    McGraw..., 2007
    Brossura

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