數(shù)學(xué)建模課件

MathematicalModeling數(shù)學(xué)建模(英文版)機(jī)械工業(yè)出版社,北京,2003.5經(jīng)典原版書庫,原書名:AFirstCourseinMathematicalModeling(ThirdEdition)byFrankR.Giordano,MauriceD.Weir,WilliamP.Fox1Chapter1ModelingChangeIntroduction Weoftendescribeaparticularphenomenonmathematically(bymeansofafunctionoranequation,forinstance). Suchamathematicalmodelisanidealizationofthereal-worldphenomenonandneveracompletelyaccuraterepresentation.2MathematicalModels Weareofteninterestedinpredictingthevalueofavariableatsometimeinthefuture.Amathematicalmodelcanhelpusunderstandabehaviorbetteroraidusinplanningforthefuture.

Let'sthinkofamathematicalmodelasamathematicalconstructdesignedtostudyaparticularreal-worldsystemorbehaviorofinterest.3 Themodelallowsustoreachmathematicalconclusionsaboutthe behavior,asillustratedinFigure1.1. Theseconclusionscanbeinterpretedtohelpadecisionmakerplanforthefuture. Inthischapterwedirectourattentiontomodelingchange.4Figure1.1Aflowofthemodelingprocessbeginningwithanexaminationofreal-worlddataReal-worlddataModelMathematicalconclusionsPredictions/explanationssimplificationAnalysisVerificationInterpretation5Simplification Mostmodelssimplifyreality.Generally,modelscanonlyapproximatereal-worldbehavior.Oneverypowerfulsimplifyingrelationshipisproportionality.6

DefinitionTwovariablesyandxareproportional(toeachother)ifoneisalwaysaconstantmultipleoftheother,thatis,ify=kxforsomenonzeroconstantk.Wewritey

x. Thedefinitionmeansthatthegraphofyversusxliesalongastraightlinethroughtheorigin.Thisgraphicalobservationisusefulintestingwhetheragivendatacollectionreasonablyassumesaproportionalityrelationship.7Example1

TestingforProportionality

Consideraspring-masssystem(Figure1.2).Weconductanexperimenttomeasurethestretchofthespringasafunctionofthemass(measuredasweight)placedonthespring.Considerthedatacollectedforthisexperiment,displayedinTable1.1.Figure1.2Spring-masssystem8Table1.1Spring-masssystemMass50100150200250Elongation1.0001.8752.7503.2504.3753003504004505005504.8755.6756.5007.2508.0008.7509

Ascatterplotgraphofthestretchorelongationofthespringversusthemassorweightplacedonitrevealsanapproximatestraightlinepassingthroughtheorigin.Figure1.3Datafromspring-masssystem10 Thedataappeartofollowtheproportionalityrulethatelongationeisproportionaltothemassm,orsymbolically,e

m. Wecalculatetheslopeofthelinejoiningthesepointsas Andthemodelisestimatedase=0.0163m.11 Byplottingthelinethemodelrepresentssuperimposedonthescatterplot(Figure1.4),thegraphrevealsthatthesimplifyingproportionalitymodelisreasonable.Figure1.4Datafromspring-masssystem12ModelingChange Apowerfulparadigmtouseinmodelingchangeis

futurevalue=presentvalue+change. Often,wewishtopredictthefutureonwhatweknownowandthechangethathasbeencarefullyobserved.Insuchcases,webeginbystudyingthechangeitselfaccordingtotheformula

change=futurevalue

presentvalue.13 Ifthebehaviorofinterestistakingplaceoverdiscretetimeperiods,theprecedingconstructleadstoadifferenceequation. Otherwise,ifthebehavioristakingplacecontinuouslywithrespecttotime,thentheconstructleadstoadifferentialequation.141.1ModelingChangewithDifferenceEquations

