Synopses & Reviews
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.
Description
Includes bibliographical references (p. 336-339) and index.
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