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1、FractionalCalculus:History,De¯nitionsandApplicationsfortheEngineerAdamLoverroDepartmentofAerospaceandMechanicalEngineeringUniversityofNotreDameNotreDame,IN46556,U.S.A.May8,2004AbstractThisreportisaimedattheengineeringand/orscienti¯cprofessionalwhowishestolearnaboutFrac-tionalCal
2、culusanditspossibleapplicationsinhis/her¯eld(s)ofstudy.Theintentisto¯rstexposethereadertotheconcepts,applicablede¯nitions,andexecutionoffractionalcalculus(includingadiscussionofnotation,operators,andfractionalorderdi®erentialequations),andsecondtoshowhowthesemaybeusedtosolveseve
3、ralmodernproblems.Alsoincludedwithinisalistofapplicablereferencesthatmayprovidethereaderwithalibraryofinformationforthefurtherstudyanduseoffractionalcalculus.1IntroductionThetraditionalintegralandderivativeare,tosaytheleast,astapleforthetechnologyprofessional,essentialasameansof
4、understandingandworkingwithnaturalandarti¯cialsystems.FractionalCalculusisa¯eldofmathematicstudythatgrowsoutofthetraditionalde¯nitionsofthecalculusintegralandderivativeoperatorsinmuchthesamewayfractionalexponentsisanoutgrowthofexponentswithintegervalue.Considerthephysicalmeaning
5、oftheexponent.Accordingtoourprimaryschoolteachersexponentsprovideashortnotationforwhatisessentiallyarepeatedmultiplicationofanumericalvalue.Thisconceptinitselfiseasytograspandstraightforward.However,thisphysicalde¯nitioncanclearlybecomeconfusedwhenconsideringexponentsofnonintege
6、rvalue.Whilealmostanyonecanverify1thatx3=x¦x¦x,howmightonedescribethephysicalmeaningofx3:4,ormoreoverthetranscendentalexponentx¼.Onecannotconceivewhatitmightbeliketomultiplyanumberorquantitybyitself3.4times,or¼times,andyettheseexpressionshaveade¯nitevalueforanyvaluex,veri¯ableby
7、in¯niteseriesexpansion,ormorepractically,bycalculator.Now,inthesamewayconsidertheintegralandderivative.Althoughtheyareindeedcon-ceptsofahighercomplexitybynature,itisstillfairlyeasytophysicallyrepresenttheirmeaning.Oncemastered,theideaofcompletingnumerousoftheseoperations,integra
8、tionsordi®erentiationsfollowsnaturally.Giventhe
