Fluent Essays

The book you will use is uploaded. Please explain the question 1.1 formula for me thx. introduction_to_the_foundations_of_applied_mathem.pdf Unformatted Attachment Preview Texts in Applied Mathematics 56 Editors J.E. Marsden L. Sirovich S.S. Antman Advisors G. Iooss P. Holmes D. Barkley M. Dellnitz P. Newton Texts in Applied Mathematics 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. Sirovich: Introduction to Applied Mathematics. Wiggins: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Hale/Koçak: Dynamics and Bifurcations. Chorin/Marsden: A Mathematical Introduction to Fluid Mechanics, 3rd ed. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Ordinary Differential Equations. Sontag: Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd ed. Perko: Differential Equations and Dynamical Systems, 3rd ed. Seaborn: Hypergeometric Functions and Their Applications. Pipkin: A Course on Integral Equations. Hoppensteadt/Peskin: Modeling and Simulation in Medicine and the Life Sciences, 2nd ed. Braun: Differential Equations and Their Applications, 4th ed. Stoer/Bulirsch: Introduction to Numerical Analysis, 3rd ed. Renardy/Rogers: An Introduction to Partial Differential Equations. Banks: Growth and Diffusion Phenomena: Mathematical Frameworks and Applications. Brenner/Scott: The Mathematical Theory of Finite Element Methods, 2nd ed. Van de Velde: Concurrent Scientific Computing. Marsden/Ratiu: Introduction to Mechanics and Symmetry, 2nd ed. Hubbard/West: Differential Equations: A Dynamical Systems Approach: Higher-Dimensional Systems. Kaplan/Glass: Understanding Nonlinear Dynamics. Holmes: Introduction to Perturbation Methods. Curtain/Zwart: An Introduction to Infinite-Dimensional Linear Systems Theory. Thomas: Numerical Partial Differential Equations: Finitc Difference Methods. Taylor: Partial Differential Equations: Basic Theory. Merkin: Introduction to the Theory of Stability of Motion. Naber: Topology, Geometry, and Gauge Fields: Foundations. Polderman/Willems: Introduction to Mathematical Systems Theory: A Behavioral Approach. Reddy: Introductory Functional Analysis with Applications to Boundary-Value Problems and Finite Elements. Gustafson/Wilcox: Analytical and Computational Methods of Advanced Engineering Mathematics. Tveito/Winther: Introduction to Partial Differential Equations: A Computational Approach. Gasquet/Witomski: Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets. (continued after index) Mark H. Holmes Introduction to the Foundations of Applied Mathematics 123 Mark H. Holmes Department of Mathematical Sciences Rensselaer Polytechnic Institute 110 8th Street Troy NY 12180-3590 USA holmes@rpi.edu Series Editors J.E. Marsden Control and Dynamical Systems, 107–81 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA lawrence.sirovich@mssm.edu S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu ISSN 0939-2475 ISBN 978-0-387-87749-5 e-ISBN 978-0-387-87765-5 DOI 10.1007/978-0-387-87765-5 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009929235 Mathematics Subject Classification (2000): 74-01; 76R50; 76A02; 76M55; 35Q30; 35Q80; 92C45; 74A05; 74A10 c Springer Science+Business Media, LLC 2009  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Colette, Matthew and Marianna Preface FOAM. This acronym has been used for over fifty years at Rensselaer to designate an upper-division course entitled, Foundations of Applied Mathematics. This course was started by George Handelman in 1956, when he came to Rensselaer from the Carnegie Institute of Technology. His objective was to closely integrate mathematical and physical reasoning, and in the process enable students to obtain a qualitative understanding of the world we live in. FOAM was soon taken over by a young faculty member, Lee Segel. About this time a similar course, Introduction to Applied Mathematics, was introduced by Chia-Ch’iao Lin at the Massachusetts Institute of Technology. Together Lin and Segel, with help from Handelman, produced one of the landmark textbooks in applied mathematics, Mathematics Applied to Deterministic Problems in the Natural Sciences. This was originally published in 1974, and republished in 1988 by the Society for Industrial and Applied Mathematics, in their Classics Series. This textbook comes from the author teaching FOAM over the last few years. In this sense, it is an updated version of the Lin and Segel textbook. The objective is definitely the same, which is the construction, analysis, and interpretation of mathematical models to help us understand the world we live in. However, there are some significant differences. Lin and Segel, like many recent modeling books, is based on a case study format. This means that the mathematical ideas are introduced in the context of a particular application. There are certainly good reasons why this is done, and one is the immediate relevance of the mathematics. There are also disadvantages, and one pointed out by Lin and Segel is the fragmentary nature of the development. However, there is another, more important reason for not following a case studies approach. Science evolves, and this means that the problems of current interest continually change. What does not change as quickly is the approach used to derive the relevant mathematical models, and the methods used to analyze the models. Consequently, this book is written in such a way as to establish the mathematical ideas underlying model development independently of a specific application. This does not mean applications are not vii viii Preface considered, they are, and connections with experiment are a staple of this book. The first two chapters establish some of the basic mathematical tools that are needed. The model development starts in Chapter 3, with the study of kinetics. The goal of this chapter is to understand how to model interacting populations. This does not account for the spatial motion of the populations, and this is the objective of Chapters 4 and 5. What remains is to account for the forces in the system, and this is done in Chapter 6. The last three chapters concern the application to specific problems and the generalization of the material to more geometrically realistic systems. The book, as well as the individual chapters, is written in such a way that the material becomes more sophisticated as you progress. This provides some flexibility in how the book is used, allowing consideration for the breadth and depth of the material covered. The principal objective of this book is the derivation and analysis of mathematical models. Consequently, after deriving a model, it is necessary to have a way to solve the resulting mathematical problem. A few of the methods developed here are standard topics in upper-division applied math courses, and in this sense there is some overlap with