دانلود کتاب Introduction to partial differential equations

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Preface......Page 8
Prerequisites......Page 9
Outline of Chapters......Page 11
Course Outlines and Chapter Dependencies......Page 14
Movies......Page 15
Conventions and Notation......Page 16
Historical Matters......Page 18
Acknowledgments......Page 19
Corrected Printing......Page 20
Table of Contents......Page 22
Chapter 1 What Are Partial Differential Equations?......Page 27
Classical Solutions......Page 30
Initial Conditions and Boundary Conditions......Page 32
Linear and Nonlinear Equations......Page 34
Exercises......Page 39
Chapter 2 Linear and Nonlinear Waves......Page 40
2.1 StationaryWaves......Page 41
Exercises......Page 43
Uniform Transport......Page 44
Transport with Decay......Page 47
Exercises......Page 48
Nonuniform Transport......Page 49
Exercises......Page 55
2.3 Nonlinear Transport and Shocks......Page 56
Shock Dynamics......Page 62
More General Wave Speeds......Page 71
Exercises......Page 72
2.4 TheWave Equation: d’Alembert’s Formula......Page 74
d’Alembert’s Solution......Page 75
External Forcing and Resonance......Page 81
Exercises......Page 85
Chapter 3 Fourier Series......Page 88
3.1 Eigensolutions of Linear Evolution Equations......Page 89
The Heated Ring......Page 94
3.2 Fourier Series......Page 97
Exercises......Page 101
Periodic Extensions......Page 102
Exercises......Page 103
Piecewise Continuous Functions......Page 104
Exercises......Page 106
The Convergence Theorem......Page 107
Exercises......Page 109
Even and Odd Functions......Page 110
Exercises......Page 112
Complex Fourier Series......Page 113
Exercises......Page 116
Integration of Fourier Series......Page 117
Differentiation of Fourier Series......Page 119
3.4 Change of Scale......Page 120
Exercises......Page 122
3.5 Convergence of Fourier Series......Page 123
Pointwise and Uniform Convergence......Page 124
Exercises......Page 128
Smoothness and Decay......Page 129
Hilbert Space......Page 131
Convergence in Norm......Page 134
Completeness......Page 137
Pointwise Convergence......Page 140
Exercises......Page 143
Chapter 4 Separation of Variables......Page 145
4.1 The Diffusion and Heat Equations......Page 146
The Heat Equation......Page 148
Smoothing and Long–Time Behavior......Page 150
The Heated Ring Redux......Page 154
Inhomogeneous Boundary Conditions......Page 157
Robin Boundary Conditions......Page 158
The Root Cellar Problem......Page 160
Exercises......Page 162
Separation of Variables and Fourier Series Solutions......Page 164
Exercises......Page 169
The d’Alembert Formula for Bounded Intervals......Page 170
Exercises......Page 173
4.3 The Planar Laplace and Poisson Equations......Page 176
Exercises......Page 178
Separation of Variables......Page 179
Exercises......Page 183
Polar Coordinates......Page 184
Averaging, the Maximum Principle, and Analyticity......Page 191
Exercises......Page 194
4.4 Classification of Linear Partial Differential Equations......Page 196
Exercises......Page 198
Characteristics and the Cauchy Problem......Page 199
Exercises......Page 203
Chapter 5 Finite Differences......Page 204
5.1 Finite Difference Approximations......Page 205
Exercises......Page 208
5.2 Numerical Algorithms for the Heat Equation......Page 209
Stability Analysis......Page 211
Implicit and Crank–Nicolson Methods......Page 213
Exercises......Page 216
5.3 Numerical Algorithms for First–Order Partial Differential Equations......Page 218
The CFL Condition......Page 219
Upwind and Lax–Wendroff Schemes......Page 221
Exercises......Page 223
5.4 Numerical Algorithms for theWave Equation......Page 224
Exercises......Page 229
5.5 Finite Difference Algorithms for the Laplace and Poisson Equations......Page 230
Solution Strategies......Page 234
Exercises......Page 237
Chapter 6 Generalized Functions and Green’s Functions......Page 238
6.1 Generalized Functions......Page 239
The Delta Function......Page 240
Calculus of Generalized Functions......Page 244
Exercises......Page 250
The Fourier Series of the Delta Function......Page 251
Exercises......Page 256
6.2 Green’s Functions for One–Dimensional Boundary Value Problems......Page 257
Exercises......Page 263
Calculus in the Plane......Page 265
The Two–Dimensional Delta Function......Page 269
The Green’s Function......Page 271
Exercises......Page 278
The Method of Images......Page 279
Exercises......Page 282
7.1 The Fourier Transform......Page 285
Exercises......Page 295
Differentiation......Page 297
Integration......Page 298
Exercises......Page 299
Solution of Boundary Value Problems......Page 300
Exercises......Page 302
Convolution......Page 303
Exercises......Page 305
7.4 The Fourier Transform on Hilbert Space......Page 306
Quantum Mechanics and the Uncertainty Principle......Page 308
Exercises......Page 310
Chapter 8 Linear and Nonlinear Evolution Equations......Page 312
8.1 The Fundamental Solution to the Heat Equation......Page 313
The Forced Heat Equation and Duhamel’s Principle......Page 317
The Black–Scholes Equation and Mathematical Finance......Page 320
Exercises......Page 324
8.2 Symmetry and Similarity......Page 326
Similarity Solutions......Page 329
Exercises......Page 331
8.3 The Maximum Principle......Page 333
