1 Heat Equation 1
1.1 Introduction 1
1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod 2
Contents 3
前言 3
1.3 Boundary Conditions 12
1.4 Equilibrium Temperature Distribution 14
1.4.1 Prescribed Temperature 14
1.4.2 Insulated Boundaries 16
1.5 Derivation of the Heat Equation in Two or Three Dimensions 21
2 Method of Separation of Variables 35
2.1 Introduction 35
2.2 Linearity 36
2.3 Heat Equation with Zero Temperatures at Finite Ends 38
2.3.1 Introduction 38
2.3.2 Separation of Variables 39
2.3.3 Time-Dependent Equation 41
2.3.4 Boundary Value Problem 42
2.3.5 Product Solutions and the Principle of Superposition 47
2.3.6 Orthogonality of Sines 50
2.3.7 Formulation,Solution,and Interpretation of an Example 51
2.3.8 Summary 54
2.4 Worked Examples with the Heat Equation:Other Boundary Value Problems 59
2.4.1 Heat Conduction in a Rod with Insulated Ends 59
2.4.2 Heat Conduction in a Thin Circular Ring 63
2.4.3 Summary of Boundary Value Problems 68
2.5.1 Laplace's Equation Inside a Rectangle 71
2.5 Laplace's Equation:Solutions and Qualitative Properties 71
2.5.2 Laplace's Equation for a Circular Disk 76
2.5.3 Fluid Flow Past a Circular Cylinder （Lift） 80
2.5.4 Qualitative Properties of Laplace's Equation 83
3 Fourier Series 89
3.1 Introduction 89
3.2 Statement of Convergence Theorem 91
3.3 Fourier Cosine and Sine Series 96
3.3.1 Fourier Sine Series 96
3.3.2 Fourier Cosine Series 106
3.3.3 Representing f（x）by Both a Sine and Cosine Series 108
3.3.4 Even and Odd Parts 109
3.3.5 Continuous Fourier Series 111
3.4 Term-by-Term Differentiation of Fourier Series 116
3.5 Term-By-Term Integration of Fourier Series 127
3.6 Complex Form of Fourier Series 131
4 Wave Equation:Vibrating Strings and Membranes 135
4.1 Introduction 135
4.2 Derivation of a Vertically Vibrating String 135
4.3 Boundary Conditions 139
4.4 Vibrating String with Fixed Ends 142
4.5 Vibrating Membrane 149
4.6 Reflection and Refraction of Electromagnetic（Light）and Acoustic（Sound）Waves 151
4.6.1 Snell's Law of Refraction 152
4.6.2 Intensity（Amplitude）of Reflected and Refracted Waves 154
4.6.3 Total Internal Refection 155
5.1 Introduction 157
5 Sturm-Liouville Eigenvalue Problems 157
5.2 Examples 158
5.2.1 Heat Flow in a Nonuniform Rod 158
5.2.2 Circularly Symmetric Heat Flow 159
5.3 Sturm-Liouville Eigenvalue Problems 161
5.3.1 General Classification 161
5.3.2 Regular Sturm-Liouville Eigenvalue Problem 162
5.3.3 Example and Illustration of Theorems 164
5.4 Worked Example:Heat Flow in a Nonuniform Rod without Sources 170
5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems 174
5.6 Rayleigh Quotient 189
5.7 Worked Example:Vibrations of a Nonuniform String 195
5.8 Boundary Conditions of the Third Kind 198
5.9 Large Eigenvalues（Asymptotic Behavior） 212
5.10 Approximation Properties 216
6 Finite Difference Numerical Methods for Partial Differential Equations 222
6.1 Introduction 222
6.2 Finite Differences and Truncated Taylor Series 223
6.3.2 A Partial Difference Equation 229
6.3.1 Introduction 229
6.3 Heat Equation 229
6.3.3 Computations 231
6.3.4 Fourier-von Neumann Stability Analysis 235
6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations 241
6.3.6 Matrix Notation 243
6.3.7 Nonhomogeneous Problems 247
6.3.8 Other Numerical Schemesv 247
6.3.9 Other Types of Boundary Conditions 248
6.4 Two-Dimensional Heat Equation 253
6.5 Wave Equation 256
