Boundary. DIFFERENTIAL EQUATIONS with Fourier Series and. Value Problems APPLIED PARTIAL. Fifth Edition. Richard Haberman PEARSON

Transcription

1 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

2 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 Equilibrium Temperature Distribution Prescribed Temperature Insulated Boundaries 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 Introduction Linearity Heat Equation with Zero Temperatures at Finite Ends Introduction Separation of Variables Time-Dependent Ordinary Differential Equation Boundary Value (Eigenvalue) Problem Product Solutions and the Principle of Superposition Orthogonality of Sines Formulation, Solution, and Interpretation of an Example Summary 50 Appendix to 23: Orthogonality of Functions Worked Examples with the Heat Equation (Other Boundary Value Problems) Heat Conduction in a Rod with Insulated Ends Heat Conduction in a Thin Insulated Circular Ring Summary of Boundary Value Problems Laplace's Equation: Solutions and Qualitative Properties Laplace's Equation Inside a Rectangle Laplace's Equation Inside a Circular Disk Fluid Flow Outside a Circular Cylinder (Lift) Qualitative Properties of Laplace's Equation 79 vii

3 viii Contents 3 Fourier Series Introduction Statement of Convergence Theorem Fourier Cosine and Sine Series Fourier Sine Series Fourier Cosine Series Representing f(x) by Both a Sine and Cosine Series Even and Odd Parts Continuous Fourier Series Term-by-Term Differentiation of Fourier Series Term-By-Term Integration of Fourier Series Complex Form of Fourier Series Wave Equation: Vibrating Strings and Membranes Introduction Derivation of a Vertically Vibrating String Boundary Conditions Vibrating String with Fixed Ends Vibrating Membrane Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves Snell's Law of Refraction Intensity (Amplitude) of Reflected and Refracted Waves 463 Total Internal Reflection Sturm-Liouville Eigenvalue Problems Introduction Examples Heat Flow in a Nonuniform Rod Circulaily Symmetric Heat Flow Sturm-Liouville Eigenvalue Problems General Classification Regular Sturm-Liouville Eigenvalue Problem Example and Illustration of Theorems 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 Rayleigh Quotient Worked Example: Vibrations of a Nonuniform String Boundary Conditions of the Third Kind Large Eigenvalues (Asymptotic Behavior) Approximation Properties 211

4 Contents ix 6 Finite Difference Numerical Methods for Partial Differential Equations Introduction Finite Differences and Truncated Taylor Series Heat Equation Introduction A Partial Difference Equation Computations Fourier-von Neumann Stability Analysis Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations Matrix Notation Nonhomogeneous Problems Other Numerical Schemes Other Types of Boundary Conditions Two-Dimensional Heat Equation Wave Equation Laplace's Equation Finite Element Method Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation) The Simplest Triangular Finite Elements Higher-Dimensional Partial Differential Equations Introduction Separation of the Time Variable Vibrating Membrane: Any Shape Heat Conduction: Any Region Summary Vibrating Rectangular Membrane 272 Appendix to 73: Outline of Alternative Method to Separate Variables Statements and Illustrations of Theorems for the Eigenvalue Problem V2c/> +A</> = Green's Formula, Self-Adjoint Operators, and Multidimensional Eigenvalue Problems 287 Appendix to 75: Gram-Schmidt Method Rayleigh Quotient and Laplace's Equation Rayleigh Quotient Time-Dependent Heat Equation and Laplace's Equation Vibrating Circular Membrane and Bessel Functions Introduction Separation of Variables Eigenvalue Problems (One-Dimensional) Bessel's Differential Equation Singular Points and Bessel's Differential Equation 299

5 x Contents Bessel Functions and Their Asymptotic Properties (Near 2 0) = Eigenvalue Problem Involving Bessel Functions Initial Value Problem for a Vibrating Circular Membrane 779 Circularly Symmetric Case More on Bessel Functions Qualitative Properties of Bessel Functions Asymptotic Formulas for the Eigenvalues Zeros of Bessel Functions and Nodal Curves Series Representation of Bessel Functions Laplace's Equation in a Circular Cylinder Introduction Separation of Variables Zero Temperature on the Lateral Sides and on the Bottom or Top Zero Temperature on the Top and Bottom Modified Bessel Functions Spherical Problems and Legendre Polynomials Introduction Separation of Variables and One-Dimensional Eigenvalue Problems Associated Legendre Functions and Legendre Polynomials Radial Eigenvalue Problems Product Solutions, Modes of Vibration, and the Initial Value Problem Laplace's Equation Inside a Spherical Cavity Nonhomogeneous Problems Introduction Heat Flow with Sources and Nonhomogeneous Boundary Conditions Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions) Method of Eigenfunction Expansion Using Green's Formula (With or Without Homogeneous Boundary Conditions) Forced Vibrating Membranes and Resonance Poisson's Equation Green's Functions for Time-Independent Problems Introduction One-Dimensional Heat Equation (Optional) Green's Functions for Boundary Value Problems for Ordinary Differential Equations One-Dimensional Steady-State Heat Equation The Method of Variation of Parameters (Optional) 379

