(cid:65)(cid:110)(cid:32)(cid:73)(cid:110)(cid:118)(cid:105)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:116)(cid:111)(cid:32)(cid:65)(cid:112)(cid:112)(cid:108)(cid:105)(cid:101)(cid:100) (cid:77)(cid:97)(cid:116)(cid:104)(cid:101)(cid:109)(cid:97)(cid:116)(cid:105)(cid:99)(cid:115) (cid:65)(cid:110)(cid:32)(cid:73)(cid:110)(cid:118)(cid:105)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110) (cid:116)(cid:111) (cid:65)(cid:112)(cid:112)(cid:108)(cid:105)(cid:101)(cid:100) Mathematics: Differential Equations, Modeling, and Computation Carmen Chicone AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO (cid:65)(cid:99)(cid:97)(cid:100)(cid:101)(cid:109)(cid:105)(cid:99)(cid:32)(cid:80)(cid:114)(cid:101)(cid:115)(cid:115)(cid:32)(cid:105)(cid:115)(cid:32)(cid:97)(cid:110)(cid:32)(cid:105)(cid:109)(cid:112)(cid:114)(cid:105)(cid:110)(cid:116)(cid:32)(cid:111)(cid:102)(cid:32)(cid:69)(cid:108)(cid:115)(cid:101)(cid:118)(cid:105)(cid:101)(cid:114) AcademicPressisanimprintofElsevier 32JamestownRoad,LondonNW17BY,UK 525BStreet,Suite1800,SanDiego,CA92101-4495,USA 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Copyright(cid:13)c 2017 ElsevierInc.Allrightsreserved. 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Every answer to this question is likely to initiateadebate.Mydefinitionistheuseofmathematicstosolveproblems or gain insight into phenomena that arise outside of mathematics. The prototypicalexampleistheuseofmathematicstosolveproblemsinphysics. Of course, the world of applied mathematics is much broader: important applications of mathematics occur in all areas of science, engineering, and technology. The concept of this book is to introduce the reader to one aspect of applied mathematics: the use of differential equations to solve physical problems. To cover the full (ever expanding) range of applications of mathematics would require a series of books, which would include invi- tations to applied mathematics using the other branches of mathematics: calculus,linearalgebra,differentialgeometry,graphtheory,combinatorics, numbertheory,thecalculusofvariations,probabilitytheory,andothers.The application of statistics (especially in experimental science) is a branch of applied mathematics of great importance, but of a different character than theappliedmathematicsconsideredhere. Although there are already many books and articles devoted to appli- cations of mathematical subjects, I believe that there is room for more introductorymaterialaccessibletoadvancedundergraduatesandbeginning graduate students. If my invitation is accepted, perhaps the reader will pursue further study, find a problem in applied mathematics, and make a contributiontotechnologyortheunderstandingofthephysicaluniverse. Myinvitationincludesatourthroughafewofthehistoricallyimportant uses of differential equations in science and technology. The relevant mathematics is presented in context where there is no question of its importance. A typical scenario in many research papers by mathematicians is an introduction that includes such phrases as “our subject is important in the study of ...,” “this problem arises in ..., ” or “our subject has many applications to ....” The authors go on to state a precise mathematical xi xii Preface problem,theyproveatheorem—perhapsaverygoodtheorem,andperhaps theygiveamathematicalexampletoillustratetheirresult,butalltoooften, theirtheoremdoesnotsolveaproblemofinterestinthescientificareathat they used to advertise their work. This is not applied mathematics. The correct approach is joint work with an expert in some area of science: a physical problem is stated, a mathematical model is proposed, a prediction ismadefromthemathematicalmodel—astepthatmightrequiresomenew mathematicsincludingmathematicaltheorems—andthepredictionistested againstaphysicalexperiment.Thispointofviewmotivatesthestyleofthe presentationinallthatfollows. Although the basics of mathematical modeling is discussed, the models tobeconsideredarisefromproblemswheretheunderlyingscienceiseasily accessible. The simple truth is that the construction of many important mathematical models requires a serious treatment of the corresponding science. This is one good reason for joint work between mathematicians and scientists or engineers on applied projects. Carefully chosen models, along with the essential science needed for their construction, are explored inthisbook. Applied mathematics requires an understanding of mathematics, some familiarity with the subject area of application, creativity, hard work, and experience. The study of (pure) mathematics is essential. As an aspiring applied mathematician approaching this book, you should know at least what constitutes a mathematical proof and have a working knowledge of basic analysis and linear algebra. To proceed further toward competence in applied mathematics, you will need to know and understand more and deeper mathematics. Along the way, part of your mathematics education should include some study in an applied context. This book is intended to provideawealthofthisvaluableexperience. Columbia,Missouri CarmenChicone March4,2016 I thank all the people who have offered valuable suggestions for cor- rections of and additions to this book, especially Oksana Bihun, Michael Heitzman,SeanSweany,andSamuelWalsh. TO THE PROFESSOR Thisbookissuitableforcoursesinappliedmathematicswithnumerics, basic fluid mechanics, basic mathematics of electromagnetism, or mathe- matical modeling. The prerequisites for students are vector calculus, basic differential equations, the rudiments of matrix algebra, knowledge of some programming language, and of course some mathematical maturity. No