An Introduction to Nonlinear Partial Differential Equations

by Logan, J. David

• ISBN13: 9780470225950
• ISBN10: 0470225955
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-04-11
• Publisher: Wiley-Interscience

Summary

This introductory textbook on nonlinear partial differential equations is technique oriented with an emphasis on applications and is designed to build a foundation for studying advanced treatises in the field. The Second Edition features an updated bibliography as well as an increase in the number of exercises. All software references have been updated with the latest version of MATLAB, the the corresponding graphics have also been updated using MATLAB. An increased focus on hydrogeology and mathematical biology is evident in the new edition. Hints and solutions to selected exercises have been added to the back of the book. The book emphasizes hyperbolic and parabolic problems and includes a range of applications in the following areas: biology, chemistry, porous media, biological problems, combustion and detonation, traffic flow, water waves, plug flow reactors, and heat transfer. Early chapters offer insight into how to understand problems involving phenomena, how specific equations describe evolutionary processes, and what the terms in such equations describe physically. When discussing wave propagination and hyperbolic problems, the text develops algorithms to solve first-order equations and highlights the concept of the weak solution. The material in the book is presented in such a manner that many of the chapters are independent, which allows for instructors flexibility to design several courses around various topics.

Author Biography

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.

Table of Contents

Preface	p. xi
Introduction to Partial Differential Equations	p. 1
Partial Differential Equations	p. 2
Equations and Solutions	p. 2
Classification	p. 5
Linear versus Nonlinear	p. 8
Linear Equations	p. 11
Conservation Laws	p. 20
One Dimension	p. 20
Higher Dimensions	p. 23
Constitutive Relations	p. 25
Initial and Boundary Value Problems	p. 35
Waves	p. 45
Traveling Waves	p. 45
Plane Waves	p. 50
Plane Waves and Transforms	p. 52
Nonlinear Dispersion	p. 54
First-Order Equations and Characteristics	p. 61
Linear First-Order Equations	p. 62
Advection Equation	p. 62
Variable Coefficients	p. 64
Nonlinear Equations	p. 68
Quasilinear Equations	p. 72
The General Solution	p. 76
Propagation of Singularities	p. 81
General First-Order Equation	p. 86
Complete Integral	p. 91
A Uniqueness Result	p. 94
Models in Biology	p. 96
Age Structure	p. 96
Structured Predator-Prey Model	p. 101
Chemotherapy	p. 103
Mass Structure	p. 105
Size-Dependent Predation	p. 106
Weak Solutions to Hyperbolic Equations	p. 113
Discontinuous Solutions	p. 114
Jump Conditions	p. 116
Rarefaction Waves	p. 118
Shock Propagation	p. 119
Shock Formation	p. 125
Applications	p. 131
Traffic Flow	p. 132
Plug Flow Chemical Reactors	p. 136
Weak Solutions: A Formal Approach	p. 140
Asymptotic Behavior of Shocks	p. 148
Equal-Area Principle	p. 148
Shock Fitting	p. 152
Asymptotic Behavior	p. 154
Hyperbolic Systems	p. 159
Shallow-Water Waves; Gas Dynamics	p. 160
Shallow-Water Waves	p. 160
Small-Amplitude Approximation	p. 163
Gas Dynamics	p. 164
Hyperbolic Systems and Characteristics	p. 169
Classification	p. 170
The Riemann Method	p. 179
Jump Conditions for Systems	p. 179
Breaking Dam Problem	p. 181
Receding Wall Problem	p. 183
Formation of a Bore	p. 187
Gas Dynamics	p. 190
Hodographs and Wavefronts	p. 192
Hodograph Transformation	p. 192
Wavefront Expansions	p. 193
Weakly Nonlinear Approximations	p. 201
Derivation of Burgers' Equation	p. 202
Diffusion Processes	p. 209
Diffusion and Random Motion	p. 210
Similarity Methods	p. 217
Nonlinear Diffusion Models	p. 224
Reaction-Diffusion; Fisher's Equation	p. 234
Traveling Wave Solutions	p. 235
Perturbation Solution	p. 238
Stability of Traveling Waves	p. 240
Nagumo's Equation	p. 242
Advection-Diffusion; Burgers' Equation	p. 245
Traveling Wave Solution	p. 246
Initial Value Problem	p. 247
Asymptotic Solution to Burgers' Equation	p. 250
Evolution of a Point Source	p. 252
Dynamical Systems	p. 257
Reaction-Diffusion Systems	p. 267
Reaction-Diffusion Models	p. 268
Predator-Prey Model	p. 270
Combustion	p. 271
Chemotaxis	p. 274
Traveling Wave Solutions	p. 277
Model for the Spread of a Disease	p. 278
Contaminant Transport in Groundwater	p. 284
Existence of Solutions	p. 292
Fixed-Point Iteration	p. 293
Semilinear Equations	p. 297
Normed Linear Spaces	p. 300
General Existence Theorem	p. 303
Maximum Principles and Comparison Theorems	p. 309
Maximum Principles	p. 309
Comparison Theorems	p. 314
Energy Estimates and Asymptotic Behavior	p. 317
Calculus Inequalities	p. 318
Energy Estimates	p. 320
Invariant Sets	p. 326
Pattern Formation	p. 333
Equilibrium Models	p. 345
Elliptic Models	p. 346
Theoretical Results	p. 352
Maximum Principle	p. 353
Existence Theorem	p. 355
Eigenvalue Problems	p. 358
Linear Eigenvalue Problems	p. 358
Nonlinear Eigenvalue Problems	p. 361
Stability and Bifurcation	p. 364
Ordinary Differential Equations	p. 364
Partial Differential Equations	p. 368
References	p. 387
Index	p. 395
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