




版权说明:本文档由用户提供并上传,收益归属内容提供方,若内容存在侵权,请进行举报或认领
文档简介
MathematicalModeling数学建模(英文版)机械工业出版社,北京,2003.5经典原版书库,原书名: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
温馨提示
- 1. 本站所有资源如无特殊说明,都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
- 2. 本站的文档不包含任何第三方提供的附件图纸等,如果需要附件,请联系上传者。文件的所有权益归上传用户所有。
- 3. 本站RAR压缩包中若带图纸,网页内容里面会有图纸预览,若没有图纸预览就没有图纸。
- 4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
- 5. 人人文库网仅提供信息存储空间,仅对用户上传内容的表现方式做保护处理,对用户上传分享的文档内容本身不做任何修改或编辑,并不能对任何下载内容负责。
- 6. 下载文件中如有侵权或不适当内容,请与我们联系,我们立即纠正。
- 7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
最新文档
- 土壤污染修复技术碳排放影响研究:2025年应用效果与成本效益报告
- 基金管理人评价-洞察及研究
- 广西贵港高一数学试卷
- 韩国高中纯韩语数学试卷
- 广东省佛山中考数学试卷
- 合肥初三中考数学试卷
- 哈尔滨初中数学试卷
- 多灾种智慧预警系统-洞察及研究
- 广丰区联考初二数学试卷
- 湖南省中考统考2024数学试卷
- 2024届黑龙江省哈师大附属中学物理高二下期末统考试题含解析
- 护士重症监护室护理的进修
- 康复科护理中的疼痛管理
- 医疗机构卫生法律制度与监督-
- 液压动力系统中英文对照外文翻译文献
- 已有设施管线的加固和保护等特殊情况下的施工措施
- 商品混凝土技术规格书
- 决策力与执行力
- 加压法制备硫酸铝新技术
- 第7课日本の秋は涼しいです课件高中日语华东理工版新编日语教程1
- 2022年FIBA篮球竞赛规则
评论
0/150
提交评论