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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:
Boundary value problems.