Introduction to Nonlinear Partial Differential Equations

J. David Logan, PhD, is Willa Cather Professor of Mathematics at the University of Nebraska–Lincoln. He has authored several texts on elementary differential equations and beginning partial differential equations, including Applied Mathematics, Third Edition, also published by Wiley. Dr. Logan's research interests include mathematical physics, combustion and detonation, hydrogeology, and mathematical biology.

• Preface xi1. Introduction to Partial Differential Equations 11.1 Partial Differential Equations 21.1.1 Equations and Solutions 21.1.2 Classification 51.1.3 Linear versus Nonlinear 81.1.4 Linear Equations 111.2 Conservation Laws 201.2.1 One Dimension 201.2.2 Higher Dimensions 231.3 Constitutive Relations 251.4 Initial and Boundary Value Problems 351.5 Waves 451.5.1 Traveling Waves 451.5.2 Plane Waves 501.5.3 Plane Waves and Transforms 521.5.4 Nonlinear Dispersion 542. First-Order Equations and Characteristics 612.1 Linear First-Order Equations 622.1.1 Advection Equation 622.1.2 Variable Coefficients 642.2 Nonlinear Equations 682.3 Quasilinear Equations 722.3.1 The General Solution 762.4 Propagation of Singularities 812.5 General First-Order Equation 862.5.1 Complete Integral 912.6 A Uniqueness Result 942.7 Models in Biology 962.7.1 Age Structure 962.7.2 Structured Predator-Prey Model 1012.7.3 Chemotherapy 1032.7.4 Mass Structure 1052.7.5 Size-Dependent Predation 1063. Weak Solutions to Hyperbolic Equations 1133.1 Discontinuous Solutions 1143.2 Jump Conditions 1163.2.1 Rarefaction Waves 1183.2.2 Shock Propagation 1193.3 Shock Formation 1253.4 Applications 1313.4.1 Traffic Flow 1323.4.2 Plug Flow Chemical Reactors 1363.5 Weak Solutions: A Formal Approach 1403.6 Asymptotic Behavior of Shocks 1483.6.1 Equal-Area Principle 1483.6.2 Shock Fitting 1523.6.3 Asymptotic Behavior 1544. Hyperbolic Systems 1594.1 Shallow-Water Waves: Gas Dynamics 1604.1.1 Shallow-Water Waves1604.1.2 Small-Amplitude Approximation 1634.1.3 Gas Dynamics 1644.2 Hyperbolic Systems and Characteristics 1694.2.1 Classification 1704.3 The Riemann Method 1794.3.1 Jump Conditions for Systems 1794.3.2 Breaking Dam Problem 1814.3.3 Receding Wall Problem 1834.3.4 Formation of a Bore 1874.3.5 Gas Dynamics 1904.4 Hodographs and Wavefronts 1924.4.1 Hodograph Transformation 1924.4.2 Wavefront Expansions 1934.5 Weakly Nonlinear Approximations 2014.5.1 Derivation of Burgers‘ Equation 2025. Diffusion Processes 2095.1 Diffusion and Random Motion 2105.2 Similarity Methods 2175.3 Nonlinear Diffusion Models 2245.4 Reaction-Diffusion: Fisher’s Equation 2345.4.1 Traveling Wave Solutions 2355.4.2 Perturbation Solution 2385.4.3 Stability of Traveling Waves 2405.4.4 Nagumo‘s Equation 2425.5 Advection-Diffusion: Burgers’ Equation 2455.5.1 Traveling Wave Solution 2465.5.2 Initial Value Problem 2475.6 Asymptotic Solution to Burgers’ Equation 2505.6.1 Evolution of a Point Source 252Appendix: Dynamical Systems 2576. Reaction-Diffusion Systems 2676.1 Reaction-Diffusion Models 2686.1.1 Predator-Prey Model 2706.1.2 Combustion 2716.1.3 Chemotaxis 2746.2 Waveling IJ1Bve Solutions 2776.2.1 Model for the Spread of a Disease 2786.2.2 Contaminant Transport in Groundwater 2846.3 Existence of Solutions 2926.3.1 Fixed-Point Iteration 2936.3.2 Semilinear Equations 2976.3.3 Normed Linear Spaces 3006.3.4 General Existence Theorem 3036.4 Maximum Principles and Comparison Theorems 3096.4.1 Maximum Principles 3096.4.2 Comparison Theorems 3146.5 Energy Estimates and Asymptotic Behavior 3176.5.1 Calculus Inequalities 3186.5.2 Energy Estimates 3206.5.3 Invariant Sets 3266.6 Pattern Formation 3337. Equilibrium Models 3457.1 Elliptic Models 3467.2 Theoretical Results 3527.2.1 Maximum Principle 3537.2.2 Existence Theorem 3557.3 Eigenvalue Problems 3587.3.1 Linear Eigenvalue Problems 3587.3.2 Nonlinear Eigenvalue Problems 3617.4 Stability and Bifurcation 3647.4.1 Ordinary Differential Equations 3647.4.2 Partial Differential Equations 368References 387Index 395

"This book is an ideal text for applied mathematics courses at the upper-undergraduate and graduate levels. It also serves as a valuable resource for researchers and professionals in the fields of mathematics, biology, engineering, and physics who would like to further their knowledge of PDEs." (Mathematical Reviews, 2009c)