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1、HindawiPublishingCorporationDiscreteDynamicsinNatureandSocietyVolume2011,ArticleID562494,15pagesdoi:10.1155/2011/562494ResearchArticleOnRiemann-LiouvilleandCaputoDerivativesChangpinLi,1DeliangQian,2andYangQuanChen31DepartmentofMathematics,ShanghaiUniversity,Shanghai200444,China2Depart
2、mentofMathematics,ZhongyuanUniversityofTechnology,Zhengzhou450007,China3DepartmentofElectricalandComputerEngineering,UtahStateUniversity,Logan,UT84322-4120,USACorrespondenceshouldbeaddressedtoChangpinLi,lcp@shu.edu.cnReceived27June2010;Accepted25January2011AcademicEditor:DanielCzamans
3、kiCopyrightq2011ChangpinLietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.Recently,manymodelsareformulatedintermsoffractionalderivatives,suchas
4、incontrolprocessing,viscoelasticity,signalprocessing,andanomalousdiffusion.Inthepresentpaper,wefurtherstudytheimportantpropertiesoftheRiemann-LiouvilleRLderivative,oneofmostlyusedfractionalderivatives.SomeimportantpropertiesoftheCaputoderivativewhichhavenotbeendiscussedelsewherearesi
5、multaneouslymentioned.Thepartialfractionalderivativesarealsointroduced.Thesediscussionsarebeneficialinunderstandingfractionalcalculusandmodelingfractionalequationsinscienceandengineering.1.IntroductionFractionalcalculusisnotanewtopic;inrealityithasalmostthesamehistoryasthatoftheclassic
6、alcalculus1.Sincetheoccurrenceoffractionalorfractional-orderderivative,thetheoriesoffractionalcalculusfractionalderivativeplusfractionalintegralhasundergoneasignificantandevenheateddevelopment,whichhasbeenprimarilycontributedbypurebutnotappliedmathematicians;thereadercanrefertoan
7、encyclopedicbook2andmanyreferencescitedtherein.Inthelastfewdecades,however,appliedscientistsandengineersrealizedthatdifferentialequationswithfractionalderivativeprovidedanaturalframeworkforthediscussionofvariouskindsofrealproblemsmodeledbytheaidoffractionalderivative,suchasviscoelast
8、icsys
