Applied Partial Differential Equations (4th Edition)

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Chinese Preface......Page 4
Contents......Page 6
1.1 Introduction......Page 15
1.2 Derivation of the Conduction of Heat in a One-Dimensional Rod......Page 16
1.3 Boundary Conditions......Page 26
1.4.1 Prescribed Temperature......Page 28
1.4.2 Insulated Boundaries......Page 30
1.5 Derivation of the Heat Equation in Two or Three Dimensions......Page 35
2.1 Introduction......Page 49
2.2 Linearity......Page 50
2.3.1 Introduction......Page 52
2.3.2 Separation of Variables......Page 53
2.3.3 Time-Dependent Equation......Page 55
2.3.4 Boundary Value Problem......Page 56
2.3.5 Product Solutions and the Principle of Superposition......Page 61
2.3.6 Orthogonality of Sines......Page 64
2.3.7 Formulation, Solution, and Interpretation of an Example......Page 65
2.3.8 Summary......Page 68
2.4.1 Heat Conduction in a Rod with Insulated Ends......Page 73
2.4.2 Heat Conduction in a Thin Circular Ring......Page 77
2.4.3 Summary of Boundary Value Problems......Page 82
2.5.1 Laplace's Equation Inside a Rectangle......Page 85
2.5.2 Laplace's Equation for a Circular Disk......Page 90
2.5.3 Fluid Flow Past a Circular Cylinder (Lift)......Page 94
2.5.4 Qualitative Properties of Laplace's Equation......Page 97
3.1 Introduction......Page 103
3.2 Statement of Convergence Theorem......Page 105
3.3.1 Fourier Sine Series......Page 110
3.3.2 Fourier Cosine Series......Page 120
3.3.3 Representing f (x) by Both a Sine and Cosine Series......Page 122
3.3.4 Even and Odd Parts......Page 123
3.3.5 Continuous Fourier Series......Page 125
3.4 Term-by-Term Differentiation of Fourier Series......Page 130
3.5 Term-By-Term Integration of Fourier Series......Page 141
3.6 Complex Form of Fourier Series......Page 145
4.2 Derivation of a Vertically Vibrating String......Page 149
4.3 Boundary Conditions......Page 153
4.4 Vibrating String with Fixed Ends......Page 156
4.5 Vibrating Membrane......Page 163
4.6 Reflection and Refraction of Electromagnetic (Light) and Acoustic (Sound) Waves......Page 165
4.6.1 Snell's Law of Refraction......Page 166
4.6.2 Intensity (Amplitude) of Reflected and Refracted Waves......Page 168
4.6.3 Total Internal Reflection......Page 169
5.1 Introduction......Page 171
5.2.1 Heat Flow in a Nonuniform Rod......Page 172
5.2.2 Circularly Symmetric Heat Flow......Page 173
5.3.1 General Classification......Page 175
5.3.2 Regular Sturm-Liouville Eigenvalue Problem......Page 176
5.3.3 Example and Illustration of Theorems......Page 178
5.4 Worked Example: Heat Flow in a Nonuniform Rod without Sources......Page 184
5.5 Self-Adjoint Operators and Sturm-Liouville Eigenvalue Problems......Page 188
5.6 Rayleigh Quotient......Page 203
5.7 Worked Example: Vibrations of a Nonuniform String......Page 209
5.8 Boundary Conditions of the Third Kind......Page 212
5.9 Large Eigenvalues (Asymptotic Behavior)......Page 226
5.10 Approximation Properties......Page 230
6.1 Introduction......Page 236
6.2 Finite Differences and Truncated Taylor Series......Page 237
6.3.2 A Partial Difference Equation......Page 243
6.3.3 Computations......Page 245
6.3.4 Fourier-von Neumann Stability Analysis......Page 249
6.3.5 Separation of Variables for Partial Difference Equations and Analytic Solutions of Ordinary Difference Equations......Page 255
6.3.6 Matrix Notation......Page 257
