APPLIED PARTIAL DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fifth Edition Richard Haberman Southern Methodist University PEARSON Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Contents Preface xvii 1 Heat Equation 1 11 Introduction 1 12 Derivation of the Conduction of Heat in a One-Dimensional Rod 2 13 Boundary Conditions 11 14 Equilibrium Temperature Distribution 14 141 Prescribed Temperature 14 142 Insulated Boundaries 16 15 Derivation of the Heat Equation in Two or Three Dimensions 19 Appendix to 15: Review of Gradient and a Derivation of Fourier's Law of Heat Conduction 30 2 Method of Separation of Variables 32 21 Introduction 32 22 Linearity 32 23 Heat Equation with Zero Temperatures at Finite Ends 35 231 Introduction 35 232 Separation of Variables 35 233 Time-Dependent Ordinary Differential Equation 37 234 Boundary Value (Eigenvalue) Problem 38 235 Product Solutions and the Principle of Superposition 43 236 Orthogonality of Sines 46 237 Formulation, Solution, and Interpretation of an Example 48 238 Summary 50 Appendix to 23: Orthogonality of Functions 54 24 Worked Examples with the Heat Equation (Other Boundary Value Problems) 55 241 Heat Conduction in a Rod with Insulated Ends 55 242 Heat Conduction in a Thin Insulated Circular Ring 59 243 Summary of Boundary Value Problems 64 25 Laplace's Equation: Solutions and Qualitative Properties 67 251 Laplace's Equation Inside a Rectangle 67 252 Laplace's Equation Inside a Circular Disk 72 253 Fluid Flow Outside a Circular Cylinder (Lift) 76 254 Qualitative Properties of Laplace's Equation 79 vii
viii Contents 3 Fourier Series 86 31 Introduction 86 32 Statement of Convergence Theorem 88 33 Fourier Cosine and Sine Series 92 331 Fourier Sine Series 92 332 Fourier Cosine Series 102 333 Representing f(x) by Both a Sine and Cosine Series 105 334 Even and Odd Parts 106 335 Continuous Fourier Series 107 34 Term-by-Term Differentiation of Fourier Series 112 35 Term-By-Term Integration of Fourier Series 123 36 Complex Form of Fourier Series 127 4 Wave Equation: Vibrating Strings and Membranes 130 41 Introduction 130 42 Derivation of a Vertically Vibrating String 130 43 Boundary Conditions 133 44 Vibrating String with Fixed Ends 137 45 Vibrating Membrane 143 46 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves 145 461 Snell's Law of Refraction 146 148 462 Intensity (Amplitude) of Reflected and Refracted Waves 463 Total Internal Reflection 149 163 5 Sturm-Liouville Eigenvalue Problems 151 51 Introduction 151 52 Examples 151 521 Heat Flow in a Nonuniform Rod 151 522 Circulaily Symmetric Heat Flow 153 53 Sturm-Liouville Eigenvalue Problems 155 531 General Classification 155 532 Regular Sturm-Liouville Eigenvalue Problem 156 533 Example and Illustration of Theorems 157 54 Worked Example: Heat Flow in a Nonuniform Rod without Sources 55 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems Appendix to 55: Matrix Eigenvalue Problem and Orthogonality 167 of Eigenvectors 178 56 Rayleigh Quotient 184 57 Worked Example: Vibrations of a Nonuniform String 189 58 Boundary Conditions of the Third Kind 192 59 Large Eigenvalues (Asymptotic Behavior) 207 510 Approximation Properties 211
Contents ix 6 Finite Difference Numerical Methods for Partial Differential Equations 217 61 Introduction 217 62 Finite Differences and Truncated Taylor Series 217 63 Heat Equation 224 631 Introduction 224 632 A Partial Difference Equation 224 633 Computations 226 634 Fourier-von Neumann Stability Analysis 228 635 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations 235 636 Matrix Notation 238 637 Nonhomogeneous Problems 242 638 Other Numerical Schemes 242 639 Other Types of Boundary Conditions 243 64 Two-Dimensional Heat Equation 247 65 Wave Equation 250 66 Laplace's Equation 253 67 Finite Element Method 260 671 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation) 260 672 The Simplest Triangular Finite Elements 263 7 Higher-Dimensional Partial Differential Equations 268 71 Introduction 268 72 Separation of the Time Variable 269 721 Vibrating Membrane: Any Shape 269 722 Heat Conduction: Any Region 271 723 Summary 272 73 Vibrating Rectangular Membrane 272 Appendix to 73: Outline of Alternative Method to Separate Variables 281 74 Statements and Illustrations of Theorems for the Eigenvalue Problem V2c/> +A</> = 0 282 75 Green's Formula, Self-Adjoint Operators, and Multidimensional Eigenvalue Problems 287 Appendix to 75: Gram-Schmidt Method 292 76 Rayleigh Quotient and Laplace's Equation 293 761 Rayleigh Quotient 293 762 Time-Dependent Heat Equation and Laplace's Equation 294 77 Vibrating Circular Membrane and Bessel Functions 295 771 Introduction 295 772 Separation of Variables 296 773 Eigenvalue Problems (One-Dimensional) 297 774 Bessel's Differential Equation 299 775 Singular Points and Bessel's Differential Equation 299