DefinitionForasequenceofnumbersA={a0,a1,a2,…},thenthfirstdifferencesare

an=an+1

an,n=0,1,2,…

NotefromFigure1.5thatthedifferencerepresentstheriseorfallbetweenconsecutivevaluesofthesequence.15Figure1.5Thefirstdifferenceofasequenceistheriseinthegraphduringonetimeperiod16Example1ASavingsCertificate Considerthevalueofasavingscertificateinitiallyworth$1000thataccumulatesinterestpaideachmonthat1%permonth.Thefollowingsequenceofnumbersrepresentsthevalueofthecertificatemonthbymonth:A={1000,1010,1020.10,1030.30,…}.17 ThefirstdifferenceofAareasfollows: Thisexpressioncanberewrittenasthedifferenceequation:whichgivesthedynamicalsystemmodel:18 Equation(1.1)representsaninfinitesetofalgebraicequations,calledadynamicalsystem. Dynamicalsystemsallowustodescribethechangefromoneperiodtothenext. Thedifferenceequationformulacomputesthenexttermknowingtheimmediatelypreviousterminthesequence,butitdoesnotcomputethevalueofaspecifictermdirectly(e.g.,thesavingsafter100periods).19 Tomodifyourexample,ifweweretowithdraw$50fromtheaccounteachmonth,thechangeduringaperiodwouldbetheinterestearnedduringthatperiodminusthemonthlywithdrawal,or20 Inmostexamples,mathematicallydescribingthechangeisnotgoingtobeaspreciseaprocedureasillustratedhere.Oftenitisnecessarytoplotthechange,observeapattern,andthendescribethechangeinmathematicalterms.Thatis,wewillbetryingtofindchange=

an=somefunctionf.21 Thechangemaybeafunctionofprevioustermsinthesequence,oritmayalsoinvolvesomeexternalterms.Thus,wewillbemodelingchangeindiscreteintervalsthisway:

change=

an=an+1

an

=f(termsinthesequence,externalterms). Modelingchangeinthiswaybecomestheartofdeterminingorapproximatingafunctionfthatrepresentsthechange.22Example2MortgagingaHome Sixyearsagoyourparentspurchasedahomebyfinancing$80000for20years,payingmonthlypaymentsof$880.87withamonthlyinterestof1%. Theyhavemade72paymentsandwishtoknowhowmuchtheyoweonthemortgage,whichtheyareconsideringpayingoffwithaninheritancetheyreceived.23 Thechangeintheamountowedeachperiodincreasesbytheamountofinterestanddecreasesbytheamountofthepayment: Solvingforbn+1andincorporatingtheinitialconditiongivesthedynamicalsystemmodel24Thus,yieldingthesequenceB={80000,79919.13,79837.45,…}. ThesequenceisgraphedinFigure1.6.ThefigureisplottedwithMatlab,b72=71532,b241=025Figure1.6ThesequenceandgraphforExample226 Inthissectionwehavediscussedbehaviorsintheworldthatcanbemodeledexactlybydifferenceequations.Inthenextsection,weusedifferenceequationtoapproximateobservedchange.Aftercollectingdataforthechangeanddiscerningpatternsofthebehavior,wewillusetheconceptofproportionalitytotestandfitmodelsthatwepropose.271.2ApproximatingChangewithDifferenceEquations

Inmostexamples,describingthechangemathematicallywillnotbeaspreciseaprocedureasinthesavingscertificateandmortgageexamplespresentedintheprevioussection.Typically,wemustplotthechange,observeapattern,andthenapproximatethechangeinmathematicalterms.28Example1GrowthofaYeastCulture Thedatainthetablebellowwascollectedfromanexperimentmeasuringthegrowthofayeastculture.TheGraph1.7representstheassumptionthatthechangeinpopulationisproportionaltothecurrentsizeofthepopulation.Thatis,

pn=pn+1

pn=kpn,wherepnrepresentsthesizeofthepopulationbiomassafternhours,andkisapositiveconstant.Thevalueofkdependsonthetimemeasurement.Inthisexamplek

0.5.29Timeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.7

30Figure1.7Growthofayeastcultureversusbiomass31 Usingtheestimatek=0.5fortheslopeoftheline,wehypothesizetheproportionalitymodel

pn=pn+1

pn=0.5pn,yieldingthepredictionpn+1=1.5pn. Thismodelpredictsapopulationthatincreasesforever,whichisquestionable.32