6.3.8 Other Numerical Schemes......Page 261
6.3.9 Other Types of Boundary Conditions......Page 262
6.4 Two-Dimensional Heat Equation......Page 267
6.5 Wave Equation......Page 270
6.6 Laplace's Equation......Page 274
6.7.1 Approximation with Nonorthogonal Functions (Weak Form of the Partial Differential Equation)......Page 281
6.7.2 The Simplest Triangular Finite Elements......Page 284
7.1 Introduction......Page 289
7.2.1 Vibrating Membrane: Any Shape......Page 290
7.2.2 Heat Conduction: Any Region......Page 292
7.2.3 Summary......Page 293
7.3 Vibrating Rectangular Membrane......Page 294
7.4 Statements and Illustrations of Theorems for the Eigenvalue Problem V20 + Aq = 0......Page 303
7.5 Green's Formula, Self-Adjoint Operators and Multidimensional Eigenvalue Problems......Page 309
7.6.1 Rayleigh Quotient......Page 314
7.6.2 Time-Dependent Heat Equation and Laplace's Equation......Page 315
7.7.2 Separation of Variables......Page 317
7.7.3 Eigenvalue Problems (One Dimensional)......Page 319
7.7.4 Bessel's Differential Equation......Page 320
7.7.5 Singular Points and Bessel's Differential Equation......Page 321
7.7.6 Bessel Functions and Their Asymptotic Properties (nearz=0)......Page 322
7.7.7 Eigenvalue Problem Involving Bessel Functions......Page 323
7.7.8 Initial Value Problem for a Vibrating Circular Membrane......Page 325
7.7.9 Circularly Symmetric Case......Page 327
7.8.1 Qualitative Properties of Bessel Functions......Page 332
7.8.2 Asymptotic Formulas for the Eigenvalues......Page 333
7.8.3 Zeros of Bessel Functions and Nodal Curves......Page 334
7.8.4 Series Representation of Bessel Functions......Page 336
7.9.2 Separation of Variables......Page 340
7.9.3 Zero Temperature on the Lateral Sides and on the Bottom or Top......Page 342
7.9.4 Zero Temperature on the Top and Bottom......Page 344
7.9.5 Modified Bessel Functions......Page 346
7.10.1 Introduction......Page 350
7.10.2 Separation of Variables and One-Dimensional Eigenvalue Problems......Page 351
7.10.3 Associated Legendre Functions and Legendre Polynomials......Page 352
7.10.4 Radial Eigenvalue Problems......Page 355
7.10.5 Product Solutions, Modes of Vibration, and the Initial Value Problem......Page 356
7.10.6 Laplace's Equation Inside a Spherical Cavity......Page 357
8.2 Heat Flow with Sources and Nonhomogeneous Boundary Conditions......Page 361
8.3 Method of Eigenfunction Expansion with Homogeneous Boundary Conditions (Differentiating Series of Eigenfunctions)......Page 368
8.4 Method of Eigenfunction Expansion Using Green's Formula (With or Without Homogeneous Boundary Conditions)......Page 373
8.5 Forced Vibrating Membranes and Resonance......Page 378
8.6 Poisson's Equation......Page 386
9.2 One-dimensional Heat Equation......Page 394
9.3.1 One-Dimensional Steady-State Heat Equation......Page 399
9.3.2 The Method of Variation of Parameters......Page 400
9.3.3 The Method of Eigenfunction Expansion for Green's Functions......Page 403
9.3.4 The Dirac Delta Function and Its Relationship to Green's Functions......Page 405
9.3.5 Nonhomogeneous Boundary Conditions......Page 411
9.3.6 Summary......Page 413
9.4.1 Introduction......Page 419
9.4.2 Fredholm Alternative......Page 421
9.4.3 Generalized Green's Functions......Page 423
9.5.1 Introduction......Page 430
9.5.2 Multidimensional Dirac Delta Function and Green's Functions......Page 431
