Mathematical Modeling, Numerical Methods, and Problem Solving

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1、9/12/2013Part1Chapter1MathematicalModeling,NumericalMethods,andProblemSolvingPowerPointsorganizedbyDr.MichaelR.GustafsonII,DukeUniversityandProf.SteveChapra,TuftsUniversityAllimagescopyright©TheMcGraw-HillCompanies,Inc.Permissionrequiredforreproductionordisp

2、lay.ChapterObjectives•Learninghowmathematicalmodelscanbeformulatedonthebasisofscientificprinciplestosimulatethebehaviorofasimplephysicalsystem.•Understandinghownumericalmethodsaffordameanstogeneralizesolutionsinamannerthatcanbeimplementedonadigitalcomputer.•

3、Understandingthedifferenttypesofconservationlawsthatliebeneaththemodelsusedinthevariousengineeringdisciplinesandappreciatingthedifferencebetweensteady-stateanddynamicsolutionsofthesemodels.•Learningaboutthedifferenttypesofnumericalmethodswewillcoverinthisboo

4、k.19/12/2013ASimpleMathematicalModel•Amathematicalmodelcanbebroadlydefinedasaformulationorequationthatexpressestheessentialfeaturesofaphysicalsystemorprocessinmathematicalterms.•Modelscanberepresentedbyafunctionalrelationshipbetweendependentvariables,indepen

5、dentvariables,parameters,andforcingfunctions.ModelFunctionDependentindependentforcingf,parameters,variablevariablesfunctions•Dependentvariable-acharacteristicthatusuallyreflectsthebehaviororstateofthesystem•Independentvariables-dimensions,such

6、astimeandspace,alongwhichthesystem’sbehaviorisbeingdetermined•Parameters-constantsreflectiveofthesystem’spropertiesorcomposition•Forcingfunctions-externalinfluencesactinguponthesystem29/12/2013ModelFunctionExample•Assumingabungeejumperisinmid-flight,ananalyt

7、icalmodelforthejumper’svelocity,accountingfordrag,isgmgcdvttanhtcdm•Dependentvariable-velocityv•Independentvariables-timet•Parameters-massm,dragcoefficientcd•Forcingfunction-gravitationalaccelerationgModelResults•Usingacomputer(oracalculator

8、),themodelcanbeusedtogenerateagraphicalrepresentationofthesystem.Forexample,thegraphbelowrepresentsthevelocityofa68.1kgjumper,assumingadragcoefficientof0.25kg/m39/12/2013NumericalModeling•Somesy