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Now in a Second Edition, this popular book on nonlinear partial differential equations (PDEs) contains expanded coverage on the central topics of applied mathematics in an elementary, highly readable format and is accessible to students and researchers in the field of pure and applied mathematics. This book provides a new focus on the increasing use of mathematical applications in the life sciences, while also addressing key topics such as linear PDEs, first-order nonlinear PDEs, classical and weak solutions, shocks, hyperbolic systems, nonlinear diffusion, and elliptic equations. Unlike comparable books that typically only use formal proofs and theory to demonstrate results, An Introduction to Nonlinear Partial Differential Equations, Second Edition takes a more practical approach to nonlinear PDEs by emphasizing how the results are used, why they are important, and how they are applied to real problems.
The intertwining relationship between mathematics and physical phenomena is discovered using detailed examples of applications across various areas such as biology, combustion, traffic flow, heat transfer, fluid mechanics, quantum mechanics, and the chemical reactor theory. New features of the Second Edition also include: Additional intermediate-level exercises that facilitate the development of advanced problem-solving skills, New applications in the biological sciences, including age-structure, pattern formation, and the propagation of diseases, An expanded bibliography that facilitates further investigation into specialized topics.
With individual, self-contained chapters and a broad scope of coverage that offers instructors the flexibility to design courses to meetspecific objectives, An Introduction to Nonlinear Partial Differential Equations, Second Edition 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.
About the Author:
J. David Logan, PhD, is Willa Cather Professor of Mathematics at the University of Nebraska-Lincoln
"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)
Preface xi
Introduction to Partial Differential Equations 1
Partial Differential Equations 2
Equations and Solutions 2
Classification 5
Linear versus Nonlinear 8
Linear Equations 11
Conservation Laws 20
One Dimension 20
Higher Dimensions 23
Constitutive Relations 25
Initial and Boundary Value Problems 35
Waves 45
Traveling Waves 45
Plane Waves 50
Plane Waves and Transforms 52
Nonlinear Dispersion 54
First-Order Equations and Characteristics 61
Linear First-Order Equations 62
Advection Equation 62
Variable Coefficients 64
Nonlinear Equations 68
Quasilinear Equations 72
The General Solution 76
Propagation of Singularities 81
General First-Order Equation 86
Complete Integral 91
A Uniqueness Result 94
Models in Biology 96
Age Structure 96
StructuredPredator-Prey Model 101
Chemotherapy 103
Mass Structure 105
Size-Dependent Predation 106
Weak Solutions to Hyperbolic Equations 113
Discontinuous Solutions 114
Jump Conditions 116
Rarefaction Waves 118
Shock Propagation 119
Shock Formation 125
Applications 131
Traffic Flow 132
Plug Flow Chemical Reactors 136
Weak Solutions: A Formal Approach 140
Asymptotic Behavior of Shocks 148
Equal-Area Principle 148
Shock Fitting 152
Asymptotic Behavior 154
Hyperbolic Systems 159
Shallow-Water Waves; Gas Dynamics 160
Shallow-Water Waves 160
Small-Amplitude Approximation 163
Gas Dynamics 164
Hyperbolic Systems and Characteristics 169
Classification 170
The Riemann Method 179
Jump Conditions for Systems 179
Breaking Dam Problem 181
Receding Wall Problem 183
Formation of a Bore 187
Gas Dynamics 190
Hodographs and Wavefronts 192
Hodograph Transformation 192
Wavefront Expansions 193
Weakly Nonlinear Approximations 201
Derivation of Burgers' Equation 202
Diffusion Processes 209
Diffusion and Random Motion 210
Similarity Methods 217
Nonlinear Diffusion Models 224
Reaction-Diffusion; Fisher's Equation 234
Traveling Wave Solutions 235
Perturbation Solution 238
Stability of Traveling Waves 240
Nagumo's Equation 242
Advection-Diffusion; Burgers' Equation 245
Traveling Wave Solution 246
Initial Value Problem 247
Asymptotic Solution to Burgers' Equation 250
Evolution of a Point Source 252
Dynamical Systems 257
Reaction-Diffusion Systems 267
Reaction-Diffusion Models 268
Predator-Prey Model 270
Combustion 271
Chemotaxis 274
Traveling Wave Solutions 277
Model for the Spread of a Disease 278
Contaminant Transport in Groundwater 284
Existence of Solutions 292
Fixed-Point Iteration 293
Semilinear Equations 297
Normed Linear Spaces 300
General Existence Theorem 303
Maximum Principles and Comparison Theorems 309
Maximum Principles 309
Comparison Theorems 314
Energy Estimates and Asymptotic Behavior 317
Calculus Inequalities 318
Energy Estimates 320
Invariant Sets 326
Pattern Formation 333
Equilibrium Models 345
Elliptic Models 346
Theoretical Results 352
Maximum Principle 353
Existence Theorem 355
Eigenvalue Problems 358
Linear Eigenvalue Problems 358
Nonlinear Eigenvalue Problems 361
Stability and Bifurcation 364
Ordinary Differential Equations 364
Partial Differential Equations 368
References 387
Index 395
Overview
Now in a Second Edition, this popular book on nonlinear partial differential equations (PDEs) contains expanded coverage on the central topics of applied mathematics in an elementary, highly readable format and is accessible to students and researchers in the field of pure and applied mathematics. This book provides a new focus on the increasing use of mathematical applications in the life sciences, while also addressing key topics such as linear PDEs, first-order nonlinear PDEs, classical and weak solutions, ...