9.5.3 Green's Functions by the Method of Eigenfunction Expansion and the Fredholm Alternative......Page 432
9.5.4 Direct Solution of Green's Functions (One-Dimensional Eigenfunctions)......Page 434
9.5.5 Using Green's Functions for Problems with Nonhomogeneous Boundary Conditions......Page 436
9.5.6 Infinite Space Green's Functions......Page 437
9.5.7 Green's Functions for Bounded Domains Using Infinite Space Green's Functions......Page 440
9.5.8 Green's Functions for a Semi-Infinite Plane (y > 0) Using Infinite Space Green's Functions: The Method of Images......Page 441
9.5.9 Green's Functions for a Circle: The Method of Images......Page 444
9.6.2 Mathematical Example......Page 452
9.6.3 Vibrating Nearly Circular Membrane......Page 454
9.7 Summary......Page 457
10.2 Heat Equation on an Infinite Domain......Page 459
10.3.1 Motivation from Fourier Series Identity......Page 463
10.3.2 Fourier Transform......Page 464
10.3.3 Inverse Fourier Transform of a Gaussian......Page 465
10.4.1 Heat Equation......Page 473
10.4.2 Fourier Transforming the Heat Equation: Transforms of Derivatives......Page 478
10.4.3 Convolution Theorem......Page 480
10.4.4 Summary of Properties of the Fourier Transform......Page 483
10.5.2 Heat Equation on a Semi-Infinite Interval I......Page 485
10.5.3 Fourier Sine and Cosine Transforms......Page 487
10.5.4 Transforms of Derivatives......Page 488
10.5.5 Heat Equation on a Semi-Infinite Interval II......Page 490
10.5.6 Tables of Fourier Sine and Cosine Transforms......Page 493
10.6.1 One-Dimensional Wave Equation on an Infinite Interval......Page 496
10.6.2 Laplace's Equation in a Semi-Infinite Strip......Page 498
10.6.3 Laplace's Equation in a Half-Plane......Page 501
10.6.4 Laplace's Equation in a Quarter-Plane......Page 505
10.6.5 Heat Equation in a Plane (Two- Dimensional Fourier Transforms)......Page 508
10.6.6 Table of Double-Fourier Transforms......Page 512
10.7 Scattering and Inverse Scattering......Page 517
11.2.1 Introduction......Page 522
11.2.2 Green's Formula......Page 524
11.2.3 Reciprocity......Page 525
11.2.4 Using the Green's Function......Page 527
11.2.6 Alternate Differential Equation for the Green's Function......Page 529
11.2.7 Infinite Space Green's Function for the One-Dimensional Wave Equation and d'Alembert's Solution......Page 530
11.2.8 Infinite Space Green's Function for the Three- Dimensional Wave Equation (Huygens' Principle)......Page 532
11.2.10 Summary......Page 534
11.3.1 Introduction......Page 537
11.3.2 Non-Self-Adjoint Nature of the Heat Equation......Page 538
11.3.3 Green's Formula......Page 539
11.3.5 Reciprocity......Page 541
11.3.6 Representation of the Solution Using Green's Functions......Page 542
11.3.8 Infinite Space Green's Function for the Diffusion Equation......Page 544
11.3.9 Green's Function for the Heat Equation (Semi-Infinite Domain)......Page 546
11.3.10 Green's Function for the Heat Equation (on a Finite Region)......Page 547
12.1 Introduction......Page 550
12.2.1 Introduction......Page 551
12.2.2 Method of Characteristics for First-Order Partial Differential Equations......Page 552
12.3.1 General Solution......Page 557
12.3.2 Initial Value Problem (Infinite Domain)......Page 559
12.3.3 D'alembert's Solution......Page 563
12.4 Semi-Infinite Strings and Reflections......Page 566
12.5 Method of Characteristics for a Vibrating String of Fixed Length......Page 571
12.6.1 Method of Characteristics......Page 575
