Existence of solutions for a class of fractional b

ComputersandMathematicswithApplications62(2011)1181–1199ContentslistsavailableatScienceDirectComputersandMathematicswithApplicationsjournalhomepage:✩FengJiaoa,YongZhoub,∗aSchoolofMathematicsandInformationSciences,GuangzhouUniversity,Guangdong510006,PRChinabSchoolofMathematicsandComputationalScience,XiangtanUniversity,Hunan411105,PRChinaarticleinfoKeywords:FractionaldifferentialequationsBoundaryvalueproblemFractionaladvection–dispersionequationCriticalpointtheoryExistenceabstractInthispaper,bythecriticalpointtheory,anewapproachisprovidedtostudytheexistenceofsolutionstothefollowingfractionalboundaryvalueproblem:ddt120D−βt(u′(t))+12tD−βT(u′(t))+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,where0D−βtandtD−βTaretheleftandrightRiemann–Liouvillefractionalintegralsoforder0≤β1respectively,F:[0,T]×RN→Risagivenfunctionand∇F(t,x)isthegradientofFatx.Ourinterestinthisproblemarisesfromthefractionaladvection–dispersionequation(seeSection2).Thevariationalstructureisestablishedandvariouscriteriaontheexistenceofsolutionsareobtained.©2011ElsevierLtd.Allrightsreserved.1.IntroductionFractionaldifferentialequationshaverecentlybeenprovedtobevaluabletoolsinthemodelingofmanyphenomenainvariousfieldsofscienceandengineering.Indeed,wecanfindnumerousapplicationsinviscoelasticity,neurons,electrochemistry,control,porousmedia,electromagnetism,etc.,(see[1–6]).Therehasbeensignificantdevelopmentinfractionaldifferentialequationsinrecentyears;seethemonographsofKilbasetal.[7],MillerandRoss[8],Podlubny[9],Samkoetal.[10]andthepapers[11–30]andthereferencestherein.Inthispaper,weconsiderthefractionalboundaryvalueproblem(BVP)ofthefollowingformddt120D−βt(u′(t))+12tD−βT(u′(t))+∇F(t,u(t))=0,a.e.t∈[0,T],u(0)=u(T)=0,(1)where0D−βtandtD−βTaretheleftandrightRiemann–Liouvillefractionalintegralsoforder0≤β1respectively,F:[0,T]×RN→Risagivenfunctionsatisfyingsomeassumptionsand∇F(t,x)isthegradientofFatx.Inparticular,ifβ=1,BVP(1)reducestothestandardsecond-orderboundaryvalueproblem.Physicalmodelscontainingfractionaldifferentialoperatorshaverecentlyrenewedattentionfromscientistswhichismainlyduetoapplicationsasmodelsforphysicalphenomenaexhibitinganomalousdiffusion.Astrongmotivationfor✩ThisprojectissupportedbytheNationalScienceFoundationofChina(No.10971173).∗Correspondingauthor.E-mailaddresses:[e-mail](F.Jiao),[e-mail](Y.Zhou).0898-1221/$–seefrontmatter©2011ElsevierLtd.Allrightsreserved.doi:10.1016/j.camwa.2011.03.0861182F.Jiao,Y.Zhou/ComputersandMathematicswithApplications62(2011)1181–1199investigatingthefractionalBVP(1)comesfromthefractionaladvection–dispersionequation(ADE).AfractionalADEisageneralizationoftheclassicalADEinwhichthesecond-orderderivativeisreplacedwithafractional-orderderivative.IncontrasttotheclassicalADE,thefractionalADEhassolutionsthatresemblethehighlyskewedandheavy-tailedbreakthroughcurvesobservedinfieldandlaboratorystudies[11,12],inparticularincontaminanttransportofground-waterflow[13].In[13],Bensonetal.statethatsolutesmovingthroughahighlyheterogeneousaquiferviolationsviolatethebasicassumptionsoflocalsecond-ordertheoriesbecauseoflargedeviationsfromthestochasticprocessofBrownianmotion.Letφ(t,x)representtheconcentrationofasoluteatapointxattimetinanarbitraryboundedconnectedsetΩ⊂RN.Accordingto[12,15],theN-dimensionalformofthefractionalADEcanbewrittenas∂φ∂t=−∇(vφ)−∇(∇−β(−k∇φ))+f,inΩ,(2)wherevisaconstantmeanvelocity,kisaconstantdispersioncoefficient,vφand−k∇φdenotethemassfluxfromadvectionanddispersionrespectively.Thecomponentsof∇−βin(2)arealinearcombinationoftheleftandrightRiemann–Liouvillefractionalintegraloperators(∇−β(−k∇φ))i=(q−∞D−βxi+(1−q)xiD−β∞)−k∂φ∂xi,i=1,...,N,(3)whereq∈[0,1]describestheskewnessofthetransportprocess,andβ∈[0,1)istheorderoftheRiemann–Liouvilleleftandrightfractionalintegraloperatorsontherealline(seeSection2,(12)and(13)).Thisequationmaybeinterpretedasstatingthatthemassfluxofaparticleisrelatedtothenegativegradientviaacombinationoftheleftandrightfractionalintegrals.Eq.(3)isphysicallyinterpretedasaFick’slawforconcentrationsofparticleswithastrongnonlocalinteraction.FordiscussionsofEq.(2),see[13,15].Whenβ=0,thedispersionoperatorsin(2)areidenticalandtheclassicalADEisrecovered.Inamoregeneralversionof(2),kisreplacedbyasymmetricpositivedefinitematrix.AspecialcaseofthefractionalADE(Eq.(2))describessymmetrictransitions,whereq=1/2.Inthiscase,∇−βisequivalenttothesymmetricoperator(∇−β)i=12−∞D−βxi+12xiD−β∞,i=1,...,N.(4)Combining(2)and(4)givesthemassbalanceequationforadvectionandsymmetricfractionaldispersion.ThefractionalADEhasbeenstudiedinonedimension[13],andinthreedimension[14],overinfinitedomainsbyusingtheFouriertransformoffractionaldifferentialoperatorstodetermineaclassicalsolution.Variationalmethods,especiallytheGalerkinapproximationhasbeeninvestigatedtofindthesolutionsoffractionalBVP[15]andfractionalADE[16]onafinitedomainbyestablishingsomesuitablefractionalderivativespaces.ALagrangianstructureforsomepartialdiffe