实用偏微分方程 第4版 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 Heat Equation1

1.1 Introduction1

1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod2

Contents3

前言3

1.3 Boundary Conditions12

1.4 Equilibrium Temperature Distribution14

1.4.1 Prescribed Temperature14

1.4.2 Insulated Boundaries16

1.5 Derivation of the Heat Equation in Two or Three Dimensions21

2 Method of Separation of Variables35

2.1 Introduction35

2.2 Linearity36

2.3 Heat Equation with Zero Temperatures at Finite Ends38

2.3.1 Introduction38

2.3.2 Separation of Variables39

2.3.3 Time-Dependent Equation41

2.3.4 Boundary Value Problem42

2.3.5 Product Solutions and the Principle of Superposition47

2.3.6 Orthogonality of Sines50

2.3.7 Formulation,Solution,and Interpretation of an Example51

2.3.8 Summary54

2.4 Worked Examples with the Heat Equation:Other Boundary Value Problems59

2.4.1 Heat Conduction in a Rod with Insulated Ends59

2.4.2 Heat Conduction in a Thin Circular Ring63

2.4.3 Summary of Boundary Value Problems68

2.5.1 Laplace's Equation Inside a Rectangle71

2.5 Laplace's Equation:Solutions and Qualitative Properties71

2.5.2 Laplace's Equation for a Circular Disk76

2.5.3 Fluid Flow Past a Circular Cylinder （Lift）80

2.5.4 Qualitative Properties of Laplace's Equation83

3 Fourier Series89

3.1 Introduction89

3.2 Statement of Convergence Theorem91

3.3 Fourier Cosine and Sine Series96

3.3.1 Fourier Sine Series96

3.3.2 Fourier Cosine Series106

3.3.3 Representing f（x）by Both a Sine and Cosine Series108

3.3.4 Even and Odd Parts109

3.3.5 Continuous Fourier Series111

3.4 Term-by-Term Differentiation of Fourier Series116

3.5 Term-By-Term Integration of Fourier Series127

3.6 Complex Form of Fourier Series131

4 Wave Equation:Vibrating Strings and Membranes135

4.1 Introduction135

4.2 Derivation of a Vertically Vibrating String135

4.3 Boundary Conditions139

4.4 Vibrating String with Fixed Ends142

4.5 Vibrating Membrane149

4.6 Reflection and Refraction of Electromagnetic（Light）and Acoustic（Sound）Waves151

4.6.1 Snell's Law of Refraction152

4.6.2 Intensity（Amplitude）of Reflected and Refracted Waves154

4.6.3 Total Internal Refection155

5.1 Introduction157

5 Sturm-Liouville Eigenvalue Problems157

5.2 Examples158

5.2.1 Heat Flow in a Nonuniform Rod158

5.2.2 Circularly Symmetric Heat Flow159

5.3 Sturm-Liouville Eigenvalue Problems161

5.3.1 General Classification161

5.3.2 Regular Sturm-Liouville Eigenvalue Problem162

5.3.3 Example and Illustration of Theorems164

5.4 Worked Example:Heat Flow in a Nonuniform Rod without Sources170

5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems174

5.6 Rayleigh Quotient189

5.7 Worked Example:Vibrations of a Nonuniform String195

5.8 Boundary Conditions of the Third Kind198

5.9 Large Eigenvalues（Asymptotic Behavior）212

5.10 Approximation Properties216

