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arXiv:0712.0168v1[math.NA]2Dec2007AnovelnumerialtehniqueusedinthesolutionofordinarydierentialequationswithamixtureofintegerandfrationalderivativesJaekS.Leszzynski,TomaszBlaszzykCzestohowaUniversityofTehnology,InstituteofMathematisandComputerSiene,ul.Dabrowskiego73,42-200Czestohowa,Polande-mail:{jalesz,tomblaszzyk}gmail.om2007AbstratUsingbothfrationalderivatives,denedintheRiemann-LiouvilleandCaputosenses,andlassialderivativesoftheintegerorderweex-aminedierentnumerialapproahestoordinarydierentialequations.GenerallyweformulatesomealgorithmswherefourdisreteformsoftheCaputoderivativeandthreedierentnumerialtehniquesofsolvingor-dinarydierentialequationsareproposed.WethenillustratehowtointroduelassialinitialonditionsintoequationswheretheRiemann-Liouvillederivativeisinluded.1IntrodutionInthepast,frationalaluluswasappliedonlyfromamathematialpointofview.Thefundamentalworkwasdonein[20,21,23,25℄.Atpresentfrationalalulusisextremelypopularduetoarapidexpansionintheeldofpratialappliations.Suhappliationshavebeenusedinphysisandmehanis[15,28℄,nane[19,26℄,hydrology[3,27℄andmanyotherdisiplines.Ordinarydierentialequationsinludingamixtureofintegerandfrationalderivativesareanaturalextensionofinteger-orderdierentialequationsandgiveanovelapproahtomathematialmodelingmanyproessesinnature.Thesolutionofaequationstronglydependsonformoftheequationandisstillonsideredbymanyauthors.ItshouldbenotedthatananalytialapproahislimitedtothelinearformofequationsandinludesspeialfuntionssuhasFoxandWrightfuntions[12,16℄ortheMittag-Leerfuntion[25℄.Thisgreatlylimitspratialimplementations,i.e.sometimesitisverydiulttoillustratethesolutioninonesimplehart.Ontheotherhand,anumerialsolution[1,7,19,10,17℄isanalternativeapproahtoanalytialone.However,thisapproahhasmanydisadvantages,i.e.theintrodutionoftheinitialonditionsinludedintheRiemann-Liouvillederivative[14℄,theunreasonableassumptionthatamethodappliedtoasingletermequationisproperforsolvingamulti-termequation[8,24℄et.AgainstthisbakgroundFord[9℄notiedthatthereanbeaonsiderablegapbetweenmethodsthatperformwellintheoryandthosewhoseimplementationsareeetive.Inthispaperwetrytoproposeanumerialapproahwhihwillbemoreonvenientinpratialappliations.WewillgiveanumerialproedureforhowtointroduelassialinitialonditionsintoanequationwheretheRiemann-Liouvillederivativeisinluded.Herewewillfousonsuhtypesofequationasfx,y(x),D1y(x),...,Dpy(x),Dα1y(x),...,Dαmy(x)
=0(1)wherey(x)isthesolutionobtainedforthelassofontinuousfuntions,D1y(x),...,Dpy(x)arederivativesoftheintegerorder,Dα1y(x),...,Dαmy(x)arederivativesofthefrationalorderandα1,...,αm∈Rarerealordersofafrationalderivative.Weassumethatthefrationalderivativeisdenedastheleft-sideCaputoderivative[2℄Cx0Dαxy(x)=1Γ(n−α)xZx0y(n)(τ)(x−τ)α−n+1dτforxx0(2)andtheleft-sideRiemman-Liouvillederivative[25℄x0Dαxy(x)=1Γ(n−α)dndxnxZx0y(τ)(x−τ)α−n+1dτforxx0(3)Inaboveformulae,thenotationn=[α]+1where[·]isanintegerpartofarealnumber.Moreover,weintrodueadenitionoftheleft-sideRiemann-Liouvillefrationalintegral[21℄asx0Iβxy(x)=1Γ(β)xZx0y(τ)(x−τ)1−βdτforxx0(4)whihwillbeusedinourfurtheralulations.Notethatβ(β0)istherealorderofEqn.(4).Onthebaseoftheory[21℄weuseanexpressionCx0Dαxy(x)=x0In−αx(Dny(x))(5)whihshowsarelationshipbetweentheCaputoderivative(2)andtheRiemann-Liouvilleintegral(3).Withregardtopapers[1,4,9℄inwhihnumerialmethodsareusedinthesolutionoffrationaldierentialequations,theauthorsmostlyusetheCaputoderivative.However,thereisasmallnumberofpapers[10℄wheretheauthorsusetheRiemann-Liouvillederivative.Thissmallnumberofpapers2wereonfrontedbytheproblemofhowtointroduelassialinitialonditionsintheRiemann-Liouvillederivativeinordertoobtainasolutionforalassofontinuousfuntions.Inthispaper,weproposeawaytoavoidthisproblem.Tobemorepreise,ineveryequationwheretheRiemann-LiouvillederivativeourswewillhangeitfortheCaputoone.FollowingthiswewilldisretizeonlytheCaputoderivative,exeptforoneasewheretherealnumberoftheRiemann-Liouvillederivativedominatesintheequation.Inthisaseweproposex0Dαxy(x)=Dnx0In−αxy(x)andthendisretizetheleft-sideRiemann-Liouvilleintegralx0In−αx.2StatementoftheproblemanditssolutionWithregardtoEqn.(1)welimitouronsiderationstotheequationwhihhasthefollowingformDpy(x)+λCx0Dαxy(x)x0Dαxy(x)=0(6)wherepdenotesanintegernumberbeingthederivativesorder,α∈h0,1)istheorderofthefrationalderivativeandλisanarbitraryrealnumber.Thissimpleformoftheequationallowsustoshowhowourmethodsworkproperlyinomparisontoanalytialsolutions.NotethatEqn.(6)isthehomogeneousordinarydierentialequationwithamixtureofderivatives.Thefuntiony(x)beingthesolutionofthisequation,stronglybelongstothelassofontinuousfuntions.Onthebasisofourpreviousresults[17℄werewriteEqn.(6)inanexpliitform.Consequentlyweobtainthethreefollowingtypesofequation:•pnforp=2,α∈h0,1),n=1D2y(x)+λCx0Dαxy(x)=0(7)D2y(x)+λx0Dαxy(x)=0(8)Itanbeseeninaboveequationsthattheintegerorderoflassialderiva-tivedominatesoverthefrationalone.•p=nforp=1,α∈h0,1),n=1D1y(x)+λCx0Dαxy(x)=0(9)D1y(x)+λx0Dαxy(x)=0(10)Inthisasewehaveequalintegerordersforthelassialandfrationalderivative.3•pnfor