knowledgeofpartialdifferentialequationsornumericalanalysisisassumed. The author has used parts of this book while teaching courses in mathematical modeling at the University of Missouri where students (un- dergraduate and graduate) of engineering, the sciences, and mathematics enrolled.Thisheterogeneousmixofstudentsshouldbeexpectedinacourse at the advanced undergraduate beginning graduate level with a title such as Mathematical Modeling I. Thus, the instructor must assess the abilities and background knowledge of the students who show up on the first day ofclass.Professorsshouldbepreparedandwillingtomodifytheirsyllabus afteraweekortwoofinstructiontoaccommodatetheirstudents.Infact,the most likely modification is to cover less material at a slower pace. Perhaps learning a few concepts and techniques well is always more valuable than exposuretoasurveyofnewideas. A typical 15-week semester course might consist of one lecture on Chapter 1, two weeks on Chapter 2 (mostly ODE), two weeks on Chapter 5 (fundamental physical modeling, reaction-diffusion systems, and basic numerics for simple parabolic PDE), one week on Chapter 6 (electrical signalsonneuronsandtravelingwavesolutions),andoneweekonChapter8 (basic PID control) to complete approximately half of the semester. Of course only parts of the material in these chapters (in particular Chapter 5) can be covered in detail in class. By this time in the semester at least three substantial homework assignments should be completed using exercises, problems,andprojectssuggestedinthetext.Ofcourse,thereisgoodreason toalsoincludeexercisesdesignedbytheinstructor.Atleast,studentsshould havewritten,tested,andreportedapplicationstoappliedproblemsofafew basic codes for approximating solutions of ODEs and PDEs. Their work shouldbepresentedin(carefully)writtenreports(inEnglishprose[orsome xv xvi TotheProfessor otherlanguage])whereanalysisanddiscussionofresultsaresupplemented withreferencestooutputfromnumericalexperimentsintabularorgraphical formats. In-class exams are possible but perhaps not as appropriate to the material as homework assignments. The book does not contain many routine problems; in fact, many problems and all of the projects are open ended.Howelsewillstudentsexperiencechallengesthatanticipaterealistic applied problems? Some of the projects introduce new concepts and are fleshed out accordingly. A list of suggested projects is given in the index (see the entry Projects). The second half of the semester might be devoted to continuum mechanics or electromagnetism. But, the usual choice is fluid mechanics. There will be sufficient time to derive the conservation of momentumequationanddiscusstheEulerandNavier–Stokesstresstensors as in Chapter 11. Standard applications include flow in a pipe (Chapter 12) followed by a discussion of potential flow with applications to circulation, lift,anddraginChapter13.Perhapstheendofthesemesterisreachedwith adiscussionoftheCorioliseffectondrainsandhurricanes.Thefinalexam can be replaced by a set of problems and projects taken from Chapters 10 and 19, with respect given to sufficient background material discussed in class. In addition, each student might be required to present a project—in thespiritofthecourse—takendirectlyfromthisbook,relatedtotheirwork insomeotherclass,orrelatedtotheirresearch. A more advanced course might be devoted entirely to continuum me- chanics with the intention of covering more sophisticated mathematics and numerics. In particular, basic water wave phenomena and free-surface flow canbeaddressedalongwithappropriatenumericalmethods.InChapter16, a complete treatment of Chorin’s projection method is given in sufficient detail for students (and perhaps their professor) to write a basic CFD code that can be applied to a diverse set of applied problems. This is followed by the most mathematically sophisticated part of the book on the boundary element method, where classical potential theory is covered and all the ingredientsofthisnumericalmethodarediscussedindetail.Thisisfollowed by a treatment of smoothed particle hydrodynamics, again with sufficient detail to write a viable code. Channel flow provides a modeling experience along with a discussion and application of Prandtl’s boundary layer theory, andasolidtreatmentofthetheoryandnumericsofhyperbolicconservation laws. All of this material is written in context with applied problems. The chapterendswithabasicdiscussionofelasticsolids,continuummechanics, the weak formulation of PDEs, and sufficient detail to write a basic finite- TotheProfessor xvii elementcodethatcanbeusedtoapproximatethesolutionsofproblemsthat ariseinmodelingelasticsolids. Likewise, an advanced course might be devoted to applied problems in electromagnetism.ThematerialinChapter20providesabasic(mathemati- callyoriented)introductiontoMaxwell’sequationsandtheelectromagnetic boundaryvalueproblem.Anenlighteningapplicationofthetheoryismade to transverse electromagnetic waves and waveguides. This is specialized to thetheoryoftransmissionlineswheretheRiemannproblemforhyperbolic conservation laws arises in context and its solution is used to construct a viable numerical method to approximate the electromagnetic waves. This theory is applied to the practical problem of time-domain reflectometry, whichservesasanintroductiontoabasicinverseproblemofwideinterest: shine radiation on some object with the intent of identifying the object by analyzingthereflectedelectromagneticwaves. The material in the book can be used to design undergraduate research projects and master’s projects. Of course, it can also be used to help PhD students gain valuable experience before approaching an applied research problem.