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REACTION-DIFFUSION ANALYSIS

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REACTION-DIFFUSION ANALYSISMATH 350 - RENATO FERESCUPPLES I - ROOM 17 - [email protected] - 5-67521. NOTE:These are rough lecture notes for a course on applied math (Math 350), with an emphasis onchemical kinetics, for advanced undergraduate and beginning graduate students in science and math-ematics. There are (Fall 2003) three students enrolled: a second year undergraduate with a majorin applied mathematics, a senior with majors in mathematics, chemical engineering, and environ-mental sciences, and a first year graduate student in chemistry. During the course of the semesterthe students prepared projects on topics related to the course material and are presently scheduledto present them to each other in a four hour “mini-conference” on exam day, December 17, 2003.Their expositions will be about: pattern formation, implementation of a reaction diffusion systemnumerically, and use of group theory to study the spectrum of the Laplacian.In parallel to this course (also in the Fall of 2003), Professor Gregory Yablonsky, of the ChemicalEngineering Department, taught a course on “Transport in the Environment,” CE 583, for chemicalengineering students. Professor Yablonsky and I communicated almost daily during the semester andconstantly exchanged ideas about the two courses. Our plan is to combine them into one course thatcontains both the mathematics at a similar depth as discussed here, but with a much more developedscientific content, emphasizing issues in biochemistry and environmental science. Our lecture noteswill be integrated into a single text. Professor Yablosnky’s lecture notes comprise the following maintopics:• Introduction. Environmental transport;• fluid mechanics;• ideal reactors;• viscosity, transport in non-uniform media, diffusion;• hydrodynamics and the Navier-Stokes equations;• realistic models of surface water systems;• non-steady state behaviors of surface water systems, system of lakes;• incompletely mixed systems, propagations of polutants;• conservation of energy, Bernoulli equation, heat balance;• dissolved oxygen;• chemical transformations in natural waters;• eutrophication;• modeling of the living matter;• ground water and the subsurface environment;• ground water motion;• atmospheric transport and processes;• transport of chemicals in the atmosphere.12 MATH 350 - RENATO FERES CUPPLES I - ROOM 17 - [email protected] - 5-6752A few of these topics are discussed below as well, but only in a very simplified and not speciallycoherent form. The unifying themes for the present notes are mathematical. The joint text will havea more developed scientific flesh for the mathematical bare bones described here.2. WHAT IS THIS COURSE ABOUT?This course is centered around the study of systems of reactions (mostly chemical but not ex-clusively), possibly coupled with transport processes, mainly diffusion. Very broadly stated, we areinterested in the temporal and spatial properties of reaction networks. This study brings together awide variety of mathematical ideas and can be applied to a wealth of scientific problems.Mathematically, much of the course will be dedicated to the study of ‘dynamical systems,’ aterm that refers to the qualitative study of systems of non-linear (ordinary and/or partial) differentialequations. Other subjects, particularly from algebra, will also be needed. To varying degrees ofdetail, we plan to cover the following topics:(1) Box models and steady state(a) Stock, flows, residence time;(b) Basic concepts in chemistry and simple applications.(2) Abstract Stoichiometry.(a) Linear algebra for complex reaction networks.(b) Graph representation.(3) Finite Dimensional Dynamical Systems. This is the qualitative study of systems of ordi-nary differential equations and is comprised of topics such as:(a) Linear systems;(b) Generalities about vector fields and flows;(c) Local theory: linearization and local stability, stable and central manifolds;(d) Global theory: limit sets and attractors; Poincar´e-Bendixon theorem, global phase por-traits;(e) Bifurcationtheory: structural stability; bifurcations at nonhyperbolic equilibrium points,Hopf bifurcations and bifurcations of limit cycles from a multiple focus, homoclinic bi-furcations and chaos, the Melnikov’s method.(4) Partial Differential Equations.(a) The linear diffusion equation;(b) Diffusion and probability;(c) General properties of reaction-diffusion equations and special systems.Reaction-diffusion equations are important to a wide range of applied areas such as cell processes,drug release, ecology, spread of diseases, industrial catalytic processes, transport of contaminants inthe environment, chemistry in interstellar media, to mention a few. Some of these applications,particularly in chemistry and biology, will be considered along the course.It is not meaningful to talk about a general theory of reaction-diffusion systems. This is a rela-tively recent subject of mathematical and applied research. Most of the work that has been done sofar is concerned with the exploration of particular aspects of very specific systems and equations.This is because such systems are generally very complicated and display a wide array of poorlyunderstood phenomena. As a result, there is no established syllabus for such a course, and no singletextbook that would cover the topics that we wish to study. Topics and exercises will thus be drawnfrom a large number of different sources. The following list of references should prove useful. I willmake clear throughout the course which sources I’m using at any particular moment.REACTION-DIFFUSION ANALYSIS 3(1) G. Nicolis and I. Prigogine, Exploring Complexity - An Introduction. W. H. Freeman andCompany, New York, 1989.(2) G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems - From Dissipa-tive Structures to Order Through Fluctuations. John Wiley & Sons, New York, 1977.This is a more technical text, of which the first book is the popular, bedside version.(3) Rutherford Aris, Elementary Chemical Reactor Analysis. Dover Publications, Mineola,1999.(4) John Harte, Consider a Spherical Cow - A Course in Environmental Problem Solving.William Kaufman, Inc, Los Angeles, 1985.(5) Howard C. Berg, Random Walks in Biology. Princeton University Press, Princeton, 1993.(6) S. I. Rubnow, Introduction to Mathematical Biology. Dover Publications, Mineola, 2002.(7) L. Perko, Differential Equation and Dynamical System.


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