x Contents 304 776 Bessel Functions and Their Asymptotic Properties (Near 2 0) = 301 777 Eigenvalue Problem Involving Bessel Functions 302 778 Initial Value Problem for a Vibrating Circular Membrane 779 Circularly Symmetric Case 305 78 More on Bessel Functions 312 781 Qualitative Properties of Bessel Functions 312 782 Asymptotic Formulas for the Eigenvalues 313 783 Zeros of Bessel Functions and Nodal Curves 314 784 Series Representation of Bessel Functions 316 79 Laplace's Equation in a Circular Cylinder 319 791 Introduction 319 792 Separation of Variables 320 793 Zero Temperature on the Lateral Sides and on the Bottom or Top 322 794 Zero Temperature on the Top and Bottom 323 795 Modified Bessel Functions 326 710 Spherical Problems and Legendre Polynomials 330 7101 Introduction 330 7102 Separation of Variables and One-Dimensional Eigenvalue Problems 330 7103 Associated Legendre Functions and Legendre Polynomials 332 7104 Radial Eigenvalue Problems 335 7105 Product Solutions, Modes of Vibration, and the Initial Value Problem 335 7106 Laplace's Equation Inside a Spherical Cavity 336 8 Nonhomogeneous Problems 341 81 Introduction 341 82 Heat Flow with Sources and Nonhomogeneous Boundary Conditions 341 83 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions) 347 84 Method of Eigenfunction Expansion Using Green's Formula (With or Without Homogeneous Boundary Conditions) 353 85 Forced Vibrating Membranes and Resonance 358 86 Poisson's Equation 366 9 Green's Functions for Time-Independent Problems 374 91 Introduction 374 92 One-Dimensional Heat Equation (Optional) 374 93 Green's Functions for Boundary Value Problems for Ordinary Differential Equations 379 931 One-Dimensional Steady-State Heat Equation 379 932 The Method of Variation of Parameters (Optional) 379
Contents xi 397 420 933 The Method of Eigenfunction Expansion for Green's Functions 382 934 The Dirac Delta Function and Its Relationship to Green's Functions 384 935 Nonhomogeneous Boundary Conditions 391 936 Limit of Time-Dependent Problem 392 Appendix to 93: Establishing Green's Formula with Dirac Delta Functions 94 Fredholm Alternative and Generalized Green's Functions 398 941 Introduction 398 942 Fredholm Alternative 400 943 Generalized Green's Functions 402 95 Green's Functions for Poisson's Equation 409 951 Introduction 409 952 Multidimensional Dirac Delta Function and Green's Functions 410 953 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative 411 954 Direct Solution of Green's Functions (One-Dimensional Eigenfunctions) (Optional) 413 955 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions 415 956 Infinite Space Green's Functions 416 957 Green's Functions for Bounded Domains Using Infinite Space Green's Functions 419 958 Green's Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green's Functions: The Method of Images 959 Green's Functions for a Circle: The Method of Images 423 96 Perturbed Eigenvalue Problems 430 961 Introduction 430 962 Mathematical Example 431 963 Vibrating Nearly Circular Membrane 432 97 Summary 435 10 Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations 437 101 Introduction 437 102 Heat Equation on an Infinite Domain 437 103 Fourier Transform Pair 441 1031 Motivation from Fourier Series Identity 441 1032 Fourier Transform 442 1033 Inverse Fourier Transform of a Gaussian 443 Appendix to 103: Derivation of the Inverse Fourier Transform of a Gaussian 445 104 Fourier Transform and the Heat Equation 450 1041 Heat Equation 450
xii Contents 473 1042 Fourier Transforming the Heat Equation: Transforms of Derivatives 455 1043 Convolution Theorem 457 1044 Summary of Properties of the Fourier Transform 461 105 Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals 463 1051 Introduction 4G3 1052 Heat Equation on a Semi-Infinite Interval I 463 1053 Fourier Sine and Cosine Transforms 465 1054 Transforms of Derivatives 466 1055 Heat Equation on a Semi-Infinite Interval II 467 1056 Tables of Fourier Sine and Cosine Transforms 469 106 Worked Examples Using Transforms 473 1061 One-Dimensional Wave Equation on an Infinite Interval 1062 Laplace's Equation in a Semi-Infinite Strip 475 1063 Laplace's Equation in a Half-Plane 479 1064 Laplace's Equation in a Quarter-Plane 482 1065 Heat Equation in a Plane (Two-Dimensional Fourier Transforms) 486 1066 Table of Double Fourier Transforms 490 107 Scattering and Inverse Scattering 495 11 Green's Functions for Wave and Heat Equations 499 111 Introduction 499 112 Green's Functions for the Wave Equation 499 1121 Introduction 499 1122 Green's Formula for the Wave Equation 500 1123 Reciprocity 502 1124 Using the Green's Function 504 1125 Green's Function for the Wave Equation 506 1126 Alternate Differential Equation for the Green's Function 1127 Infinite Space Green's Function for the One-Dimensional 506 Wave Equation and d'alembert's Solution 507 1128 Infinite Space Green's Function for the Three-Dimensional Wave Equation (Huygens' Principle) 509 1129 Two-Dimensional Infinite Space Green's Function 511 11210 Summary 511 113 Green's Functions for the Heat Equation 514 1131 Introduction 514 1132 Non-Self-Adjoint Nature of the Heat Equation 515 1133 Green's Formula for the Heat Equation 516 1134 Adjoint Green's Function 517 1135 Reciprocity 518 1136 Representation of the Solution Using Green's Functions 518 1137 Alternate Differential Equation for the Green's Function 520
