The Wright functions as solutions of the time-fractional diffusion equation

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1、AppliedMathematicsandComputation141(2003)51–62www.elsevier.com/locate/amcTheWrightfunctionsassolutionsofthetime-fractionaldiffusionequationa,*bFrancescoMainardi,GianniPagniniaDipartimentodiFisica,UniversitadiBolognaandINFN,SezionediBologna,ViaIrnerio46,I-40126Bo

2、logna,ItalybIstitutoperleScienzedell’AtmosferaedelClimadelCNR,ViaGobetti101,I-40129Bologna,ItalyAbstractWerevisittheCauchyproblemforthetime-fractionaldiffusionequation,whichisobtainedfromthestandarddiffusionequationbyreplacingthefirst-ordertimederiv-ativewithafract

3、ionalderivativeoforderb2ð0;2.ByusingtheFourier–Laplacetransformsthefundamentalssolutions(Greenfunctions)areshowntobehightran-scendentalfunctionsoftheWright-typethatcanbeinterpretedasspatialprobabilitydensityfunctionsevolvingintimewithsimilarityproperties.Weprovi

4、deageneralrepresentationofthesefunctionsintermsofMellin–Barnesintegralsusefulfornu-mericalcomputation.Ó2002ElsevierScienceInc.Allrightsreserved.Keywords:Fractionalderivatives;Laplacetransforms;Fouriertransforms;Mellin–Barnesinte-grals;Mittag–Lefflerfunctions;Wrigh

5、tfunctions;FoxH-functions1.IntroductionTime-fractionaldiffusionequations,obtainedfromthestandarddiffusionequationbyreplacingthefirst-ordertimederivativebyafractionalderivative(oforder0

6、ofauthors,see,e.g.thereviewsin[1,13,19],andreferencestherein.Inthispaperweintendtoprovidemoreinsightsforthe*Correspondingauthor.E-mailaddress:mainardi@bo.infn.it(F.Mainardi).0096-3003/02/$-seefrontmatterÓ2002ElsevierScienceInc.Allrightsreserved.doi:10.1016/S0096

7、-3003(02)00320-X52F.Mainardi,G.Pagnini/Appl.Math.Comput.141(2003)51–62fundamentalsolutionsofthegeneraltime-fractionaldiffusionequation,basedontherecentresultsbyMainardietal.[15].Bytime-fractionaldiffusionequationwemeantheevolutionequationobo2þuðx;tÞ¼uðx;tÞ;0

8、2R;t2R;ð1:1Þotbox20wherethetime-fractionalderivativeisintendedintheCaputosense.Foradetaileddiscussiononthisfractionalderivativewereferthereadertoe.g.[8,20].Whenbisnot