Applied Mathematics

Praise for the Third Edition

“Future mathematicians, scientists, and engineers should find the book to be an excellent introductory text for coursework or self-study as well as worth its shelf space for reference.”
—MAA Reviews

Applied Mathematics, Fourth Edition is a thoroughly updated and revised edition on the applications of modeling and analyzing natural, social, and technological processes. The book covers a wide range of key topics in mathematical methods and modeling and highlights the connections between mathematics and the applied and natural sciences.

The Fourth Edition covers both standard and modern topics, including scaling and dimensional analysis; regular and singular perturbation; calculus of variations; Green’s functions and integral equations; nonlinear wave propagation; and stability and bifurcation. The book provides extended coverage of mathematical biology, including biochemical kinetics, epidemiology, viral dynamics, and parasitic disease. In addition, the new edition features:

• Expanded coverage on orthogonality, boundary value problems, and distributions, all of which are motivated by solvability and eigenvalue problems in elementary linear algebra
• Additional MATLAB® applications for computer algebra system calculations
• Over 300 exercises and 100 illustrations that demonstrate important concepts
• New examples of dimensional analysis and scaling along with new tables of dimensions and units for easy reference
• Review material, theory, and examples of ordinary differential equations
• New material on applications to quantum mechanics, chemical kinetics, and modeling diseases and viruses

Written at an accessible level for readers in a wide range of scientific fields, Applied Mathematics, Fourth Edition is an ideal text for introducing modern and advanced techniques of applied mathematics to upper-undergraduate and graduate-level students in mathematics, science, and engineering. The book is also a valuable reference for engineers and scientists in government and industry.

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 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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