6 Finite Difference Numerical Methods for Partial Differential Equations222

6.1 Introduction222

6.2 Finite Differences and Truncated Taylor Series223

6.3.2 A Partial Difference Equation229

6.3.1 Introduction229

6.3 Heat Equation229

6.3.3 Computations231

6.3.4 Fourier-von Neumann Stability Analysis235

6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations241

6.3.6 Matrix Notation243

6.3.7 Nonhomogeneous Problems247

6.3.8 Other Numerical Schemesv247

6.3.9 Other Types of Boundary Conditions248

6.4 Two-Dimensional Heat Equation253

6.5 Wave Equation256

6.6 Laplace's Equation260

6.7 Finite Element Method267

6.7.1 Approximation with Nonorthogonal Functions（Weak Form of the Partial Differential Equation）267

6.7.2 The Simplest Triangular Finite Elements270

7 Higher Dimensional Partial Differential Equations275

7.1 Introduction275

7.2.1 Vibrating Membrane:Any Shape276

7.2 Separation of the Time Variable276

7.2.2 Heat Conduction:Any Region278

7.2.3 Summary279

7.3 Vibrating Rectangular Membrane280

7.4 Statements and Illustrations of Theorems289

for the Eigenvalue Problem ？2φ＋？φ＝0289

7.5 Green's Formula,Self-Adjoint Operators and Multidimensional Eigenvalue Problems295

7.6 Rayleigh Quotient and Laplace's Equation300

7.6.1 Rayleigh Quotient300

7.6.2 Time-Dependent Heat Equation and Laplace's Equation301

7.7 Vibrating Circular Membrane and Bessel Functions303

7.7.1 Introduction303

7.7.2 Separation of Variables303

7.7.3 Eigenvalue Problems（One Dimensional）305

7.7.4 Bessel's Differential Equation306

7.7.5 Singular Points and Bessel's Differential Equation307

7.7.6 Bessel Functions and Their Asymptotic Properties（near z＝0）308

7.7.7 Eigenvalue Problem Involving Bessel Functions309

7.7.8 Initial Value Problem for a Vibrating Circular Membrane311

7.7.9 Circularly Symmetric Case313

7.8 More on Bessel Functions318

7.8.1 Qualitative Properties of Bessel Functions318

7.8.2 Asymptotic Formulas for the Eigenvalues319

7.8.3 Zeros of Bessel Functions and Nodal Curves320

7.8.4 Series Representation of Bessel Functions322

7.9 Laplace's Equation in a Circular Cylinder326

7.9.1 Introduction326

7.9.2 Separation of Variables326

7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top328

7.9.4 Zero Temperature on the Top and Bottom330

7.9.5 Modified Bessel Functions332

7.10 Spherical Problems and Legendre Polynomials336

7.10.1 Introduction336

7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems337

7.10.3 Associated Legendre Functions and Legendre Polynomials338

7.10.4 Radial Eigenvalue Problems341

7.10.5 Product Solutions,Modes of Vibration,and the Initial Value Problem342

7.10.6 Laplace's Equation Inside a Spherical Cavity343

8 Nonhomogeneous Problems347

8.1 Introduction347

8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions347

8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions（Differentiating Series of Eigenfunctions）354

