Fourier Series and Boundary Value Problems

This text is designed to be 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 mathematics 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. This book has been published by McGraw-Hill since 1941.

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 Jo(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