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Preface xiii
1. Dimensional Analysis and One-Dimensional Dynamics 1
1.1 Dimensional Analysis 2
1.2 Scaling 30
1.3 Differential Equations 46
2. Two-Dimensional Dynamical Systems 77
2.1 Phase Plane Phenomena. 77
2.2 Linear Systems 87
2.3 Nonlinear Systems 94
2.4 Bifurcations 103
2.5 Reaction Kinetics112
2.6 Pathogens 126
3. Perturbation Methods and Asymptotic Expansions 149
3.1 Regular Perturbation 150
3.2 Singular Perturbation 170
3.3 Boundary Layer Analysis. 179
3.4 Initial Layers 191
3.5 The WKB Approximation202
3.6 Asymptotic Expansion of Integrals 210
4. Calculus of Variations221
4.1 Variational Problems221
4.2 Necessary Conditions for Extrema 227
4.3 The Simplest Problem 236
4.4 Generalizations 245
4.5 Hamilton’s Principle253
4.6 Isoperimetric Problems 266
5. Eigenvalue Problems, Integral Equations, and Green’s Functions 275
5.1 Boundary-Value Problems277
5.2 Sturm–Liouville Problems284
5.3 Classical Fourier Series 310
5.4 Integral Equations 317
5.5 Green’s Functions 339
5.6 Distributions 352
6. Partial Differential Equations 365
6.1 Basic Concepts 365
6.2 Conservation Laws 375
6.3 Equilibrium Equations 397
6.4 Eigenfunction Expansions404
6.5 Integral Transforms 415
6.7 Distributions 443
7. Wave Phenomena 457
7.1 Waves 457
7.2 Nonlinear Waves470
7.3 Quasi-linear Equations 488
7.4 The Wave Equation497
8. Mathematical Models of Continua 523
8.1 Kinematics and Mass Conservation524
8.2 Momentum and Energy 534
8.3 Gas Dynamics 551
8.4 Fluid Motions in R3560
9. Discrete Models 587
9.1 One-Dimensional Models. 588
9.2 Systems of Difference Equations601
9.3 Stochastic Models 621
9.4 Probability-Based Models639
Index 655
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