Logan An Introduction to Nonlinear Partial Differential Equations

2. Auflage 2008
ISBN: 978-0-470-28708-8
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 416 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

Praise for the First Edition:
"This book is well conceived and well written. The author hassucceeded in producing a text on nonlinear PDEs that is not onlyquite readable but also accessible to students from diversebackgrounds."
--SIAM Review
A practical introduction to nonlinear PDEs and their real-worldapplications
Now in a Second Edition, this popular book on nonlinear partialdifferential equations (PDEs) contains expanded coverage on thecentral topics of applied mathematics in an elementary, highlyreadable format and is accessible to students and researchers inthe field of pure and applied mathematics. This book provides a newfocus on the increasing use of mathematical applications in thelife sciences, while also addressing key topics such as linearPDEs, first-order nonlinear PDEs, classical and weak solutions,shocks, hyperbolic systems, nonlinear diffusion, and ellipticequations. Unlike comparable books that typically only use formalproofs and theory to demonstrate results, An Introduction toNonlinear Partial Differential Equations, Second Edition takes amore practical approach to nonlinear PDEs by emphasizing how theresults are used, why they are important, and how they are appliedto real problems.
The intertwining relationship between mathematics and physicalphenomena is discovered using detailed examples of applicationsacross various areas such as biology, combustion, traffic flow,heat transfer, fluid mechanics, quantum mechanics, and the chemicalreactor theory. New features of the Second Edition alsoinclude:
* Additional intermediate-level exercises that facilitate thedevelopment of advanced problem-solving skills
* New applications in the biological sciences, includingage-structure, pattern formation, and the propagation ofdiseases
* An expanded bibliography that facilitates further investigationinto specialized topics
With individual, self-contained chapters and a broad scope ofcoverage that offers instructors the flexibility to design coursesto meet specific objectives, An Introduction to Nonlinear PartialDifferential Equations, Second Edition is an ideal text for appliedmathematics courses at the upper-undergraduate and graduate levels.It also serves as a valuable resource for researchers andprofessionals in the fields of mathematics, biology, engineering,and physics who would like to further their knowledge of PDEs.

Autoren/Hrsg.

Logan, J. David

Inhaltsverzeichnis

Preface.
1. Partial Differential Equations.
1.1 Partial Differential Equations.
1.1.1 PDEs and Solutions.
1.1.2 Classification.
1.1.3 Linear vs. Nonlinear.
1.1.4 Linear Equations.
1.2 Conservation Laws.
1.2.1 One Dimension.
1.2.2 Higher Dimensions.
1.3 Constitutive Relations.
1.4 Initial and Boundary Value Problems.
1.5 Waves.
1.5.1 Traveling Waves.
1.5.2 Plane Waves.
1.5.3 Plane Waves and Transforms.
1.5.4 Nonlinear Dispersion.
2. First-Order Equations and Characteristics.
2.1 Linear First-Order Equations.
2.1.1 Advection Equation.
2.1.2 Variable Coefficients.
2.2 Nonlinear Equations.
2.3 Quasi-linear Equations.
2.3.1 The general solution.
2.4 Propagation of Singularities.
2.5 General First-Order Equation.
2.5.1 Complete Integral.
2.6 Uniqueness Result.
2.7 Models in Biology.
2.7.1 Age-Structure.
2.7.2 Structured predator-prey model.
2.7.3 Chemotherapy.
2.7.4 Mass structure.
2.7.5 Size-dependent predation.
3. Weak Solutions To Hyperbolic Equations.
3.1 Discontinuous Solutions.
3.2 Jump Conditions.
3.2.1 Rarefaction Waves.
3.2.2 Shock Propagation.
3.3 Shock Formation.
3.4 Applications.
3.4.1 Traffic Flow.
3.4.2 Plug Flow Chemical Reactors.
3.5 Weak Solutions: A Formal Approach.
3.6 Asymptotic Behavior of Shocks.
3.6.1 Equal-Area Principle.
3.6.2 Shock Fitting.
3.6.3 Asymptotic Behavior.
4. Hyperbolic Systems.
4.1 Shallow Water Waves; Gas Dynamics.
4.1.1 Shallow Water Waves.
4.1.2 Small-Amplitude Approximation.
4.1.3 Gas Dynamics.
4.2 Hyperbolic Systems and Characteristics.
4.2.1 Classification.
4.3 The Riemann Method.
4.3.1 Jump Conditions for Systems.
4.3.2 Breaking Dam Problem.
4.3.3 Receding Wall Problem.
4.3.4 Formation of a Bore.
4.3.5 Gas Dynamics.
4.4 Hodographs and Wavefronts.
4.4.1 Hodograph Transformation.
4.4.2 Wavefront Expansions.
4.5 Weakly Nonlinear Approximations.
4.5.1 Derivation of Burgers' Equation.
5. Diffusion Processes.
5.1 Diffusion and Random Motion.
5.2 Similarity Methods.
5.3 Nonlinear Diffusion Models.
5.4 Reaction-Diffusion; Fisher's Equation.
5.4.1 Traveling Wave Solutions.
5.4.2 Perturbation Solution.
5.4.3 Stability of Traveling Waves.
5.4.4 Nagumo's Equation.
5.5 Advection-Diffusion; Burgers' Equation.
5.5.1 Traveling Wave Solution.
5.5.2 Initial Value Problem.
5.6 Asymptotic Solution to Burgers' Equation.
5.6.1 Evolution of a Point Source.
6. Reaction-Diffusion Systems.
6.1 Reaction-Diffusion Models.
6.1.1 Predator-Prey Model.
6.1.2 Combustion.
6.1.3 Chemotaxis.
6.2 Traveling Wave Solutions.
6.2.1 Model for the Spread of a Disease.
6.2.2 Contaminant transport in groundwater.
6.3 Existence of Solutions.
6.3.1 Fixed-Point Iteration.
6.3.2 Semi-Linear Equations.
6.3.3 Normed Linear Spaces.
6.3.4 General Existence Theorem.
6.4 Maximum Principles.
6.4.1 Maximum Principles.
6.4.2 Comparison Theorems.
6.5 Energy Estimates and Asymptotic Behavior.
6.5.1 Calculus Inequalities.
6.5.2 Energy Estimates.
6.5.3 Invariant Sets.
6.6 Pattern Formation.
7. Equilibrium Models.
7.1 Elliptic Models.
7.2 Theoretical Results.
7.2.1 Maximum Principle.
7.2.2 Existence Theorem.
7.3 Eigenvalue Problems.
7.3.1 Linear Eigenvalue Problems.
7.3.2 Nonlinear Eigenvalue Problems.
7.4 Stability and Bifurcation.
7.4.1 Ordinary Differential Equations.
7.4.2 Partial Differential Equations.
References.
Index.

Über Autor(innen)

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.