Fourier Series and Boundary Value Problems

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 Jn(x) Solutions When v = 0,1,2,... 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 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 Pn(cos theta) 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