Numerical treatment of differential equations of f

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2020-01-20

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NumericalTreatmentofDierentialEquationsofFractionalOrderLuiseBlankAbstract:Thecollocationapproximationwithpolynomialsplinesisappliedtodierentialequationsoffractionalorderandthesystemsofequationscharacterizingthenumericalsolutionaredetermined.Inparticular,theweightmatricesresultingfromthefractionalderivativeofthesplinearededucedanddecomposedfornumericalimplementation.Themainresultofthispaperisthesimplicationofthenumericalmethodunderaspecicsmoothnessconditiononthechosensplines.Numericalresultsconcludetheworkandshowthattheexpectedqualitativebehaviourisgivenbycollocationapproximation.Moreover,amethodofdecompositionissuggestedformultiple{termfractionaldierentialequations.1IntroductionThequestion’whatcouldbeaderivativeofnonintegerorder’aroseintheyear1695,whenL’H^opitalaskedLeibnizforaninterpretationofDfwhereisafraction.Thetheoryforderivativesoffractionalorderwasdevelopedinthe19thcentury.Twodierentapproachesexist.Theyareequivalentforawideclassoffunctions(formoredetailswerefertoMillerandRoss[1993]).TheGrunwald{LetnikovdenitiongeneralizestheexpressionoftheN-thderiva-tiveasalimitofbackwarddierencequotientsandgivesforpositiveDy(t)=limn!1tnnXj=0(1)jj!ytjtn:OntheotherhandtheRiemann{LiouvilledenitionhasasastartingpointCauchy’sintegralformulawhichleadstothedenitionofDforNN+1,N2INDy(t)=DN+11(N+1)Zt0(ts)Ny(s)ds:ConsequentlyDistheleftinverseoperatoroftheAbel{integraloperatorJy(t)=1()Zt0(ts)1y(s)ds;1(i.e.DJ=Id)andDy(t)=DN+1JN+1y(t).Thisworkisbasedonthelatterdenition.Incontrasttoderivativesofintegerorder,whichdependonlyonthelocalbe-haviourofthefunction,derivativesoffractionalorderinvolvethewholehistoryofthefunctioninaweightedform.Thismemoryeectleadstomanyapplicationsofdierentialequationsoffractionalorder.Forexample,phenomenainelectromagnet-ics,acoustics,viscoelasticity,electrochemistryandmaterialssciencearedescribedbydierentialequationsoffractionalorder(seeforexampleBeyerandKempe[1993];OchmanandMakarov[1993];Michalski[1993];GlockleandNonnenmacher[1991];Babenko[1994a,1994b];Mainardi[1994];Gaul,KleinandKempe[1991];MannandWolf[1951]).Fractionalcalculusalsohasapplicationsinprobabilitytheory,classicalanalysisandforsolvingpartialdierentialequationsofintegerorder(seeOldhamandSpanier[1974]).2ResultsforbasiclineardierentialequationsoffractionalorderTogiveaninsightintodierentialequationsoffractionalorderletusgeneralizetherelaxationequationDy(t)=y(t)withy(0)=y0withthesolutiony(t)=y0exp(t)andtheoscillationequationD2y(t)=2y(t)withy(0)=y0andy0(0)=y00withthesolutiony(t)=y0cos(t)+y001sin(t).Inviewoftheapplicationsinscience,wherethevaluesoffractionalderivativesattheinitialpoint0aremostunlikelytobeknownbutwherederivativesy(s)(0)ofintegerordersaregiven,weincorporatetheinitialdata.ForarbitrarypositivewithNN+1weobtaintheequationDy(t)NXs=01s!tsy(s)(0)!=y(t)(1)withgiveninitialdatay(s)(0)(s=0;;N).NoticethatforintegervalueswehaveDy(t)PNs=01s!tsy(s)(0)=D(y(t)).GorenoandRutman[1995]analysedthesolutionof(1)inmoredetailforvalues2(0;2):Theorem1(GorenoandRutman[1995])For01theuniquesolutionoftheequationD(y(t)y0)=y(t)withy(0)=y0(2)2isgivenbyy(t)=y0w0(t)withw0(t)=E((t)):Furthermore,w0(t)=1(t)(1+)+O((t)2)ast!0whilew0(t)1(1)(t)ast!1.HereE(z)denotestheMittag{LeerfunctionE(z)=1Xk=0zk(k+1)(formoredetailsseeErdelyietc.[1955]),whichgeneralizestheexponentialfunction.Wecouldsay(2)’extrapolates’therelaxationequationtosocalled’ultraslow’processes.Thismeansthesmallerthevalueof1,thefastertherateofdecreaseofthesolutionforsmalltandtheslowertherateofdecreaseinthesolutionast!1(seealsonumericalresultslateron).Theorem2(GorenoandRutman[1995])For12theuniquesolutionoftheequationD(y(t)y0ty00)=y(t)withy(0)=y0andy0(0)=y00(3)isgivenbyy(t)=y0w0(t)+y00w1(t)withw1(t)=Zt0w0(s)ds:w0(t)1(1)(t)andw1(t)1(2)(t)1ast!1.Furthermore,w0andw1haveanitenumberofpositivezeros(w0hasanoddnumberwhilew1hasanevennumberofzeros)andw0tendsto0frombelowast!1.Theequation(3)’interpolates’betweentherelaxationequationandtheoscillationequation(seenumericalresultsinsection6).FurtherparticularsaboutthedecayandthelocationofthezeroscanbefoundinGorenoandMainardi[1996].Mathematicianshaveconcentratedsofaronthetheoreticalanalysisofthesolu-tionsofdierentialequationsofnonintegerorder(Babenko[1994a,1994b],GorenoandRutman[1995],Samko,KilbasandMarichev[1993],Podlubny[1994b]etc.).However,inspiteofalargenumberofrecentlyformulatedapplications,thestateoftheartisfarlessadvancedinthenumericaltreatment.ForfractionalderivativesandintegralsLubich[1986a]introducedandanalysedfractionalmultistepmethods.Inhisfollowingpapers[1985,1986b]thismethodisappliedtoAbel{integralequationsandisthoroughlyinvestigatedconcerningstabil-itypropertiesandconvergence.Ontheotherhand,BrunnerappliedthecollocationmethodtoAbel{integralequationsinseveralmodiedwaysandanalyseditsconver-gence(Brunner[1983,1985],BrunnerandvanderHouwen[1986]).Stabilityanalysis3hereofcanbefoundinBlank[1995,1996].Yet,totheauthor’sknowledge