Table of contents for Fourier series and boundary value problems / James Ward Brown, Ruel Churchill.

Bibliographic record and links to related information available from the Library of Congress catalog.

Note: Contents data are machine generated based on pre-publication provided by the publisher. Contents may have variations from the printed book or be incomplete or contain other coding.


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CONTENTS
Preface xv
Partial Differential 1
Equations of Physics 3
Linear Boundary Value Problems 2 Conduction of Heat 4 Higher Dimensions 6 
Cylindrical Coordinates 8 Spherical Coordinates 11 
Boundary Conditions 13 A Vibrating String 17
Vibrations of Bars and Membranes 21
Types of Equations and Boundary Conditions 24 
Methods of Solution 26
The Fourier Method Linear Operators 30
Principle of Superposition 35 
A Generalization 38 
A Temperature Problem 41 
The Nonhomogeneous Condition 45 
A Vibrating String Problem 48 
The Nonhomogeneous Condition 50 Historical Development 54
Orthonormal Sets and Fourier Series 56
Piecewise Continuous Functions 56 Inner Products and Orthonormal Sets 59
Examples 61
Generalized Fourier Series 62 
Fourier Cosine Series 66 
Fourier Sine Series 68 
Fourier Series 73 
Examples 75
Best Approximation in the Mean 77
Bessel?s Inequality and a Property of Fourier Constants 80
Convergence of Fourier Series 84
One-Sided Derivatives 84 
Two Lemmas 87 
A Fourier Theorem 89
Discussion of the Theorem and Its Corollary 92 
Fourier Series on Other Intervals 98
A Lemma 105
Uniform Convergence of Fourier Series 107 
Differentiation of Fourier Series 110 
Integration of Fourier Series 111 
Convergence in the Mean 113
Boundary Value Problems 120
A Slab with Faces at Prescribed Temperatures 121
Related Problems 124
A Slab with Internally Generated Heat 130 
A Dirichlet Problem 135 
Cylindrical Coordinates 137
A String with Prescribed Initial Velocity 143 
Resonance 145 
An Elastic Bar 149 
Double Fourier Series 152
Periodic Boundary Conditions 155
Sturm-Liouville Problems and Applications 159
Regular Sturm-Liouville Problems 159 
Modifications 161
Orthogonality of Eigenfunctions 162
Real-Valued Eigenfunctions and Nonnegative Eigenvalues 167 
Methods of Solution 170
Examples of Eigenfunction Expansions 175 Surface Heat Transfer 181
A Dirichlet Problem 185 
Modifications of the Method 190 
A Vertically Hung Elastic Bar 197
Fourier Integrals and Applications 200
The Fourier Integral Formula 204 
Dirichlet?s Integral 206
Two Lemmas 208
A Fourier Integral Theorem 211 The Cosine and Sine Integrals 215
More on Superposition of Solutions 219 
Temperatures in a Semi-Infinite Solid 221 
Temperatures in an Unlimited Medium 227
Bessel Functions and Applications 230
General Solutions of Bessel?s Equation 233 
Recurrence Relations 239 
Bessel?s Integral Form 241 
The Zeros of J0(x) 246
Zeros of Related Functions 248 
Orthogonal Sets of Bessel Functions 250 
Proof of Theorem 252
The Orthonormal Functions 256
Fourier-Bessel Series 258 
Temperatures in a Long Cylinder 264 
Internally 268
Generated Heat 270 
Vibration of a Circular Membrane 275
Legendre Polynomials and Applications 279
Solutions of Legendre?s Equation 279 
Legendre Polynomials 281
Orthogonality of Legendre Polynomials 286 
Rodrigues Formula and Norms 288 
Legendre Series 292
The Eigenfunctions Pn(cos) 297
Dirichlet Problems in Spherical Regions 299 Steady Temperatures in a Hemisphere 
301
Verification of Solutions and Uniqueness 307
Abel?s Test for Uniform Convergence 307 Verification of Solution of Temperature 
Problem 310 
Uniqueness of Solutions of the Heat Equation 313 
Verification of Solution of Vibrating String Problem 317 
Uniqueness of Solutions of the Wave Equation 320
On Laplace?s and Poisson?s Equations 322 
An Example 324
Appendixes 329
Bibliography 329
Some Fourier Series Expansions 333
Solutions of Some Regular Sturm-Liouville Problems 335
Index 339

Library of Congress Subject Headings for this publication:

Fourier series.
Functions, Orthogonal.
Boundary value problems.