ModelRefinement:ModelingBirths,Deaths,andResources Ifbothbirthsanddeathsduringaperiodareproportionaltothepopulation,thenthechangeinpopulationshouldbeproportionaltothepopulation,aswasillustratedinExample1.However,certainresources(e.g.,food)cansupportonlyamaximumpopulationlevelratherthanonethatincreasesindefinitely.Asthesemaximumlevelsareapproached,growthshouldslow.33Example2GrowthofaYeastCultureRevisited FindingaModelThedatainFigure1.8showwhatactuallyhappenstotheyeastculturegrowinginarestrictedareaastimeincreasesbeyondtheeightobservationsgiveninFigure1.734Timeinhoursn01234567Observedyeastbiomasspn9.618.329.047.271.1119.1174.6257.3Changeinbiomasspn+1

pn8.710.718.223.948.055.582.793.489101112131415161718350.7441.0513.3559.7594.8629.4640.8651.1655.9659.6661.190.372.346.435.134.611.410.34.83.72.2

35Figure1.8Yeastbiomassapproachesalimitingpopulationlevel36 Fromthethirdrowofthedatatablenotethatthechangeinpopulationperhourbecomessmallerastheresourcesbecomemorelimitedorconstrained.Fromthegraphofpopulationversustime,thepopulationappearstobeapproachingalimitingvalueorcarryingcapacity.Basedonourgraphweestimatethecarryingcapacitytobe665.37 Because665

pngetssmalleraspnapproaches665,weproposethemodel

pn=pn+1

pn=k(665

pn)pn,whichcausesthechange

ptobecomeincreasinglysmallaspnapproaches665. Mathematically,thishypothesizedmodelstatesthatthechange

pisproportionaltotheproduct(665

pn)pn. Totestthemodel,plot(pn+1

pn)versus(665

pn)pntoseeifthereisareasonableproportionality.Thenestimatetheproportionalityconstantk.38pn+1

pn8.710.718.223.948.055.5pn(665

pn)6291.8411834.6118444.0029160.1642226.2965016.6911.410.34.83.72.222406.6415507.369050.295968.693561.8434.641754.9682.793.490.372.346.435.185623.84104901.21110225.0198784.0077867.6158936.41Dataofpn+1

pnversus(665

pn)pn39Figure1.9Testingtheconstrainedgrowthmodel40 ExaminingFigure1.9,weseethattheplotdoesreasonablyapproximateastraightlineprojectedthroughtheorigin.Weestimatetheslopeofthelineapproximatingthedatatobeaboutk

0.00082,whichgivesthemodelpn+1

pn=0.00082(665

pn)pn.(1.2)SolvingtheModelNumerically

SolvingEquation(1.2)forpn+1givespn+1=pn+0.00082(665

pn)pn,(1.3)whichgivesadynamicalsystemmodelwiththeinitialvaluep0=9.6.41 ThisnumericalsolutionofmodelpredictionsispresentedinFigure1.10.Thepredictionsandobservationsareplottedtogetherversustimeonthesamegraph.Notethatthemodelcapturesfairlywellthetrendoftheobserveddata.42Figure1.10Modelpredictionsandobservations43Example3SpreadofaContagiousDisease Supposethereare400studentsinacollegedormitoryandthatoneormorestudentshaveaseverecaseoftheflu.Letinrepresentthenumberofinfectedstudentsafterntimeperiods. Assumesomeinteractionbetweenthoseinfectedandthosenotinfectedisrequiredtopassonthedisease.44 Ifallaresusceptibletothedisease,then400

inrepresentsthosesusceptiblebutnotyetinfected.Ifthoseinfectedremaincontagious,wecanmodelthechangeofthoseinfectedasaproportionalitytotheproductofthoseinfectedbythosesusceptiblebutnotyetinfected,or

in=in+1

in=kin(400

in).(1.4)45 Inthismodeltheproductin(400

in)representsthenumberofpossibleinteractionsbetweenthoseinfectedandthosenotinfectedattimen. Afractionkoftheseinteractionswouldcauseadditionalinfections,representedby

in.46 Equation(1.4)hasthesameformasEquation(1.2),butintheabsenceofanydatawecannotdetermineavaluefortheproportionalityconstantk.Nevertheless,agraphofthepredictionsdeterminedbyEquation(1.4)wouldhavethesameSshapeastheyeastpopulationinFigure1.10.47Example4

DecayofDigoxinintheBloodstream Digoxinisusedinthetreatmentofheartdisease.Doctormustprescribeanamountofmedicinethatkeepstheconcentrationofdigoxininthebloodstreamaboveaneffectivelevelwithoutexceedingasafelevel(thereisvariationamongpatients).48 Foraninitialdosageof0.5mginthebloodstream,tablebelowshowstheamountof