6.6 Laplace's Equation 260
6.7 Finite Element Method 267
6.7.1 Approximation with Nonorthogonal Functions（Weak Form of the Partial Differential Equation） 267
6.7.2 The Simplest Triangular Finite Elements 270
7 Higher Dimensional Partial Differential Equations 275
7.1 Introduction 275
7.2.1 Vibrating Membrane:Any Shape 276
7.2 Separation of the Time Variable 276
7.2.2 Heat Conduction:Any Region 278
7.2.3 Summary 279
7.3 Vibrating Rectangular Membrane 280
7.4 Statements and Illustrations of Theorems 289
for the Eigenvalue Problem ？2φ＋？φ＝0 289
7.5 Green's Formula,Self-Adjoint Operators and Multidimensional Eigenvalue Problems 295
7.6 Rayleigh Quotient and Laplace's Equation 300
7.6.1 Rayleigh Quotient 300
7.6.2 Time-Dependent Heat Equation and Laplace's Equation 301
7.7 Vibrating Circular Membrane and Bessel Functions 303
7.7.1 Introduction 303
7.7.2 Separation of Variables 303
7.7.3 Eigenvalue Problems（One Dimensional） 305
7.7.4 Bessel's Differential Equation 306
7.7.5 Singular Points and Bessel's Differential Equation 307
7.7.6 Bessel Functions and Their Asymptotic Properties（near z＝0） 308
7.7.7 Eigenvalue Problem Involving Bessel Functions 309
7.7.8 Initial Value Problem for a Vibrating Circular Membrane 311
7.7.9 Circularly Symmetric Case 313
7.8 More on Bessel Functions 318
7.8.1 Qualitative Properties of Bessel Functions 318
7.8.2 Asymptotic Formulas for the Eigenvalues 319
7.8.3 Zeros of Bessel Functions and Nodal Curves 320
7.8.4 Series Representation of Bessel Functions 322
7.9 Laplace's Equation in a Circular Cylinder 326
7.9.1 Introduction 326
7.9.2 Separation of Variables 326
7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top 328
7.9.4 Zero Temperature on the Top and Bottom 330
7.9.5 Modified Bessel Functions 332
7.10 Spherical Problems and Legendre Polynomials 336
7.10.1 Introduction 336
7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems 337
7.10.3 Associated Legendre Functions and Legendre Polynomials 338
7.10.4 Radial Eigenvalue Problems 341
7.10.5 Product Solutions,Modes of Vibration,and the Initial Value Problem 342
7.10.6 Laplace's Equation Inside a Spherical Cavity 343
8 Nonhomogeneous Problems 347
8.1 Introduction 347
8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions 347
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 Resonance 364
8.6 Poisson's Equation 372
9.2 One-dimensional Heat Equation 380
9.1 Introduction 380
9 Green's Functions for Time-Independent Problems 380
9.3 Green's Functions for Boundary Value Problems for Ordinary Dif-ferential Equations 385
9.3.1 One-Dimensional Steady-State Heat Equation 385
9.3.2 The Method of Variation of Parameters 386
9.3.3 The Method of Eigenfunction Expansion for Green's Functions 389
9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions 391
9.3.5 Nonhomogeneous Boundary Conditions 397
9.3.6 Summary 399
9.4.1 Introduction 405
9.4 Fredholm Alternative and Generalized Green's Functions 405
9.4.2 Fredholm Alternative 407
9.4.3 Generalized Green's Functions 409
9.5 Green's Functions for Poisson's Equation 416
9.5.1 Introduction 416
9.5.2 Multidimensional Dirac Delta Function and Green's Functions 417
9.5.3 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative 418
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 Conditions 422
9.5.6 Infinite Space Green's Functions 423
