Applied Mathematics

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 xiii1. Dimensional Analysis and One-Dimensional Dynamics 1 1.1 Dimensional Analysis 21.2 Scaling 301.3 Differential Equations 462. Two-Dimensional Dynamical Systems 77 2.1 Phase Plane Phenomena 772.2 Linear Systems 872.3 Nonlinear Systems 942.4 Bifurcations 1032.5 Reaction Kinetics1122.6 Pathogens 1263. Perturbation Methods and Asymptotic Expansions 1493.1 Regular Perturbation 1503.2 Singular Perturbation 1703.3 Boundary Layer Analysis 1793.4 Initial Layers 1913.5 The WKB Approximation 2023.6 Asymptotic Expansion of Integrals 2104. Calculus of Variations 221 4.1 Variational Problems 2214.2 Necessary Conditions for Extrema 2274.3 The Simplest Problem 2364.4 Generalizations 2454.5 Hamilton's Principle 2534.6 Isoperimetric Problems 2665. Eigenvalue Problems, Integral Equations, and Green's Functions 275 5.1 Boundary-Value Problems 2775.2 Sturm--Liouville Problems 2845.3 Classical Fourier Series 3105.4 Integral Equations 3175.5 Green's Functions 3395.6 Distributions 3526. Partial Differential Equations 365 6.1 Basic Concepts 3656.2 Conservation Laws 3756.3 Equilibrium Equations 3976.4 Eigenfunction Expansions 4046.5 Integral Transforms 4156.6 Stability of Solutions 4356.7 Distributions 4437. Wave Phenomena 457 7.1 Waves 4577.2 Nonlinear Waves 4707.3 Quasi-linear Equations 4887.4 The Wave Equation 4978. Mathematical Models of Continua 523 8.1 Kinematics and Mass Conservation 5248.2 Momentum and Energy 5348.3 Gas Dynamics 5518.4 Fluid Motions in R3 5609. Discrete Models 5859.1 One-Dimensional Models. 5869.2 Systems of Difference Equations 5999.3 Stochastic Models 6199.4 Probability-Based Models 636Index 653