ODE Architect text

287 Pages·2003·2.52 MB·English

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O D E A R C H I T E C T
Companion
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O D E A R C H I T E C T
Companion
. .
C ODE E
(Consortium for ODE Experiments)
JOHNWILEY&SONS,INC.
NewYork/Chichester/Weinheim/Brisbane/Singapore/Toronto
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Copyright c 1999,byJohnWiley&Sons,Inc.
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Allrightsreserved.
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beyondthatpermittedbySections107and108of
the1976UnitedStatesCopyrightActwithoutthe
permissionofthecopyrightownerisunlawful.
Requestsforpermissionorfurtherinformation
shouldbeaddressedtothePermissionsDepartment,
JohnWiley&Sons,Inc.
ISBN0-471-
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PREFACE
This workbook was designed to accompanythe software package ODE Ar-
chitect, and that’s why we call it a Companion. Each of the 13 Companion
chapters corresponds to a multimedia module in the Architect and provides
background and opportunities for you to extend the ideas contained in the
module. Each chapter ends with several problem sets, called Explorations,
related to the chapter and module topics. The Explorations are designed so
thatyoucanwritein answersandderivations,and,sincetheyareprintedon
perforated pages, they can be removed and handed-in along with printouts
of graphs produced by the Architect. There is also a notepad facility in the
Architect which, with the cut and paste features, makes it possible to write
reports.
ODE Architect
ODEArchitectprovidesahighlyinteractiveenvironmentforconstructingand
exploringyourownmathematicalmodelsofreal-worldphenomena,whether
theylead to linear or nonlinearsystems of ODEs. The Architect’s multime-
diafrontendguidesyouthroughexperimentstobuildandexploreyourown
ODEs. The software has numerical solvers, 2- and 3-D graphics, and the
ability to build physical representations of systems such as pendulums and
spring-masssystemsaswellastheabilitytoanimatethem. Togetherwithits
libraryofODEs,theODEArchitectbringsawealthofopportunitiestogain
insightsaboutsolutionstoODEs.
The overall guiding feature is for the software to be easy to use. Nav-
igational paths are clearly marked and simple to follow. When starting the
software, you are presentedwith a title screenfollowed by a main menu al-
lowing selection of a specific module. You may prefer to go directly to the
ArchitectTooltorunyourownexperiments.Atanyplaceinthesoftware,you
willbeabletocallupthecontentsmenusandaccessthematerialinanyorder.
Weexpectthatmostwillworkthoughthemultimediamodules. Let’slookat
each of the three principal parts of the ODE Architect in more detail: The
MultimediaODEArchitect,TheODEArchitectTool,andtheODELibrary.
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Multimedia ODE Architect
C.ODE.E members and colleagues have authored the multimedia modules,
each with its own theme. The modeling process is detailed, supported by
highlyinteractivesimulations. Youcanexploretheproblem-solvingprocess
via “what-if” scenarios and exercises. They are guided to build their own
ODEsandsolvethemnumericallyandgraphically,andcomparethepredicted
resultstoempiricaldatawhenappropriate.
Eachmodulehasuptofoursubmodules,andtheyrangefromthestraight-
forward to the advanced. The animations are often funny, the voice-overs
and text informal, but the modeling and the mathematics are the real thing.
Most submodules go through a model building process and several experi-
mentscreens,andthenendwithsomequestions(Things-to-Think-About,or
TTAs). These questionsextendthe topicsof the submoduleandtake youto
thesolvertooltoproducesolutioncurvesandorbits,writeareportconnecting
themathematics,themodelsandthepictures. WhenyouopentheToolusing
aTTAlink,thepertinentequationsandparametersettingswillautomatically
beenteredintotheequationquadrantoftheTool. Youarethenpoisedtothink
about,andwithoutconstraint,explorethemodelintroducedinthesubmodule.
ODE Architect Tool
TheODEArchitectToolisafirst-rate,research-qualitynumericalODEsolver
and graphics package. The ODE Architect Tool employs a graphical user
interfacetoenterandeditequations,controlsolversettingsandfeatures,and
tocreateandeditawidevarietyofgraphics.Asecondmodeofoperation,the
ExpertMode,providesaccesstomoreadvancedfeatures.
The Tool is the heart of the software, and it is a workspace where you
can:
• Construct,solveandexploreODEs
• Inputdatatables
• Graph and animate solution curves, phase plane graphs, 3D graphs,
Poincare´ sections,discretemaps,directionfields,etc.
• Build,analyze,andanimatephysicalrepresentationsofdynamicalsys-
tems.
