An Introduction to Nonlinear Partial Differential Equations [NOOK Book]

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, ...

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An Introduction to Nonlinear Partial Differential Equations

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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, 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

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Editorial Reviews

From the Publisher
"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)
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Product Details

Meet the Author

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.

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Table of Contents

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

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