The Laplace Transform Method for Linear Differenti

arXiv:funct-an/9710005v130Oct1997SlovakAcademyofSciencesInstituteofExperimentalPhysicsTheLaplaceTransformMethodforLinearDifferentialEquationsoftheFractionalOrderIgorPodlubnyDepartmentofControlEngineeringFacultyofMining,UniversityofTechnologyB.Nemcovej3,04200Kosice,SlovakiaPhone:([phone]Fax:[phone]E-mail:[e-mail]-02-94,June1994Thisisthee-Printversionofthepre-printprintedin1994.Misprintswerecorrected.Presentaddress:DepartmentofManagementandControlEngineering,FacultyofB.E.R.G.,TechnicalUniversityofKosice,B.Nemcovej3,04200Kosice,SlovakRepublic.Phone:[phone];fax:[phone];e-mail:asabove.c1994,RNDr.IgorPodlubny,CSc.ThispublicationwastypesetbyLaTEX.ContentsPreface31Introduction51.1Mittag-Lefflerfunctionintwoparameters.............51.2TheLaplacetransformoftheMittag-Lefflerfunctionintwoparameters.............61.3TheWrightfunction.........................71.4Toolsfortestingcandidatesolutions................92Standardfractionaldifferentialequations102.1Ordinarylinearfractionaldifferentialequations.........................102.2Partiallinearfractionaldifferentialequations.........................123Sequentialfractionaldifferentialequations153.1TheLaplacetransformofasequentialfractionaldifferentialoperator....................153.2Ordinarylinearfractionaldifferentialequations.........................163.3Partiallinearfractionaldifferentialequations.........................174FractionalGreen’sfunction194.1Definitionandsomeproperties...................194.2FractionalGreen’sfunctionfortheone-termfractionaldifferentialequation....................224.3FractionalGreen’sfunctionforthetwo-termfractionaldifferentialequation....................224.4FractionalGreen’sfunctionforthethree-termfractionaldifferentialequation....................234.5FractionalGreen’sfunctionforthefour-termfractionaldifferentialequation....................244.6FractionalGreen’sfunctionforthegenerallinearfractionaldifferentialequation................25Bibliography271PrefaceDifferentialequationsofthefractionalorderappearmoreandmorefrequentlyindifferentresearchareasandengineeringapplications.Aneffectiveandeasy-to-usemethodforsolvingsuchequationsisneeded.However,knownmethodshavecertaindisadvantages.Methods,describedindetailsin[1,2,3]forfractionaldifferentialequationsoftherationalorder,donotworkinthecaseofanarbitraryrealorder.Ontheotherhand,thereisaniterationmethoddescribedin[5],whichallowssolutionoffractionaldifferentialequationsofanarbitraryrealorder,butitworkseffectivelyonlyforrelativelysimpleequations,aswellastheseriesmethod[1,17].Otherauthors(e.g.[3,4])usedintheirinvestigationstheone-parameterMittag-LefflerfunctionEα(z)=P∞k=0zkΓ(αk+1).Stillotherauthors[6,7]prefertheFoxH-function[8],whichseemstobetoogeneraltobefrequentlyusedinapplications.Insteadofthisvarietyofdifferentmethods,weintroduceamethodwhichisfreeofthementioneddisadvantagesandsuitableforawideclassofinitialvalueproblemsforfractionaldifferentialequations.ThemethodusestheLaplacetransformtechniqueandisbasedontheformulaoftheLaplacetransformoftheMittag-LefflerfunctionintwoparametersEα,β(z).Wehopethatthede-scribedmethodcouldbeusefulforobtainingsolutionsofdifferentappliedprob-lems,appearinginphysics,chemistry,electrochemistry,engineering,financingandbanking,etc.Tooutlinetheareaofthemethod’sapplicability,wehaveincludedinthebibliographyalsotheworksbydifferentauthors[30]–[54],inwhichfractionallineardifferentialequationsappearsorwhichcouldserveasabasisforobtainingsuchequations.Thisworkdealswithsolutionofthefractionallineardifferentialequationswithconstantcoefficientsandconsistsoffourshortchapters.InChapter1wepresentsomeauxillarytoolswhicharenecessaryforus-ingthemethod.ThereadercanfindthedefinitionthereandsomeimportantpropertiesoftheMittag-LefflerfunctionintwoparametersandtheWrightfunc-tion.Thebasicresult,presentedinChapter1,istheLaplacetransformoftheMittag-Lefflerfunctionanditsderivatives.Besidesthat,weintroducetwotoolsnecessaryfortestingcandidatesolutionsbydirectsubstitutionincorrespondingequations:fractionalderivativesoftheMittag-Lefflerfunctionandaruleforthefractionaldifferentiationofintegralsdependingonaparameter.InChapter2wegivesolutionstosomeinitial-valueproblemsfor”standard”fractionaldifferentialequations.Someofthemweresolvedbyotherauthorsearlierbyothermethods,andthecomparisoninsuchcasesjustunderlinethe3simplicityandthepowerofourapproach.InChapter3weextendtheproposedmethodforthecaseofso-called”se-quential”fractionaldifferentialequations(weadoptedtheconvenientterminol-ogyofMillerandRoss).Forthispurpose,weobtainedtheLaplacetransformforthe”sequential”fractionalderivative.The”sequential”analoguesoftheproblems,solvedinChapter2,areconsidered.Naturally,wearriveatsolutionswhicharedifferentfromthoseobtainedinChapter2.However,thereissomethingcommoninsolutionsofthecorresponding”standard”and”sequential”fractionaldifferentialequations:theybothhavethesamefractionalGreen’sfunction.InChapter4wegiveourdefinitionofthefractionalGreen’sfunctionandsomeofitsproperties,necessaryforconstruct-ingsolutionsofinitial-valueproblemsforfractiona