6 Contents xi The Method of Eigenfunction Expansion for Green's Functions The Dirac Delta Function and Its Relationship to Green's Functions Nonhomogeneous Boundary Conditions 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 Introduction Fredholm Alternative Generalized Green's Functions Green's Functions for Poisson's Equation Introduction Multidimensional Dirac Delta Function and Green's Functions Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative Direct Solution of Green's Functions (One-Dimensional Eigenfunctions) (Optional) Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions Infinite Space Green's Functions Green's Functions for Bounded Domains Using Infinite Space Green's Functions 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 Perturbed Eigenvalue Problems Introduction Mathematical Example Vibrating Nearly Circular Membrane Summary Infinite Domain Problems: Fourier Transform Solutions of Partial Differential Equations Introduction Heat Equation on an Infinite Domain Fourier Transform Pair Motivation from Fourier Series Identity Fourier Transform Inverse Fourier Transform of a Gaussian 443 Appendix to 103: Derivation of the Inverse Fourier Transform of a Gaussian Fourier Transform and the Heat Equation Heat Equation 450

7 xii Contents Fourier Transforming the Heat Equation: Transforms of Derivatives Convolution Theorem Summary of Properties of the Fourier Transform Fourier Sine and Cosine Transforms: The Heat Equation on Semi-Infinite Intervals Introduction 4G Heat Equation on a Semi-Infinite Interval I Fourier Sine and Cosine Transforms Transforms of Derivatives Heat Equation on a Semi-Infinite Interval II Tables of Fourier Sine and Cosine Transforms Worked Examples Using Transforms One-Dimensional Wave Equation on an Infinite Interval 1062 Laplace's Equation in a Semi-Infinite Strip Laplace's Equation in a Half-Plane Laplace's Equation in a Quarter-Plane Heat Equation in a Plane (Two-Dimensional Fourier Transforms) Table of Double Fourier Transforms Scattering and Inverse Scattering Green's Functions for Wave and Heat Equations Introduction Green's Functions for the Wave Equation Introduction Green's Formula for the Wave Equation Reciprocity Using the Green's Function Green's Function for the Wave Equation 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 Infinite Space Green's Function for the Three-Dimensional Wave Equation (Huygens' Principle) Two-Dimensional Infinite Space Green's Function Summary Green's Functions for the Heat Equation Introduction Non-Self-Adjoint Nature of the Heat Equation Green's Formula for the Heat Equation Adjoint Green's Function Reciprocity Representation of the Solution Using Green's Functions Alternate Differential Equation for the Green's Function 520

8 Contents xiii 1138 Infinite Space Green's Function for the Diffusion Equation Green's Function for the Heat Equation (Semi-Infinite Domain) Green's Function for the Heat Equation (on a Finite Region) The Method of Characteristics for Linear and Quasilinear Wave Equations Introduction Characteristics for First-Order Wave Equations Introduction Method of Characteristics for First-Order Partial Differential Equations Method of Characteristics for the One-Dimensional Wave Equation 1231 General Solution Initial Value Problem (Infinite Domain) D'Alembert's Solution Semi-Infinite Strings and Reflections Method of Characteristics for a Vibrating String of Fixed Length The Method of Characteristics for Quasilinear Partial Differential Equations Method of Characteristics Traffic Flow Method of Characteristics (Q 0) = Shock Waves Quasilinear Example First-Order Nonlinear Partial Differential Equations Eikonal Equation Derived from the Wave Equation Solving the Eikonal Equation in Uniform Media and Reflected Waves First-Order Nonlinear Partial Differential Equations Laplace Transform Solution of Partial Differential Equations Introduction Properties of the Laplace Transform Introduction Singularities of the Laplace Transform Transforms of Derivatives Convolution Theorem Green's Functions for Initial Value Problems for Ordinary Differential Equations A Signal Problem for the Wave Equation A Signal Problem for a Vibrating String of Finite Length The Wave Equation and Its Green's Function 600

9 xiv Contents 137 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane Solving the Wave Equation Using Laplace Transforms (with Complex Variables) Dispersive Waves: Slow Variations, Stability, Nonlinearity, and Perturbation Methods Introduction Dispersive Waves and Group Velocity Traveling Waves and the Dispersion Relation Group Velocity I Wave Guides Response to Concentrated Periodic Sources with Frequency ujf Green's Function If Mode Propagates Green's Function If Mode Does Not Propagate Design Considerations Fiber Optics Group Velocity II and the Method of Stationary Phase Method of Stationary Phase Application to Linear Dispersive 146 Slowly Varying Dispersive Waves (Group Velocity and Caustics) 1461 Approximate Solutions of Dispersive Partial Waves Differential Equations Formation of a Caustic Wave Envelope Equations (Concentrated Wave Number) Schrodinger Equation Linearized Korteweg-de Vries Equation Nonlinear Dispersive Waves: Korteweg-de Vries Equation Solitons and Inverse Scattering Nonlinear Schrodinger Equation Stability and Instability Brief Ordinary Differential Equations and Bifurcation Theory Elementary Example of a Stable Equilibrium for a Partial Differential Equation Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation Ill-posed Problems Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation Nonlinear Complex Ginzburg-Landau Equation Long-Wave Instabilities Pattern Formation for Reaction-Diffusion Equations and the Turing Instability, 676

10 Contents xv 149 Singular Perturbation Methods: Multiple Scales Ordinary Differential Equation: Weakly Nonlinearly Damped Oscillator Ordinary Differential Equation: Slowly Varying Oscillator Slightly Unstable Partial Differential Equation on Fixed Spatial Domain Slowly Varying Medium for the Wave Equation Slowly Varying Linear Dispersive Waves (Including Weak Nonlinear Effects) Singular Perturbation Methods: Boundary Layers Method of Matched Asymptotic Expansions Boundary Layer in an Ordinary Differential Equation Diffusion of a Pollutant Dominated by Convection 705 Bibliography 712 Answers to Starred Exercises 716 Index 737