An Introduction to Nonlinear Partial Differential Equations, 2nd Edition

by J. David Logan (Univ. of Nebraska, Lincoln, Nebraska )

• ISBN13: 9780470287088
• ISBN10: 047028708X
• Format: eBook
• Copyright: 2008-06-01
• Publisher: Wiley-Interscience

Summary

An Introduction to Nonlinear Partial Differential Equations is a textbook on nonlinear partial differential equations. It 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 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.

Table of Contents

Preface

Partial Differential Equations

Partial Differential Equations

PDEs and Solutions

Classification

Linear vs. Nonlinear

Linear Equations

Conservation Laws

One Dimension

Higher Dimensions

Constitutive Relations

Initial and Boundary Value Problems

Waves

Traveling Waves

Plane Waves

Plane Waves and Transforms

Nonlinear Dispersion

First-Order Equations and Characteristics

Linear First-Order Equations

Advection Equation

Variable Coefficients

Nonlinear Equations

Quasi-linear Equations

The general solution

Propagation of Singularities

General First-Order Equation

Complete Integral

Uniqueness Result

Models in Biology

Age-Structure

Structured predator-prey model

Chemotherapy

Mass structure

Size-dependent predation

Weak Solutions To Hyperbolic Equations

Discontinuous Solutions

Jump Conditions

Rarefaction Waves

Shock Propagation

Shock Formation

Applications

Traffic Flow

Plug Flow Chemical Reactors

Weak Solutions: A Formal Approach

Asymptotic Behavior of Shocks

Equal-Area Principle

Shock Fitting

Asymptotic Behavior

Hyperbolic Systems

Shallow Water Waves; Gas Dynamics

Shallow Water Waves

Small-Amplitude Approximation

Gas Dynamics

Hyperbolic Systems and Characteristics

Classification

The Riemann Method

Jump Conditions for Systems

Breaking Dam Problem

Receding Wall Problem

Formation of a Bore

Gas Dynamics

Hodographs and Wavefronts

Hodograph Transformation

Wavefront Expansions

Weakly Nonlinear Approximations

Derivation of BurgersÆ Equation

Diffusion Processes

Diffusion and Random Motion

Similarity Methods

Nonlinear Diffusion Models

Reaction-Diffusion; FisherÆs Equation

Traveling Wave Solutions

Perturbation Solution

Stability of Traveling Waves

NagumoÆs Equation

Advection-Diffusion; BurgersÆ Equation

Traveling Wave Solution

Initial Value Problem

Asymptotic Solution to BurgersÆ Equation

Evolution of a Point Source

Reaction-Diffusion Systems

Reaction-Diffusion Models

Predator-Prey Model

Combustion

Chemotaxis

Traveling Wave Solutions

Model for the Spread of a Disease

Contaminant transport in groundwater

Existence of Solutions

Fixed-Point Iteration

Semi-Linear Equations

Normed Linear Spaces

General Existence Theorem

Maximum Principles

Maximum Principles

Comparison Theorems

Energy Estimates and Asymptotic Behavior

Calculus Inequalities

Energy Estimates

Invariant Sets

Pattern Formation

Equilibrium Models

Elliptic Models

Theoretical Results

Maximum Principle

Existence Theorem

Eigenvalue Problems

Linear Eigenvalue Problems

Nonlinear Eigenvalue Problems

Stability and Bifurcation

Ordinary Differential Equations

Partial Differential Equations

References

Index

Table of Contents provided by Publisher. All Rights Reserved.