The robust Tool will solve systems of up to 10 first-order ODEs which
can be entered using a simple, natural scripting language. Auxiliary func-
tions involving the state variables can be defined. A solver/grapher feature
for discrete dynamicalsystems is also available from the Tool. A variety of
engineeringfunctions such as square waves, sawtooth waves and step func-
tions are included in the Tool function library. Two and three dimensional
graphicsaresupported,aswellastimeandparameteranimationsofsolution
data. Initial conditionscanbe enteredby clicking in a graphwindow or via
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the keyboard. Graph scales can be set automatically or manually. Numer-
ical values of solutions can be viewed in tabular form. Parameter-sensitive
analysis is made easy with a built-in parameter-sweeptool. You can do pa-
rameter and initial-value sweeps to see the effects of data changeson orbits
and solution curves. Graphs are editable and you can scale and label axes,
mark equidistant-in-time orbital points, color the graphs, changeline styles,
overlaygraphsoffunctionsandsolutioncurvesfordifferentODEs—allwith
noprogrammingorspecialcommandstoremember.
The solvers in the ODE Architect are state-of-the-art numerical solvers
basedonthosedevelopedbyDr.L.F.ShampineandDr.I.GladwellatSouth-
ern Methodist University. For a delightfully readable account on using nu-
mericalODEsolversinteachingODEs,pleaserefertotheirpaper:
Shampine, L.F., and Gladwell, I., “TeachingNumerical Methods in
ODECourses”
inthebookDifferentialEquationsfortheNextMillennium,editedbyMichael
J.KallaherintheMAANotesseries.
Module1, “Modelingwiththe ODEArchitect”isanon-linetutorialfor
manyofthefeaturesoftheTool. TheArchitectalsohashelpfacilitiesandthe
multimediasideisselfdocumenting.
ODE Library
The ODE Library has dozens of pre-programmed, editable, and interactive
ODEfilescoveringawiderangeoftopicsfrommathematics,physics,chem-
istry, population biology, and epidemiology. There are also many ODEs to
illustrate points such as data compression, ODEs with singular coefficients,
bifurcations, limit cycles, and so on. Each Library file has explanatorytext
alongwiththeequationsandincludesanillustrativegraphorgraphs.TheLi-
braryfilesareorganizedintofoldersbytopicandtheyhavedescriptivetitles
tofacilitatebrowsing. Thesefilesalsoprovideamarvelouswaytolearnhow
tousethetool.
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ACKNOWLEDGMENTS
ODE Architect was developedwith partial support from the NSF/DUE,1 by
. .
the Consortium for ODE Experiments (CODEE), Intellipro, Inc., and John
Wiley & Sons, Inc. C.ODE.E saw to the mathematical side of things, Intel-
liprorenderedC.ODE.E’sworkintoanattractivemultimediasoftwarepack-
age,andJohnWileycoordinatedtheeffortsofbothteams.
As in any project like this; we owe a debt of gratitude to many people;
reviewers, beta testers, students, programmers and designers. Specifically,
we want to thank the other members of the C.ODE.E Evaluation Commit-
tee, Barbara Holland (John Wiley & Sons), Philippe Marchal (Intellipro),
MichaelMoody(HarveyMuddCollege),andBeverlyWest(CornellUniver-
sity).Withoutthemanyhoursofhardworktheyputinonthisproject,itcould
nothavebeendone.
We especially want to thank Professor L.F. Shampine for providing the
excellentsolvercodes(developedbyhimselfandhiscolleague,IanGladwell),
and for his continuing support of the project. Thanks to Mark DeMichele
at Intellipro, who wrote the code for the Architect and the implementation
of the Shampine/Gladstone solver codes, and who put up with our constant
“advice”. Another very special thanks to David Richards for designing and
implementingtheLATEXmacrosfortheCompanionbookandforhispatience
and meticulous attention to detail during the many revisions. We also very
muchappreciatethereviewers,editors,andevaluatorsSusanGerstein,Zaven
Karian, Mario Martelli, Lang Moore, Douglas Quinney, David Cook, and
RobertStyerfortheirmanyhelpfulcommentsandsuggestions.
Finally,a“thankyou”toourstudentsTiffanyArnal,ClaireLaunay,Nathan
Jakubiak,JoelMiller,JustinRadick,TreasaSweek,andmanyotherswhoread
chapters, tested modules, and commented freely (even favorably) on what
theyexperienced.