8.4 Method of Eigenfunction Expansion Using Green's Formula（With or Without Homogeneous Boundary Conditions）359

8.5 Forced Vibrating Membranes and Resonance364

8.6 Poisson's Equation372

9.2 One-dimensional Heat Equation380

9.1 Introduction380

9 Green's Functions for Time-Independent Problems380

9.3 Green's Functions for Boundary Value Problems for Ordinary Dif-ferential Equations385

9.3.1 One-Dimensional Steady-State Heat Equation385

9.3.2 The Method of Variation of Parameters386

9.3.3 The Method of Eigenfunction Expansion for Green's Functions389

9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions391

9.3.5 Nonhomogeneous Boundary Conditions397

9.3.6 Summary399

9.4.1 Introduction405

9.4 Fredholm Alternative and Generalized Green's Functions405

9.4.2 Fredholm Alternative407

9.4.3 Generalized Green's Functions409

9.5 Green's Functions for Poisson's Equation416

9.5.1 Introduction416

9.5.2 Multidimensional Dirac Delta Function and Green's Functions417

9.5.3 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative418

9.5.4 Direct Solution of Green's Functions（One-Dimensional Eigenfunctions）420

9.5.5 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions422

9.5.6 Infinite Space Green's Functions423

9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions426

9.5.8 Green's Functions for a Semi-Infinite Plane（y＞0）Using Infinite Space Green's Functions:The Method of Images427

9.5.9 Green's Functions for a Circle:The Method of Images430

9.6 Perturbed Eigenvalue Problens438

9.6.1 Introduction438

9.6.2 Mathematical Example438

9.6.3 Vibrating Nearly Circular Membrane440

9.7 Summary443

10.2 Heat Equation on an Infinite Domain445

10 Infinite Domain Problems:Fourier Transform Solutions of Partial Differential Equations445

10.1 Introduction445

10.3 Fourier Transform Pair449

10.3.1 Motivation from Fourier Series Identity449

10.3.2 Fourier Transform450

10.3.3 Inverse Fourier Transform of a Gaussian451

10.4 Fourier Transform and the Heat Equation459

10.4.1 Heat Equation459

10.4.2 Fourier Transforming the Heat Equation:Transforms of Derivatives464

10.4.3 Convolution Theorem466

10.4.4 Summary of Properties ofthe Fourier Transform469

10.5 Fourier Sine and Cosine Transforms:The Heat Equation on Semi-Infinite Intervals471

10.5.1 Introduction471

10.5.2 Heat Equation on a Semi-Infinite Interval Ⅰ471

10.5.3 Fourier Sine and Cosine Transforms473

10.5.4 Transforms of Derivatives474

10.5.5 Heat Equation on a Semi-Infinite Interval Ⅱ476

10.5.6 Tables of Fourier Sine and Cosine Transforms479

10.6.1 One-Dimensional Wave Equation on an Infinite Interval482

10.6 Worked Examples Using Transforms482

10.6.2 Laplace's Equation in a Semi-Infinite Strip484

10.6.3 Laplace's Equation in a Half-Plane487

10.6.4 Laplace's Equation in a Quarter-Plane491

10.6.5 Heat Equation in a Plane（Two-Dimensional Fourier Transforms）494

10.6.6 Table of Double-Fourier Transforms498

10.7 Scattering and Inverse Scattering503

11.2 Green's Functions for the Wave Equation508

11.2.1 Introduction508

11 Green's Functions for Wave and Heat Equations508

11.1 Introduction508

11.2.2 Green's Formula510

11.2.3 Reciprocity511

11.2.4 Using the Green's Function513

11.2.5 Green's Function for the Wave Equation515

11.2.6 Alternate Differential Equation for the Green's Function515

11.2.7 Infinite Space Green's Function for the One-Dimensional Wave Equation and d'Alembert's Solution516

11.2.8 Infinite Space Green's Function for the Three-Dimensional Wave Equation（Huygens'Principle）518

11.2.9 Two-Dimensional Infinite Space Green's Function520

11.2.10 Summary520

11.3 Green's Functions for the Heat Equation523

11.3.1 Introduction523

11.3.2 Non-Self-Adjoint Nature of the Heat Equation524

11.3.3 Green's Formulav525

11.3.5 Reciprocity527

11.3.4 Adjoint Green's Function527

11.3.6 Representation of the Solution Using Green's Functions528

11.3.7 Alternate Differential Equation for the Green's Function530

11.3.8 Infinite Space Green's Function for the Diffusion Equation530

11.3.9 Green's Function for the Heat Equation（Semi-Infinite Domain）532