9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions 426
9.5.8 Green's Functions for a Semi-Infinite Plane（y＞0）Using Infinite Space Green's Functions:The Method of Images 427
9.5.9 Green's Functions for a Circle:The Method of Images 430
9.6 Perturbed Eigenvalue Problens 438
9.6.1 Introduction 438
9.6.2 Mathematical Example 438
9.6.3 Vibrating Nearly Circular Membrane 440
9.7 Summary 443
10.2 Heat Equation on an Infinite Domain 445
10 Infinite Domain Problems:Fourier Transform Solutions of Partial Differential Equations 445
10.1 Introduction 445
10.3 Fourier Transform Pair 449
10.3.1 Motivation from Fourier Series Identity 449
10.3.2 Fourier Transform 450
10.3.3 Inverse Fourier Transform of a Gaussian 451
10.4 Fourier Transform and the Heat Equation 459
10.4.1 Heat Equation 459
10.4.2 Fourier Transforming the Heat Equation:Transforms of Derivatives 464
10.4.3 Convolution Theorem 466
10.4.4 Summary of Properties ofthe Fourier Transform 469
10.5 Fourier Sine and Cosine Transforms:The Heat Equation on Semi-Infinite Intervals 471
10.5.1 Introduction 471
10.5.2 Heat Equation on a Semi-Infinite Interval Ⅰ 471
10.5.3 Fourier Sine and Cosine Transforms 473
10.5.4 Transforms of Derivatives 474
10.5.5 Heat Equation on a Semi-Infinite Interval Ⅱ 476
10.5.6 Tables of Fourier Sine and Cosine Transforms 479
10.6.1 One-Dimensional Wave Equation on an Infinite Interval 482
10.6 Worked Examples Using Transforms 482
10.6.2 Laplace's Equation in a Semi-Infinite Strip 484
10.6.3 Laplace's Equation in a Half-Plane 487
10.6.4 Laplace's Equation in a Quarter-Plane 491
10.6.5 Heat Equation in a Plane（Two-Dimensional Fourier Transforms） 494
10.6.6 Table of Double-Fourier Transforms 498
10.7 Scattering and Inverse Scattering 503
11.2 Green's Functions for the Wave Equation 508
11.2.1 Introduction 508
11 Green's Functions for Wave and Heat Equations 508
11.1 Introduction 508
11.2.2 Green's Formula 510
11.2.3 Reciprocity 511
11.2.4 Using the Green's Function 513
11.2.5 Green's Function for the Wave Equation 515
11.2.6 Alternate Differential Equation for the Green's Function 515
11.2.7 Infinite Space Green's Function for the One-Dimensional Wave Equation and d'Alembert's Solution 516
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 Function 520
11.2.10 Summary 520
11.3 Green's Functions for the Heat Equation 523
11.3.1 Introduction 523
11.3.2 Non-Self-Adjoint Nature of the Heat Equation 524
11.3.3 Green's Formulav 525
11.3.5 Reciprocity 527
11.3.4 Adjoint Green's Function 527
11.3.6 Representation of the Solution Using Green's Functions 528
11.3.7 Alternate Differential Equation for the Green's Function 530
11.3.8 Infinite Space Green's Function for the Diffusion Equation 530
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 Equations 536
12.1 Introduction 536
12.2 Characteristics for First-Order Wave Equations 537
12.2.1 Introduction 537
12.2.2 Method of Characteristics for First-Order Partial Differential Equations 538
12.3 Method of Characteristics for the One-Dimensional Wave Equation 543
12.3.1 General Solution 543
12.3.2 Initial Value Problem（Infinite Domain） 545
12.3.3 D'alembert's Solution 549
12.4 Semi-Infinite Strings and Reflections 552
12.5 Method of Characteristics for a Vibrating String of Fixed Length 557
12.6 The Method of Characteristics for Quasilinear Partial Differential Equations 561
12.6.1 Method of Characteristics 561