Contents xiii 1138 Infinite Space Green's Function for the Diffusion Equation 521 1139 Green's Function for the Heat Equation (Semi-Infinite Domain) 522 11310 Green's Function for the Heat Equation (on a Finite Region) 523 534 12 The Method of Characteristics for Linear and Quasilinear Wave Equations 527 121 Introduction 527 122 Characteristics for First-Order Wave Equations 528 1221 Introduction 528 1222 Method of Characteristics for First-Order Partial Differential Equations 529 123 Method of Characteristics for the One-Dimensional Wave Equation 1231 General Solution 534 1232 Initial Value Problem (Infinite Domain) 536 1233 D'Alembert's Solution 540 124 Semi-Infinite Strings and Reflections 543 125 Method of Characteristics for a Vibrating String of Fixed Length 548 126 The Method of Characteristics for Quasilinear Partial Differential Equations 552 1261 Method of Characteristics 552 1262 Traffic Flow 553 1263 Method of Characteristics (Q 0) = 555 1264 Shock Waves 558 1265 Quasilinear Example 570 127 First-Order Nonlinear Partial Differential Equations 575 1271 Eikonal Equation Derived from the Wave Equation 575 1272 Solving the Eikonal Equation in Uniform Media and Reflected Waves 576 1273 First-Order Nonlinear Partial Differential Equations 579 13 Laplace Transform Solution of Partial Differential Equations 581 131 Introduction 581 132 Properties of the Laplace Transform 581 1321 Introduction 581 1322 Singularities of the Laplace Transform 582 1323 Transforms of Derivatives 586 1324 Convolution Theorem 587 133 Green's Functions for Initial Value Problems for Ordinary Differential Equations 591 134 A Signal Problem for the Wave Equation 593 135 A Signal Problem for a Vibrating String of Finite Length 597 136 The Wave Equation and Its Green's Function 600
xiv Contents 137 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane 603 138 Solving the Wave Equation Using Laplace Transforms (with Complex Variables) 608 14 Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods 611 141 Introduction 611 142 Dispersive Waves and Group Velocity 612 1421 Traveling Waves and the Dispersion Relation 612 1422 Group Velocity I 615 143 Wave Guides 617 1431 Response to Concentrated Periodic Sources with Frequency ujf 620 1432 Green's Function If Mode Propagates 620 1433 Green's Function If Mode Does Not Propagate 621 1434 Design Considerations 622 144 Fiber Optics 623 145 Group Velocity II and the Method of Stationary Phase 627 1451 Method of Stationary Phase 628 1452 Application to Linear Dispersive 146 Slowly Varying Dispersive Waves (Group Velocity and Caustics) 1461 Approximate Solutions of Dispersive Partial Waves 630 634 Differential Equations 634 1462 Formation of a Caustic 636 147 Wave Envelope Equations (Concentrated Wave Number) 642 1471 Schrodinger Equation 643 1472 Linearized Korteweg-de Vries Equation 645 1473 Nonlinear Dispersive Waves: Korteweg-de Vries Equation 647 1474 Solitons and Inverse Scattering 650 1475 Nonlinear Schrodinger Equation 652 148 Stability and Instability 656 1481 Brief Ordinary Differential Equations and Bifurcation Theory 650 1482 Elementary Example of a Stable Equilibrium for a Partial Differential Equation 663 1483 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation 664 1484 Ill-posed Problems 667 1485 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation 668 1486 Nonlinear Complex Ginzburg-Landau Equation 670 1487 Long-Wave Instabilities 676 1488 Pattern Formation for Reaction-Diffusion Equations and the Turing Instability, 676
Contents xv 149 Singular Perturbation Methods: Multiple Scales 683 1491 Ordinary Differential Equation: Weakly Nonlinearly Damped Oscillator 683 1492 Ordinary Differential Equation: Slowly Varying Oscillator 686 1493 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain 690 1494 Slowly Varying Medium for the Wave Equation 692 1495 Slowly Varying Linear Dispersive Waves (Including Weak Nonlinear Effects) 695 1410 Singular Perturbation Methods: Boundary Layers Method of Matched Asymptotic Expansions 700 14101 Boundary Layer in an Ordinary Differential Equation 700 14102 Diffusion of a Pollutant Dominated by Convection 705 Bibliography 712 Answers to Starred Exercises 716 Index 737