微分方程,英文版3

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1、Modelingwithfirstorderequations(Sect.2.3).IMainexample:Saltinawatertank.ITheexperimentaldevice.IThemainequations.IAnalysisofthemathematicalmodel.IPredictionsforparticularsituations.Saltinawatertank.Problem:Describethesaltconcentrationinatankwithwaterifsaltywatercomesin

2、andgoesoutofthetank.Mainideasofthetest:ISincethemassofsaltandwaterisconserved,weconstructamathematicalmodelforthesaltconcentrationinwater.ITheamountofsaltinthetankdependsonthesaltconcentrationcominginandgoingoutofthetank.IThesaltinthetankalsodependsonthewaterratescomi

3、nginandgoingoutofthetank.IToconstructamodelmeanstofindthedifferentialequationthattakesintoaccounttheabovepropertiesofthesystem.IFindingthesolutiontothedifferentialequationwithaparticularinitialconditionmeanswecanpredicttheevolutionofthesaltinthetankifweknowthetankinitial

4、condition.Modelingwithfirstorderequations(Sect.2.3).IMainexample:Saltinawatertank.ITheexperimentaldevice.IThemainequations.IAnalysisofthemathematicalmodel.IPredictionsforparticularsituations.Theexperimentaldevice.rq(t)iiwatersaltpipetankV(t)Q(t)roq(t)oinstantaneouslymi

5、xedTheexperimentaldevice.Definitions:Iri(t),ro(t):Ratesinandoutofwaterenteringandleavingthetankatthetimet.Iqi(t),qo(t):Saltconcentrationofthewaterenteringandleavingthetankatthetimet.IV(t):Watervolumeinthetankatthetimet.IQ(t):Saltmassinthetankatthetimet.Units:Volume

6、Massri(t)=ro(t)=,qi(t)=qo(t)=.TimeVolumeV(t)=Volume,Q(t)=Mass.Modelingwithfirstorderequations(Sect.2.3).IMainexample:Saltinawatertank.ITheexperimentaldevice.IThemainequations.IAnalysisofthemathematicalmodel.IPredictionsforparticularsituations.Themainequations.R

7、emark:Themassconservationprovidesthemainequationsofthemathematicaldescriptionforsaltinwater.Mainequations:dV(t)=ri(t)−ro(t),Volumeconservation,(1)dtdQ(t)=ri(t)qi(t)−ro(t)qo(t),Massconservation,(2)dtQ(t)qo(t)=,Instantaneouslymixed,(3)V(t)ri,ro:Constants.(4)Themainequat

8、ions.Remarks:hihidVVolume==ri−ro,dtTimehihidQMass==riqi−roqo,dtTimehiVolumeMassMassriqi−roqo==.TimeVolumeTimeModelingwithfirs