Ode Architect Companion

286 Pages·1999·2.3 MB·English

O D E A R C H I T E C T
Companion
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
Copyright c 1999,byJohnWiley&Sons,Inc.
(cid:13)
Allrightsreserved.
Reproductionortranslationofanypartofthiswork
beyondthatpermittedbySections107and108of
the1976UnitedStatesCopyrightActwithoutthe
permissionofthecopyrightownerisunlawful.
Requestsforpermissionorfurtherinformation
shouldbeaddressedtothePermissionsDepartment,
JohnWiley&Sons,Inc.
ISBN0-471-17007-0
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Printedandboundby
PREFACE
This workbook was designed to accompany the 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,
relatedtothechapterandmoduletopics. TheExplorationpagescanbepho-
tocopiedso that youcan write in answers andderivations,and handthem in
along with printouts of graphs produced by the Architect. There is also a
notepadfacilityintheArchitectwhich,withthecutandpastefeatures,makes
itpossibletowritereports.
ODE Architect
ODEArchitectprovidesahighlyinteractiveenvironmentforconstructingand
exploringyourownmathematicalmodelsofreal-worldphenomena,whether
they lead to linear or nonlinear systems 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
libraryof ODEs, the ODE Architectbringsa wealthof opportunitiesto gain
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 presented with a title screen followed by a main menu al-
lowing selection of a specific module. You may prefer to go directly to the
ArchitectTooltorunyourownexperiments.Atanyplaceinthesoftware,you
willbeabletocallupthecontentsmenusandaccessthematerialinanyorder.
We expect that most will work through the multimedia modules. Let’s look
ateachof thethreeprincipalpartsoftheODE Architectinmoredetail: The
MultimediaODEArchitect,TheODEArchitectTool,andtheODELibrary.
vi
Multimedia ODE Architect
. .
CODEE members and colleagues have authored the multimedia modules,
each with its own theme. The modeling process is detailed, supported by
highlyinteractivesimulations. Youwillexploretheproblem-solvingprocess
via“what-if”scenariosandexercises.YouareguidedtobuildyourownODEs
andsolvethemnumericallyandgraphically,andcomparethepredictedresults
toempiricaldatawhenappropriate.
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,andthenendwith somequestions(Things-to-Think-About,or
TTAs). These questionsextend the topics of the submoduleand take you to
the solver tool to produce solution curves and orbits, or write a report con-
necting the mathematics, the models, and the pictures. When you open the
ToolusingaTTAlink,thepertinentequationsandparametersettingswillau-
tomatically be entered into the equation quadrant of the Tool. You are then
poisedtothinkabout,andwithoutconstraint,explorethemodelintroducedin
thesubmodule.
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,solve,andexploreODEs
(cid:15)
Inputdatatables
(cid:15)
Graph and animate solution curves, phase plane graphs, 3D graphs,
(cid:15)
Poincare´ sections,discretemaps,directionfields,etc.
Build,analyze,andanimatephysicalrepresentationsofdynamicalsys-
(cid:15)
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 dynamical systems is also available from the Tool. A variety of
engineeringfunctionssuchas squarewaves, sawtoothwaves, andstep func-
tions are included in the Tool function library. Two- and three-dimensional
graphicsaresupported,aswellastimeandparameteranimationsofsolution
vii
data. Initial conditions can be entered by clicking in a graph window or via
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-sweep tool. You can do pa-
rameter and initial-value sweeps to see the effects of data changes on orbits
and solution curves. Graphs are editable and you can scale and label axes,
mark equidistant-in-time orbital points, color the graphs, change line 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., “Teaching Numerical Methods in
ODECourses”
inthebookRevolutionsinDifferentialEquations,editedbyMichaelJ.Kalla-
herintheMAANotesseries.
Module1,“ModelingwiththeODEArchitect”,isanon-linetutorialfor
manyofthefeaturesoftheTool. TheArchitectalsohashelpfacilitiesandthe
multimediasideisself-documenting.
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 explanatory text
alongwiththeequationsandincludesanillustrativegraphorgraphs. TheLi-
braryfilesareorganizedintofoldersbytopicandtheyhavedescriptivetitles
tofacilitatebrowsing. Thesefilesalsoprovideamarvelouswaytolearnhow
tousethetool.
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. CODEE saw to the mathematical side of things, Intel-
. .
liprorenderedCODEE’s workintoanattractivemultimediasoftwarepack-
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 CODEE 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 big “thank you” to our students Tiffany Arnal, Claire Lau-
nay, NathanJakubiak,JohnLu, Joel Miller, Justin Radick,Paul SanGiorgio,
TreasaSweek,andmanyotherswhoreadchapters,testedmodules,andcom-
mentedfreely(evenfavorably)onwhattheyexperienced.
RobertL.Borrelli
CourtneyS.Coleman
Claremont,CA
1TheworkonODEArchitectanditsCompanionbookwassupportedinpartbytheNationalSci-
enceFoundationunderGrantNumbersDMI-9509135andDUE-9450742. Anyopinions, findings,
conclusions,orrecommendationsexpressedinthismaterialarethoseoftheauthorsanddonotneces-
sarilyreflecttheviewsoftheNationalScienceFoundation.
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 involve higher-order ODEs and systems and their
applications. Once students understand how to deal with two-dimen-
sional systems graphically, any of these modules/chaptersis easily ac-
cessible.
Modules/Chapters10–12 applytwo-dimensionalsystemstomodelsthatil-
lustratemoreadvancedtechniquesandtheory;themultimediaapproach
makesthemneverthelessquiteaccessible. Themodulesareintendedto
enablestudentstogetmuchfurtherwiththetechnicalaspectsexplained
inthechaptersthanwouldbeotherwisepossible.
Module/Chapter13 treats discrete dynamical systems in an introductory
fashion 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 modules can be accessedin different orders. It is not expectedthat
theywill beassignedinnumericalorder. Consequently,wehavetriedto ex-
plain each concept wherever it appears, or to indicate where an explanation
isprovided. For example,Newton’ssecondlaw, F =ma,is describedevery
timeitisinvoked.
x
ThereisfarmorematerialinODEArchitectthancouldpossiblyfitintoa
singlecourse.
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(andaslopefieldgame).
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
realworldcasesthatareeasilyhandledbyODE Architect(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 chapter explain from scratch all the traditional details of
anoscillatingsystemsuchasamplitude,period,frequency,damping,forcing,
andbeats.
The Seismographsubmoduleis a real world application. The derivation
oftheequationofmotionis notsimple,butthemultimediamodulegivesin-
sightintotheworkingsofaseismograph,anditisnotnecessarytounderstand
thedetailsofthederivationtouseandexplorethemodelingODE.