the material covered in those courses. Examples are the Fourier and Laplace transforms, and the method of characteristics. On the other hand, other methods that are used here are not standard, and this includes perturbation approximations and similarity solutions. There are also unique methods, not found in traditional textbooks, that rely on both the mathematical and physical characteristics of the problem. The prerequisite for this text is a lower-division course in differential equations. The implication is that you have also taken two or three semesters of calculus, which includes some component of matrix algebra. The one topic from calculus that is absolutely essential is Taylor’s theorem, and for this reason a short summary is included in the appendix. Some of the more sophisticated results from calculus, related to multidimensional integral theorems, are not needed until Chapter 8. To learn mathematics you must work out problems, and for this reason the exercises in the text are important. They vary in their difficulty, and cover most of the topics in the chapter. Some of the answers are available, and can be found at www.holmes.rpi.edu. This web page also contains a typos list. I would like to express my gratitude to the many students who have taken my FOAM course at Rensselaer. They helped me immeasurably in understanding the subject, and provided much-needed encouragement to write this book. It is also a pleasure to acknowledge the suggestions of John Ringland, and his students, who read an early version of the manuscript. Troy, New York March, 2009 Mark H. Holmes Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Maximum Height of a Projectile . . . . . . . . . . . . . . . . . . . . 1.2.2 Drag on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Toppling Dominoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Pattern Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Similarity Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Nondimensionalization and Scaling . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Projectile Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Weakly Nonlinear Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 5 6 13 15 16 19 22 25 26 30 32 33 2 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Regular Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 How to Find a Regular Expansion . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Given a Specific Function . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Given an Algebraic or Transcendental Equation . . . . . . 2.2.3 Given an Initial Value Problem . . . . . . . . . . . . . . . . . . . . 2.3 Introduction to Singular Perturbations . . . . . . . . . . . . . . . . . . . . 2.4 Introduction to Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Multiple Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Multiple Scales and Two-Timing . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 43 48 48 51 53 58 60 66 68 72 79 ix x Contents 3 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Predator-Prey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Epidemic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Kinetic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Steady-States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 General Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . 3.4 Michaelis-Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Quasi-Steady-State Approximation . . . . . . . . . . . . . . . . . 3.4.3 Perturbation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Assorted Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Elementary and Nonelementary Reactions . . . . . . . . . . . 3.5.2 Reverse Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Steady-States and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Reaction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Geometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 87 87 88 88 89 91 92 94 94 96 97 100 102 103 105 111 111 113 114 115 115 118 126 128 132 4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Random Walks and Brownian Motion . . . . . . . . . . . . . . . . . . . . . 4.2.1 Calculating w(m, N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Large N Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Continuous Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 What Does D Signify? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Solving the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Point Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Continuum Formulation of Diffusion . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Fick’s Law of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Reaction-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . 4.6 Random Walks and Diffusion in Higher Dimensions . . . . . . . . . 4.6.1 Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Properties of the Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . 141 141 142 145 148 149 151 153 154 157 169 169 171 177 179 182 185 188 Contents xi 4.7.2 Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5 Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Continuum Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Balance Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Velocity Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Linear Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 General Velocity Formulation . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Flux and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Reality Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Nonconstant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Small Disturbance Approximation . . . . . . . . . . . . . . . . . . 5.6.2 Method of Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Rankine-Hugoniot Condition . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Expansion Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.5 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.6 Return of Phantom Traffic Jams . . . . . . . . . . . . . . . . . . . 5.6.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Cellular Automata Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 205 206 207 208 209 210 211 212 213 214 216 217 218 221 225 226 229 233 236 241 245 247 248 254 6 Continuum Mechanics: One Spatial Dimension . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Material Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Spatial Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 End Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Material Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Material Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary of the Equations of Motion . . . . . . . . . . . . . . . . . . . . ... Purchase answer to see full attachment