Numerical treatment of an initial-boundary value p

arXiv:math/0607103v2[math.NA]14Jul2006MANUSCRIPTpublishedin:SignalProcessing86(10),2006,pp2503-3094availableat:fferentialequationsMariuszCiesielski,JacekLeszczynskiCzestochowaUniversityofTechnologyInstituteofMathematicsandComputerScienceul.Dabrowskiego73,42-200Czestochowa,PolandReceived27April2005;receivedinrevisedform21November2005;accepted6Decemeber2005AbstractThispaperdealswithnumericalsolutionstoapartialdifferentialequationoffrac-tionalorder.Generallythistypeofequationdescribesatransitionfromanomalousdiffusiontotransportprocesses.Fromaphenomenologicalpointofview,theequa-tionincludesatleasttwofractionalderivatives:spatialandtemporal.Inthispaperweproposedanewnumericalschemeforthespatialderivative,thesocalledRiesz-Felleroperator.Moreover,usingthefinitedifferencemethod,weshowhowtoemploythisschemeinthenumericalsolutionoffractionalpartialdifferentialequations.Inotherwords,weconsideredaninitial-boundaryvalueprobleminonedimensionalspace.Inthefinalpartofthispapersomenumericalresultsandplotsofsimulationsareshownasexamples.Keywords:Anomalousdiffusion,Fractionalcalculus,Finitedifferencemethod,Riesz-Felleroperator,Boundaryconditions1IntroductionInthelastyearsfractionalderivativeshavefoundnumerousapplicationsinmanyfieldsofphysics,mathematics,mechanicalengineering,biology,electricalengineer-Emailaddresses:[e-mail](MariuszCiesielski),[e-mail](JacekLeszczynski).PreprintsubmittedtoElsevierScience2February2008ing,controltheoryandfinance[1,2,3,4,5].Onecanfindinterestingpropertiesandinterpretationsoffractionalcalculusin[4,5,6,7,8,9].Fractionalcalculusinmathe-maticsisanaturalextensionofinteger-ordercalculusandgivesausefulmathemat-icaltoolformodellingmanyprocessesinnature.Oneoftheseprocesses,inwhichfractionalderivativeshavebeensuccessfullyapplied,iscalleddiffusion.Phenomenawhichdeviatefromclassicaldiffusionaredescribedinmanypapers.Thedeviationiscalledanomalousdiffusion.Thistypeofdiffusionischaracterizedbythenonlineardependenceofthemeansquaredisplacementx(t)ofadiffusingparticleovertimet:x2(t)∼kγtγfor0γ≤2.Thisistheoppositeofclassicaldiffusionwherethelineardependencex2(t)∼k1toccurs.Analysingchangesintheparameterγitmaybesaidthattransportphenomenainsystemsexhibitingsub-diffusionhave0γ1,and1γ2inthesystemsexhibitingsuper-diffusion.Howeverthedependencex2(t)=∞fort→∞characterizesrarebutextremelylargejumpsofadiffusingparticle-knownasL´evymotionorL´evyflights[10].ClassicaldiffusionfollowsGaussianstatisticsandFick’ssecondlawforrunningprocessesattimet,whereasanomalousdiffusionfollowsnon-GaussianstatisticsorcanbeinterpretedastheL´evystabledensities.Theeverincreasingamountofliteratureinwhichthebehaviourofanomalousdiffusionisobservedpresentssomeexamplesofthisdiffu-sionin:disorderedvortexlatticeforsuperconductors[11],supercooledliquidsandglasses[12],disorderedfractalmedia[13],liquidcrystalpolymers[14]andmanyothers.Usingequationswithinteger-orderderivativestomodelanomalousdiffusionintheaboveprocessesmaynotreflecttheirrealrealbehaviourThereforeanomalousdiffu-sionisdescribedbyaspatio-temporalfractionalpartialdifferentialequationinwhichclassicalspatialandtemporalderivativesarereplacedbyderivativesoffractionalor-der.Theanomalousdiffusionequationincludestheclassicaldiffusionequation,andtherefore,thisequationhasageneralform.Moreover,theanomalousdiffusionequa-tionmayalsodescribewavepropagationoradvectionprocesses.Equationsofanomalousdiffusionwithtimeand/orspacefractionalderivativeshavebeenproposedandanalysedbynumerousauthors,forexampleNigmatullinin[15,16],Bouchaudin[17],WyssandSchneiderin[18,19,20],Westin[21],Goren-flo,FabritiisandMainardiin[22],GorenfloandMainardiin[23],Mainardiin[24],MetzlerandKlafter[25],Hilferin[1]andrecentlybyAgrawalin[26].Nevertheless,thetheoreticalanalysisandnumericalmethodsappliedtosolvefractionaldiffusionequationspresentdifficulties.Inmanypapers,theautorshaveconsideredandsolvedproblemsintheinfinitedomain.Hilfer[1]andKlafterandMetzler[27]describedtheanalyticalsolutiontotheseequationsintermsofFox’sH-function.In[26]Agrawalpresentedananalyticalsolutionovertimefortheanomalousdiffusionequationwithboundaryconditionsofthefirstkind.HebasedhisapproachontheLaplacetrans-formintermsoftheMittag-Lefferfunction.However,thenumericalapproximationfortheseriesofexpansionsofthesefunctionsarealittleproblematic,especiallyforgreatervaluesoffunctionarguments.Somenumericalmethodssuchasthefinitedifferencemethod(FDM)andthefi-niteelementmethod(FEM)aremoresuitableforsolvingtheanomalousdiffusionequationinmoregeneral(non-linear)form.ThenumerousworksbyGorenfloandMainardi[22,23],Podlubny[5]andmanyothersshouldbenoted.ThedifferenceschemeforfractionalderivativesisbasedonthedefinitionintheGr˝unwald-Letnikovform[5,9].Thisfromcanmaketheschememoreflexibleandstraightforward.Itisalsoadifficulttasktosolvetheboundaryvalueproblemoftheseequations.CiesielskiandLeszczynskiin[28],Yustein[29]proposedandanalyseddifferentcasesofnumer-icalsolutionstothefractionaldiffusionequationwherespecifick