12.6.2 Traffic Flow 562
12.6.3 Method of Characteristics （Q＝0） 564
12.6.4 Shock Waves 567
12.6.5 Quasilinear Example 579
12.7 First-Order Nonlinear Partial Differential Equations 585
12.7.1 Eikonal Equation Derived from the Wave Equation 585
12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves 586
12.7.3 First-Order Nonlinear Partial Differential Equations 589
13.1 Introduction 591
13 Laplace Transform Solution of Partial Differential Equations 591
13.2 Properties of the Laplace Transform 592
13.2.1 Introduction 592
13.2.2 Singularities of the Laplace Transform 592
13.2.3 Transforms of Derivatives 596
13.2.4 Convolution Theorem 597
13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations 601
13.4 A Signal Problem for the Wave Equation 603
13.5 A Signal Problem for a Vibrating String of Finite Length 606
13.6 The Wave Equation and its Green's Function 610
13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane 613
13.8 Solving the Wave Equation Using Laplace Transforms （with Complex Variables） 618
14 Dispersive Waves:Slow Variations,Stability,Nonlinearity,and Perturbation Methods 621
14.1 Introduction 621
14.2 Dispersive Waves and Group Velocity 622
14.2.1 Traveling Waves and the Dispersion Relation 622
14.2.2 Group Velocity Ⅰ 625
14.3 Wave Guides 628
14.3.1 Response to Concentrated Periodic Sources with Frequency ωf 630
14.3.2 Green's Function If Mode Propagates 631
14.3.3 Green's Function If Mode Does Not Propagate 632
14.3.4 Design Considerations 632
14.4 Fiber Optics 634
14.5 Group Velocity Ⅱ and the Method of Stationary Phase 638
14.5.1 Method of Stationary Phase 639
14.5.2 Application to Linear Dispersive Waves 641
14.6 Slowly Varying Dispersive Waves（Group Velocity and Caustics） 645
14.6.1 Approximate Solutions of Dispersive Partial Differential Equations 645
14.6.2 Formation of a Caustic 648
14.7 Wave Envelope Equations（Concentrated Wave Number） 654
14.7.1 Schr？dinger Equation 655
14.7.2 Linearized Korteweg-de Vries Equation 657
14.7.3 Nonlinear Dispersive Waves:Korteweg-deVries Equation 659
14.7.4 Solitons and Inverse Scattering 662
14.7.5 Nonlinear Schr？dinger Equation 664
14.8 Stability and Instability 669
14.8.1 Brief Ordinary Differential Equations and Bifurcation Theory 669
14.8.2 Elementary Example of a Stable Equilibrium for a Partial Differential Equation 676
14.8.3 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation 677
14.8.4 Ill posed Problems 679
14.8.5 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation 680
14.8.6 Nonlinear Complex Ginzburg-Landau Equation 682
14.8.7 Long Wave Instabilities 688
14.8.8 Pattern Formation for Reaction-Diffusion Equations and the Turing Instability 689
14.9 Singular Perturbation Methods:Multiple Scales 696
14.9.1 Ordinary Differential Equation:Weakly Nonlinearly Damped Oscillator 696
14.9.2 Ordinary Differential Equation:Slowly Varying Oscillator 699
14.9.3 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain 703
14.9.4 Slowly Varying Medium for the Wave Equation 705
14.9.5 Slowly Varying Linear Dispersive Waves（Including Weak Nonlinear Effects） 708
14.10 Singular Perturbation Methods:Boundary Layers Method of Matched Asymptotic Expansions 713
14.10.1 Boundary Layer in an Ordinary Differential Equation 713
14.10.2 Diffusion of a Pollutant Dominated by Convection 719
Bibliography 726
Answers to Starred Exercises 731
Index 751