Applied Mathematics (eBook)

J. DAVID LOGAN, PHD, is Willa Cather Professor of Mathematics at the University of Nebraska, Lincoln. He is also the author of An Introduction to Nonlinear Partial Differential Equations, Second Edition and Mathematical Methods in Biology, both published by Wiley. Dr. Logan has served on the faculties at the University of Arizona, Kansas State University, and Rensselaer Polytechnic Institute, and he has been affiliated with Los Alamos Scientific Laboratory, Lawrence Livermore National Laboratory, and the Aerospace Research Laboratory.

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 Approximation 202

3.6 Asymptotic Expansion of Integrals 210

4. Calculus of Variations 221

4.1 Variational Problems 221

4.2 Necessary Conditions for Extrema 227

4.3 The Simplest Problem 236

4.4 Generalizations 245

4.5 Hamilton's Principle 253

4.6 Isoperimetric Problems 266

5. Eigenvalue Problems, Integral Equations, and Green's Functions 275

5.1 Boundary-Value Problems 277

5.2 Sturm--Liouville Problems 284

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 Expansions 404

6.5 Integral Transforms 415

6.6 Stability of Solutions 435

6.7 Distributions 443

7. Wave Phenomena 457

7.1 Waves 457

7.2 Nonlinear Waves 470

7.3 Quasi-linear Equations 488

7.4 The Wave Equation 497

8. Mathematical Models of Continua 523

8.1 Kinematics and Mass Conservation 524

8.2 Momentum and Energy 534

8.3 Gas Dynamics 551

8.4 Fluid Motions in R¯3 560

9. Discrete Models 585

9.1 One-Dimensional Models. 586

9.2 Systems of Difference Equations 599

9.3 Stochastic Models 619

9.4 Probability-Based Models 636

Index 653