Applied Mathematics, Third Edition

Contents......Page 6
Preface......Page 12
1. Dimensional Analysis, Scaling, and Differential Equations  ......Page 14
1.1.1 The Program of Applied Mathematics  ......Page 15
1.1.2 Dimensional Methods ......Page 18
1.1.3 The Pi Theorem ......Page 21
1.1.4 Proof of the Pi Theorem  ......Page 26
1.2.1 Characteristic Scales  ......Page 32
1.2.2 A Chemical Reactor Problem ......Page 35
1.2.3 The Projectile Problem ......Page 38
1.3 Differential Equations ......Page 48
1.3.1 Review of Elementary Methods  ......Page 49
1.3.2 Stability and Bifurcation ......Page 57
1.4.1 Phase Plane Phenomena  ......Page 67
1.4.2 Linear Systems ......Page 76
1.4.3 Nonlinear Systems  ......Page 81
1.4.4 Bifurcation  ......Page 89
2. Perturbation Methods  ......Page 98
2.1 Regular Perturbation ......Page 100
2.1.1 Motion in a Resistive Medium ......Page 101
2.1.2 Nonlinear Oscillations ......Page 103
2.1.3 The Poincare-Lindstedt Method ......Page 106
2.1.4 Asymptotic Analysis  ......Page 108
2.2.1 Algebraic Equations  ......Page 117
2.2.2 Differential Equations ......Page 120
2.2.3 Boundary Layers ......Page 121
2.3.1 Inner and Outer Approximations ......Page 125
2.3.2 Matching  ......Page 127
2.3.3 Uniform Approximations  ......Page 129
2.3.4 General Procedures ......Page 132
2.4.1 Damped Spring-Mass System  ......Page 136
2.4.2 Chemical Reaction Kinetics  ......Page 140
2.5 The WKB Approximation ......Page 148
2.5.1 The Non-oscillatory Case  ......Page 150
2.5.2 The Oscillatory Case ......Page 151
2.6.1 Laplace Integrals ......Page 155
2.6.2 Integration by Parts  ......Page 159
2.6.3 Other Integrals ......Page 160
3.1.1 Functionals  ......Page 166
3.1.2 Examples ......Page 168
3.2.1 Normed Linear Spaces  ......Page 172
3.2.2 Derivatives of Functionals ......Page 176
3.2.3 Necessary Conditions ......Page 178
3.3.1 The Euler Equation  ......Page 181
3.3.2 Solved Examples ......Page 184
3.3.3 First Integrals ......Page 185
3.4.1 Higher Derivatives  ......Page 190
3.4.2 Several Functions ......Page 192
3.4.3 Natural Boundary Conditions  ......Page 194
3.5.1 Hamilton's Principle  ......Page 198
3.5.2 Hamilton's Equations ......Page 204
3.5.3 The Inverse Problem  ......Page 207
3.6 Isoperimetric Problems  ......Page 212
4.1.1 Orthogonality  ......Page 220
4.1.2 Classical Fourier Series  ......Page 229
4.2 Sturm-Liouville Problems ......Page 233
4.3.1 Introduction ......Page 239
4.3.2 Volterra Equations ......Page 243
4.3.3 Fredholm Equations with Degenerate Kernels  ......Page 249
4.3.4 Symmetric Kernels ......Page 252
4.4.1 Inverses of Differential Operators ......Page 260
4.4.2 Physical Interpretation  ......Page 263
4.4.3 Green's Function via Eigenfunctions  ......Page 268
4.5.1 Test Functions ......Page 271
4.5.2 Distributions  ......Page 274
4.5.3 Distribution Solutions to Differential Equations  ......Page 278
5. Discrete Models  ......Page 284
5.1.1 Linear and Nonlinear Models  ......Page 285
5.1.2 Equilibria, Stability, and Chaos  ......Page 290
5.2.1 Linear Models ......Page 298
5.2.2 Nonlinear Interactions  ......Page 309
5.3.1 Elementary Probability ......Page 316
5.3.2 Stochastic Processes  ......Page 323
5.3.3 Environmental and Demographic Models  ......Page 327
5.4.1 Markov Processes  ......Page 334
5.4.2 Random Walks ......Page 340
5.4.3 The Poisson Process  ......Page 344
6.1 Basic Concepts ......Page 350
6.1.1 Linearity and Superposition ......Page 354
6.2 Conservation Laws ......Page 359
6.2.1 One Dimension  ......Page 360
6.2.2 Several Dimensions ......Page 362
6.2.3 Constitutive Relations  ......Page 367
6.2.4 Probability and Diffusion  ......Page 371
6.2.5 Boundary Conditions ......Page 374
6.3.1 Laplace's Equation ......Page 380
6.3.2 Basic Properties  ......Page 383
6.4.1 Spectrum of the Laplacian ......Page 387
6.4.2 Evolution Problems ......Page 390
6.5.1 Laplace Transforms ......Page 396
6.5.2 Fourier Transforms ......Page 400
6.6.1 Reaction-Diffusion Equations  ......Page 411
6.6.2 Pattern Formation  ......Page 413
6.7.1 Elliptic Problems ......Page 419
6.7.2 Tempered Distributions ......Page 424
6.7.3 Diffusion Problems ......Page 425
7.1.1 Waves  ......Page 432
7.1.2 The Advection Equation  ......Page 438
7.2.1 Nonlinear Advection  ......Page 443
7.2.2 Traveling Wave Solutions  ......Page 448
7.2.3 Conservation Laws ......Page 453
7.3 Quasi-linear Equations  ......Page 458
7.3.1 Age-Structured Populations  ......Page 462
7.4.1 The Acoustic Approximation  ......Page 467
7.4.2 Solutions to the Wave Equation  ......Page 471
7.4.3 Scattering and Inverse Problems ......Page 476
8. Mathematical Models of Continua  ......Page 484
8.1 Kinematics  ......Page 485
8.1.1 Mass Conservation ......Page 490
8.1.2 Momentum Conservation  ......Page 491
8.1.3 Thermodynamics and Energy Conservation  ......Page 495
8.1.4 Stress Waves in Solids  ......Page 500
8.2.1 Riemann's Method ......Page 506
8.2.2 The Rankine-Hugoniot Conditions ......Page 512
8.3.1 Kinematics  ......Page 515
8.3.2 Dynamics ......Page 521
8.3.3 Energy  ......Page 528
Index ......Page 538