PARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS NAKHLE H. ASMAR University of Missouri PRENTICE HALL, Upper Saddle River, New Jersey 07458
Contents Preface vii A Preview of Applications and Techniques 1 1.1 What Is a Partial Differential Equation? 2 1.2 Solving and Interpreting a Partial Differential Equation 7 Fourier Series 16 2.1 Periodic Functions 17 2.2 Fourier Series 22 2.3 Fourier Series of Functions with Arbitrary Periods 35 2.4 Half-Range Expansions: The Cosine, and Sine Series 46 2.5 Mean Square Approximation and Parseval's Identity 49 2.6 Complex Form of Fourier Series 56 Supplement on Convergence 2.7 Uniform Convergence of Sequences and Series of Functions 63 2.8 Dirichlet Test and Convergence of Fourier Series 72 Partial Differential Equations in Rectangular Coordinates 81 3.1 Partial Differential Equations in Physics and Engineering 82 3.2 Modeling: Vibrating Strings and the Wave Equation 87 3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables 92 3.4 D'Alembert's Method 104 3.5 The One Dimensional Heat Equation 110 3.6 Heat Conduction in Bars: Varying the Boundary Conditions 120 3.7 The Two Dimensional Wave and Heat Equations 129 3.8 Laplace's Equation in Rectangular Coordinates 138
iv Contents 3.9 Poisson's Equation: The Method of Eigenfunction Expansions 145 3.10 The Maximum Principle 154 Partial Differential Equations in Polar and Cylindrical Coordinates 160 J 4.1 The Laplacian in Various Coordinate Systems 161 4.2 Vibrations of a Circular Membrane: Symmetric Case 165 4.3 Vibrations of a Circular Membrane: General Case 174 4.4 Steady-State Temperature in a Disk 183 4.5 Steady-State Temperature in a Cylinder 192 4.6 The Helmholtz and Poisson Equations 195 Supplement on Bessel Functions 4.7 Bessel's Equation and Bessel Functions 201 4.8 Bessel Series Expansions 212 Partial Differential Equations in Spherical Coordinates 226 5.1 Preview of Problems and Methods 227 5.2 Dirichlet Problems with Symmetry 231 5.3 Spherical Harmonics and the General Dirichlet Problem 238 5.4 The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations 248 Supplement on Legendre Functions 5.5 Legendre's Differential Equation 256 5.6 Legendre Polynomials and Legendre Series Expansions 265 5.7 Associated Legendre Functions and Series Expansions 275 Sturm Liouville Theory with Engineering Applications 281 6.1 Orthogonal Functions 282 6.2 Sturm-Liouville Theory 289 6.3 The Hanging Chain 299 6.4 Fourth Order Sturm-Liouville Theory 306 6.5 Elastic Vibrations and Buckling of Beams 313
Contents V The Fourier Transform and its Applications 325 / 7.1 The Fourier Integral Representation 326 7.2 The Fourier Transform 334 7.3 The Fourier Transform Method 345 7.4 The Heat Equation and Gauss's Kernel 353 7.5 A Dirichlet Problem and the Poisson Integral Formula 361 7.6 The Fourier Cosine and Sine Transforms 366 7.7 Problems Involving Semi-Infinite Intervals 372 The Laplace and Hankel Transforms with Applications 377 8.1 The Laplace Transform 378 8.2 Further Properties of the Laplace Transform 386 8.3 The Laplace Transform Method 397 8.4 The Hankel Transform with Applications 404 Finite Difference Numerical Methods 411 9.1 The Finite Difference Method for the Heat Equation 412 9.2 The Finite Difference Method for the Wave Equation 421 9.3 The Finite Difference Method for Laplace's Equation 429 9.4 Iteration Methods for Laplace's Equation 437 Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations 442 10.1 The Sampling Theorem 443 10.2 Partial Differential Equations and the Sampling Theorem 452 10.3 The Discrete and Fast Fourier Transforms 455 10.4 The Fourier and Discrete Fourier Transforms 464 2 ~\_ An Introduction to Quantum Mechanics 470 11.1 Schrodinger's Equation 471 11.2 The Hydrogen Atom 478 11.3 Heisenberg's Uncertainty Principle 487 Supplement on Orthogonal Polynomials 11.4 Hermite and Laguerre Polynomials 494
vi Contents APPENDIXES Ordinary Differential Equations: Review of Concepts and Methods 507 A.I Linear Ordinary Differential Equations 508 A.2 Linear Ordinary Differential Equations with Constant Coefficients 516 A.3 Methods for Solving Ordinary Differential Equations 525 A.4 The Method of Power Series 532 A.5 The Method of Frobenius 540 Tables of Transforms B.I Fourier Transforms 555 B.2 Fourier Cosine Transforms 557 B.3 Fourier Sine Transforms 558 B.4 Laplace Transforms 559 References 562 Answers to Selected Exercises 566 Index 587