12.6.2 Traffic Flow......Page 576
12.6.3 Method of Characteristics (Q = 0)......Page 578
12.6.4 Shock Waves......Page 581
12.6.5 Quasilinear Example......Page 593
12.7.1 Eikonal Equation Derived from the Wave Equation......Page 599
12.7.2 Solving the Eikonal Equation in Uniform Media and Reflected Waves......Page 600
12.7.3 First-Order Nonlinear Partial Differential Equations......Page 603
13.1 Introduction......Page 605
13.2.2 Singularities of the Laplace Transform......Page 606
13.2.3 Transforms of Derivatives......Page 610
13.2.4 Convolution Theorem......Page 611
13.3 Green's Functions for Initial Value Problems for Ordinary Differential Equations......Page 615
13.4 A Signal Problem for the Wave Equation......Page 617
13.5 A Signal Problem for a Vibrating String of Finite Length......Page 620
13.6 The Wave Equation and its Green's Function......Page 624
13.7 Inversion of Laplace Transforms Using Contour Integrals in the Complex Plane......Page 627
13.8 Solving the Wave Equation Using Laplace Transforms (with Complex Variables)......Page 632
14.1 Introduction......Page 635
14.2.1 `Raveling Waves and the Dispersion Relation......Page 636
14.2.2 Group Velocity I......Page 639
14.3 Wave Guides......Page 642
14.3.1 Response to Concentrated Periodic Sources with Frequency w f......Page 644
14.3.2 Green's Function If Mode Propagates......Page 645
14.3.4 Design Considerations......Page 646
14.4 Fiber Optics......Page 648
14.5 Group Velocity II and the Method of Stationary Phase......Page 652
14.5.1 Method of Stationary Phase......Page 653
14.5.2 Application to Linear Dispersive Waves......Page 655
14.6.1 Approximate Solutions of Dispersive Partial Differential Equations......Page 659
14.6.2 Formation of a Caustic......Page 662
14.7 Wave Envelope Equations (Concentrated Wave Number)......Page 668
14.7.1 Schrodinger Equation......Page 669
14.7.2 Linearized Korteweg-de Vries Equation......Page 671
14.7.3 Nonlinear Dispersive Waves: Korteweg-deVries Equation......Page 673
14.7.4 Solitons and Inverse Scattering......Page 676
14.7.5 Nonlinear Schrodinger Equation......Page 678
14.8.1 Brief Ordinary Differential Equations and Bifurcation Theory......Page 683
14.8.2 Elementary Example of a Stable Equilibrium for a Partial Differential Equation......Page 690
14.8.3 Typical Unstable Equilibrium for a Partial Differential Equation and Pattern Formation......Page 691
14.8.4 Ill posed Problems......Page 693
14.8.5 Slightly Unstable Dispersive Waves and the Linearized Complex Ginzburg-Landau Equation......Page 694
14.8.6 Nonlinear Complex Ginzburg-Landau Equation......Page 696
14.8.7 Long Wave Instabilities......Page 702
14.8.8 Pattern Formation for Reaction-Diffusion Equations and the Turing Instability......Page 703
14.9.1 Ordinary Differential Equation: Weakly Nonlinearly Damped Oscillator......Page 710
14.9.2 Ordinary Differential Equation: Slowly Varying Oscillator......Page 713
14.9.3 Slightly Unstable Partial Differential Equation on Fixed Spatial Domain......Page 717
14.9.4 Slowly Varying Medium for the Wave Equation......Page 719
14.9.5 Slowly Varying Linear Dispersive. Waves (Including Weak Nonlinear Effects)......Page 722
14.10.1 Boundary Laver in an Ordinary Differential Equation......Page 727
14.10.2 Diffusion of a Pollutant Dominated by Convection......Page 733
Bibliography......Page 740
Answers to Starred Exercises......Page 745
Index......Page 765