11.3.10 Green's Function for the Heat Equation（on a Finite Region）533

12 The Method of Characteristics for Linear and Quasilinear Wave Equations536

12.1 Introduction536

12.2 Characteristics for First-Order Wave Equations537

12.2.1 Introduction537

12.2.2 Method of Characteristics for First-Order Partial Differential Equations538

12.3 Method of Characteristics for the One-Dimensional Wave Equation543

12.3.1 General Solution543

12.3.2 Initial Value Problem（Infinite Domain）545

12.3.3 D'alembert's Solution549

12.4 Semi-Infinite Strings and Reflections552

12.5 Method of Characteristics for a Vibrating String of Fixed Length557

12.6 The Method of Characteristics for Quasilinear Partial Differential Equations561

12.6.1 Method of Characteristics561

12.6.2 Traffic Flow562

12.6.3 Method of Characteristics （Q＝0）564

12.6.4 Shock Waves567

12.6.5 Quasilinear Example579

12.7 First-Order Nonlinear Partial Differential Equations585

12.7.1 Eikonal Equation Derived from the Wave Equation585

12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves586

12.7.3 First-Order Nonlinear Partial Differential Equations589

13.1 Introduction591

13 Laplace Transform Solution of Partial Differential Equations591

13.2 Properties of the Laplace Transform592

13.2.1 Introduction592

13.2.2 Singularities of the Laplace Transform592

13.2.3 Transforms of Derivatives596

13.2.4 Convolution Theorem597

13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations601

13.4 A Signal Problem for the Wave Equation603

13.5 A Signal Problem for a Vibrating String of Finite Length606

13.6 The Wave Equation and its Green's Function610

13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane613

13.8 Solving the Wave Equation Using Laplace Transforms （with Complex Variables）618

14 Dispersive Waves:Slow Variations,Stability,Nonlinearity,and Perturbation Methods621

14.1 Introduction621

14.2 Dispersive Waves and Group Velocity622

14.2.1 Traveling Waves and the Dispersion Relation622

14.2.2 Group Velocity Ⅰ625

14.3 Wave Guides628

14.3.1 Response to Concentrated Periodic Sources with Frequency ωf630

14.3.2 Green's Function If Mode Propagates631

14.3.3 Green's Function If Mode Does Not Propagate632

14.3.4 Design Considerations632

14.4 Fiber Optics634

14.5 Group Velocity Ⅱ and the Method of Stationary Phase638

14.5.1 Method of Stationary Phase639

14.5.2 Application to Linear Dispersive Waves641

14.6 Slowly Varying Dispersive Waves（Group Velocity and Caustics）645

14.6.1 Approximate Solutions of Dispersive Partial Differential Equations645

14.6.2 Formation of a Caustic648

14.7 Wave Envelope Equations（Concentrated Wave Number）654

14.7.1 Schr？dinger Equation655

14.7.2 Linearized Korteweg-de Vries Equation657

14.7.3 Nonlinear Dispersive Waves:Korteweg-deVries Equation659

14.7.4 Solitons and Inverse Scattering662

14.7.5 Nonlinear Schr？dinger Equation664

14.8 Stability and Instability669

14.8.1 Brief Ordinary Differential Equations and Bifurcation Theory669

14.8.2 Elementary Example of a Stable Equilibrium for a Partial Differential Equation676

14.8.3 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation677

14.8.4 Ill posed Problems679

14.8.5 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation680

14.8.6 Nonlinear Complex Ginzburg-Landau Equation682

14.8.7 Long Wave Instabilities688

14.8.8 Pattern Formation for Reaction-Diffusion Equations and the Turing Instability689

14.9 Singular Perturbation Methods:Multiple Scales696

14.9.1 Ordinary Differential Equation:Weakly Nonlinearly Damped Oscillator696

14.9.2 Ordinary Differential Equation:Slowly Varying Oscillator699

14.9.3 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain703

14.9.4 Slowly Varying Medium for the Wave Equation705

14.9.5 Slowly Varying Linear Dispersive Waves（Including Weak Nonlinear Effects）708

14.10 Singular Perturbation Methods:Boundary Layers Method of Matched Asymptotic Expansions713

14.10.1 Boundary Layer in an Ordinary Differential Equation713

14.10.2 Diffusion of a Pollutant Dominated by Convection719

Bibliography726

Answers to Starred Exercises731

Index751