Exercises......Page 335
Burgers’ Equation......Page 336
The Hopf–Cole Transformation......Page 338
8.5 Dispersion and Solitons......Page 344
Linear Dispersion......Page 345
The Dispersion Relation......Page 351
Exercises......Page 353
The Korteweg–deVries Equation......Page 354
Exercises......Page 358
Chapter 9 A General Framework for Linear Partial Differential Equations......Page 360
9.1 Adjoints......Page 361
Differential Operators......Page 363
Higher–Dimensional Operators......Page 366
Exercises......Page 369
The Fredholm Alternative......Page 371
Exercises......Page 373
9.2 Self–Adjoint and Positive Definite Linear Functions......Page 374
Self–Adjointness......Page 375
Positive Definiteness......Page 376
Two–Dimensional Boundary Value Problems......Page 380
Exercises......Page 381
9.3 Minimization Principles......Page 383
Sturm–Liouville Boundary Value Problems......Page 384
Exercises......Page 387
The Dirichlet Principle......Page 389
Exercises......Page 390
Self–Adjoint Operators......Page 392
The Rayleigh Quotient......Page 396
Eigenfunction Series......Page 399
Green’s Functions and Completeness......Page 400
Exercises......Page 405
9.5 A General Framework for Dynamics......Page 406
Evolution Equations......Page 407
Vibration Equations......Page 409
Forcing and Resonance......Page 410
Exercises......Page 414
The Schrödinger Equation......Page 415
Exercises......Page 417
Chapter 10 Finite Elements and Weak Solutions......Page 419
10.1 Minimization and Finite Elements......Page 420
Exercises......Page 422
10.2 Finite Elements for Ordinary Differential Equations......Page 423
Exercises......Page 429
10.3 Finite Elements in Two Dimensions......Page 430
Triangulation......Page 431
Exercises......Page 435
The Finite Element Equations......Page 436
Assembling the Elements......Page 438
The Coefficient Vector and the Boundary Conditions......Page 442
Inhomogeneous Boundary Conditions......Page 444
Exercises......Page 446
10.4 Weak Solutions......Page 447
Weak Formulations of Linear Systems......Page 448
Finite Elements Based on Weak Solutions......Page 450
Shock Waves as Weak Solutions......Page 451
Exercises......Page 454
11.1 Diffusion in Planar Media......Page 455
Derivation of the Diffusion and Heat Equations......Page 456
Separation of Variables......Page 459
Qualitative Properties......Page 460
Inhomogeneous Boundary Conditions and Forcing......Page 462
The Maximum Principle......Page 463
Exercises......Page 464
Heating of a Rectangle......Page 465
Exercises......Page 469
Heating of a Disk — Preliminaries......Page 470
11.3 Series Solutions of Ordinary Differential Equations......Page 472
The Gamma Function......Page 473
Regular Points......Page 475
The Airy Equation......Page 479
Exercises......Page 481
Regular Singular Points......Page 483
Bessel’s Equation......Page 486
Exercises......Page 492
11.4 The Heat Equation in a Disk, Continued......Page 494
Exercises......Page 499
11.5 The Fundamental Solution to the Planar Heat Equation......Page 501
Exercises......Page 505
11.6 The PlanarWave Equation......Page 506
Separation of Variables......Page 507
Vibration of a Rectangular Drum......Page 508
Vibration of a Circular Drum......Page 510
Exercises......Page 512
Scaling and Symmetry......Page 514
Exercises......Page 516
Chladni Figures and Nodal Curves......Page 517
Exercises......Page 520
Chapter 12 Partial Differential Equations in Space......Page 522
12.1 The Three–Dimensional Laplace and Poisson Equations......Page 523
Self–Adjoint Formulation and Minimum Principle......Page 524
Exercises......Page 525
12.2 Separation of Variables for the Laplace Equation......Page 526
Laplace’s Equation in a Ball......Page 527
The Legendre Equation and Ferrers Functions......Page 529
Spherical Harmonics......Page 536
Harmonic Polynomials......Page 538
Averaging, the Maximum Principle, and Analyticity......Page 540
Exercises......Page 543
12.3 Green’s Functions for the Poisson Equation......Page 546
The Free–Space Green’s Function......Page 547
Bounded Domains and the Method of Images......Page 550
Exercises......Page 553
12.4 The Heat Equation for Three–Dimensional Media......Page 554
Exercises......Page 555
Heating of a Ball......Page 556
Spherical Bessel Functions......Page 557
Exercises......Page 561
The Fundamental Solution to the Heat Equation in Space......Page 562
12.5 TheWave Equation for Three–Dimensional Media......Page 564
Vibration of Balls and Spheres......Page 566
Exercises......Page 569
Spherical Waves......Page 570
Kirchhoff’s Formula and Huygens’ Principle......Page 577
Exercises......Page 579
Descent to Two Dimensions......Page 580
Exercises......Page 582
12.7 The Hydrogen Atom......Page 583
Bound States......Page 584
Atomic Eigenstates and Quantum Numbers......Page 586
Exercises......Page 588
Appendix A Complex Numbers......Page 590
B.1 Vector Spaces and Subspaces......Page 594
B.2 Bases and Dimension......Page 595
B.3 Inner Products and Norms......Page 597
B.4 Orthogonality......Page 600
B.5 Eigenvalues and Eigenvectors......Page 601
B.6 Linear Iteration......Page 602
B.7 Linear Functions and Systems......Page 604
References......Page 608
Symbol Index......Page 614
Author Index......Page 622
Subject Index......Page 626