An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation

855 Pages·2016·19.16 MB·English

by  Carmen Chicone

(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.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicor
mechanical,includingphotocopying,recording,oranyinformationstorageandretrievalsystem,
withoutpermissioninwritingfromthepublisher.Detailsonhowtoseekpermission,further
informationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuchas
theCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite:
www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe
Publisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience
broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedicaltreatment
maybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluating
andusinganyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuch
informationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyofothers,including
partiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assume
anyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproductsliability,
negligenceorotherwise,orfromanyuseoroperationofanymethods,products,instructions,orideas
containedinthematerialherein.
BritishLibraryCataloguinginPublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
ISBN:978-0-12-804153-6
(cid:70)(cid:111)(cid:114)(cid:32)(cid:105)(cid:110)(cid:102)(cid:111)(cid:114)(cid:109)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:111)(cid:110)(cid:32)(cid:97)(cid:108)(cid:108)(cid:32)(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:112)(cid:117)(cid:98)(cid:108)(cid:105)(cid:99)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:118)(cid:105)(cid:115)(cid:105)(cid:116)(cid:32)(cid:111)(cid:117)(cid:114)(cid:32)(cid:119)(cid:101)(cid:98)(cid:115)(cid:105)(cid:116)(cid:101)(cid:32)(cid:97)(cid:116)(cid:32)(cid:104)(cid:116)(cid:116)(cid:112)(cid:115)(cid:58)(cid:47)(cid:47)(cid:119)(cid:119)(cid:119)(cid:46)(cid:101)(cid:108)(cid:115)(cid:101)(cid:118)(cid:105)(cid:101)(cid:114)(cid:46)(cid:99)(cid:111)(cid:109)(cid:47)
(cid:80)(cid:117)(cid:98)(cid:108)(cid:105)(cid:115)(cid:104)(cid:101)(cid:114)(cid:58)(cid:32)(cid:78)(cid:105)(cid:107)(cid:107)(cid:105)(cid:32)(cid:76)(cid:101)(cid:118)(cid:121)
(cid:65)(cid:99)(cid:113)(cid:117)(cid:105)(cid:115)(cid:105)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:69)(cid:100)(cid:105)(cid:116)(cid:111)(cid:114)(cid:58)(cid:32)(cid:71)(cid:114)(cid:97)(cid:104)(cid:97)(cid:109)(cid:32)(cid:78)(cid:105)(cid:115)(cid:98)(cid:101)(cid:116)
(cid:69)(cid:100)(cid:105)(cid:116)(cid:111)(cid:114)(cid:105)(cid:97)(cid:108)(cid:32)(cid:80)(cid:114)(cid:111)(cid:106)(cid:101)(cid:99)(cid:116)(cid:32)(cid:77)(cid:97)(cid:110)(cid:97)(cid:103)(cid:101)(cid:114)(cid:58)(cid:32)(cid:83)(cid:117)(cid:115)(cid:97)(cid:110)(cid:32)(cid:73)(cid:107)(cid:101)(cid:100)(cid:97)
(cid:80)(cid:114)(cid:111)(cid:100)(cid:117)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:80)(cid:114)(cid:111)(cid:106)(cid:101)(cid:99)(cid:116)(cid:32)(cid:77)(cid:97)(cid:110)(cid:97)(cid:103)(cid:101)(cid:114)(cid:58)(cid:32)(cid:80)(cid:111)(cid:117)(cid:108)(cid:111)(cid:117)(cid:115)(cid:101)(cid:32)(cid:74)(cid:111)(cid:115)(cid:101)(cid:112)(cid:104)
(cid:68)(cid:101)(cid:115)(cid:105)(cid:103)(cid:110)(cid:101)(cid:114)(cid:58)(cid:32)(cid:77)(cid:97)(cid:116)(cid:116)(cid:104)(cid:101)(cid:119)(cid:32)(cid:76)(cid:105)(cid:109)(cid:98)(cid:101)(cid:114)(cid:116)
(cid:84)(cid:121)(cid:112)(cid:101)(cid:115)(cid:101)(cid:116)(cid:32)(cid:98)(cid:121)(cid:32)(cid:83)(cid:80)(cid:105)(cid:32)(cid:66)(cid:111)(cid:111)(cid:107)(cid:115)(cid:32)(cid:97)(cid:110)(cid:100)(cid:32)(cid:74)(cid:111)(cid:117)(cid:114)(cid:110)(cid:97)(cid:108)(cid:115)
PREFACE
What is applied mathematics? 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.