RobertL.Borrelli
CourtneyS.Coleman
Claremont,CA
1TheworkonODEArchitectanditsCompanionbookwassupportedinpartbytheNationalSci-
enceFoundationunderGrantNumbersDMI-9509135andDUE-9450742. Anyopinions, findings,
conclusions,orrecommendationsexpressedinthismaterialarethoseoftheauthorsanddonotneces-
sarilyreflecttheviewsoftheNationalScienceFoundation.
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INFORMATION
ABOUT MODULES/CHAPTERS
Overview
Modules/Chapters1–3 are all introductory modules for first order ODEs
and simple systems of ODEs. Any of these modules/chapters can be
used at the beginning of an ODE course, or at appropriate places in
elementarycalculuscourses.
Modules/Chapters4–9 involvehigherorderODEsandsystemsandtheirap-
plications. Oncestudentsunderstandhowtodealwithtwo-dimensional
systemsgraphically,anyofthesemodules/chaptersiseasilyaccessible.
Modules/Chapters10–12 applytwo-dimensionalsystemstomodelsthatil-
lustratemoreadvancedtechniquesandtheory;themultimediaapproach
makesthemneverthelessquiteaccessible. Themodulesareintendedto
enablestudentstogetmuchfurtherwiththetechnicalaspectsexplained
inthechaptersthanwouldbeotherwisepossible.
Module/Chapter13 treatsdiscretedynamicalsystemsinanintroductoryfash-
ion that could be used in a course in ODEs, calculus, or even a non-
calculuscourse.
AMultimediaappendixonnumericalmethods givesinsightintotheways
inwhichnumericalsolutionsareconstructed.
Description/Prerequisites for Individual Modules/Chapters
WelistbelowforeachModule/Chapteritsprerequisitesandsomecomments
on its level and goals. In general, each module progresses from easier to
hardersubmodules,butthefirstsectionofnearlyeverymoduleisatanintro-
ductorylevel.
The modulescan be accessedin differentorders. It is not expectedthat
theywill beassignedinnumericalorder. Consequently,wehavetriedtoex-
plain each concept wherever it appears, or to indicate where an explanation
isprovided. Forexample,Newton’ssecondlaw, F =ma,isdescribedevery
timeitisinvoked.
ThereisfarmorematerialinODEArchitectthancouldpossiblyfitintoa
singlecourse.
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Module/Chapter 1: Modeling with the ODE Architect
Assumedconcepts: Precalculus;derivativeasarateofchange
Thismoduleisunlikealltheothersinthatitisnotdividedintosubmod-
ules,anditprovidesatutorialforlearninghowtonavigateODEArchitect. It
carriesthattutorialprocessalongintandemwithanintroductiontomodeling
thatassumesverylittlebackground.
Module/Chapter 2: Introductionto ODEs
Assumedconcepts: Derivatives;slopes;slopefields
Themodulebeginswithsomesimplefirst-orderODEsandtheirsolutions
andcontinueswithslopefields(andaslopefiledgame).
TheJugglerandtheSkyDiversubmodulesusesecond-orderdifferential
equations,butboththechapterandthemoduleexplainthetransformationto
systemsoftwofirst-orderdifferentialequations.
Module/Chapter 3: Some Cool ODEs
Assumed concepts: Basic concepts of first-order ODEs, solutions, and
solutioncurves
Newton’s law of cooling, and solving the resulting ODEs by separation
ofvariablesoraslinearequationswithintegratingfactors,arepresentedthor-
oughlyenoughthatthereneedbenoprerequisites.
ThesubmoduleforCoolingaHouseextendsNewton’slawofcoolingto
realworldcasesthatareeasilyhandledbyODEArchitect(andnotsoeasily
bytraditionalmethods). Thissectionmakesthepointthatrateequationsand
numericalsolutionsareoftenamuchsmarterwaytogothantotrudgetoward
asolutionformula.
Module/Chapter 4: Second-Order Linear Equations
Assumedconcepts: Euler’sformulaforcomplexexponentials
ThemoduleandchaptertreatonlyconstantcoefficientODEs. Thechap-
ter begins by demonstrating how to treat a second-order ODE as a system
of first-order ODEs which can be entered in ODE Architect. Both the first
submoduleand the chapterexplainfrom scratchall the traditional details of
anoscillatingsystemsuchasamplitude,period,frequency,damping,forcing,
andbeats.
TheSeismographsubmoduleis arealworld application. Thederivation
oftheequationofmotionisnotsimple,butthemultimediamodulegivesin-
sightintotheworkingsofaseismograph,anditisnotnecessarytounderstand
thedetailsofthederivationtouseandexplorethemodelingODE.