Numerical study for the fractional differential eq

The4thInternationalConferenceonComputationalMethods(ICCM2012),GoldCoast,Australia*,M.M.Khader2,andA.M.S.Mahdy31DepartmentofMathematics,FacultyofScience,CairoUniversity,Giza,Egypt2DepartmentofMathematics,FacultyofScience,BenhaUniversity,Benha,Egypt3DepartmentofMathematics,FacultyofScience,ZagazigUniversity,Zagazig,Egypt*Correspondingauthor:[e-mail],numericalstudiesforfractionalorderdifferentialequations(FDEs)whicharegeneratedbyoptimizationproblemarestudiedusingtheChebyshevcollocationmethodandfractionalfinitedifferencemethod(FDM).ThefractionalderivativesarepresentedintermsofCaputosense.TheapplicationoftheproposedmethodstothegeneratedsystemofFDEsleadstoalgebraicsystemwhichcanbesolvedbytheNewtoniterationmethod.Themethodsintroducepromisingtoolforsolvingmanysystemsofthenon-linearFDEs.Twonumericalexamplesareprovidedtoconfirmtheaccuracyandtheeffectivenessoftheproposedmethods.AcomparisonwiththefourthorderRunge-Kutta(RK4)isgiven.Keywords:Nonlinearprogramming,penaltyfunction,dynamicsystem,Caputofractionalderivatives,Chebyshevapproximations,finitedifferencemethod,Runge-Kuttamethod.IntroductionFractionaldifferentialequationshavebeenthefocusofmanystudiesduetotheirfrequentappearanceinvariousapplicationsinfluidmechanics,biology,physicsandengineering(BagleyandTorvik,1984).Consequently,considerableattentionhasbeengiventothesolutionsofFDEsandintegralequationsofphysicalinterest.MostFDEsdonothaveexactsolutions,soapproximateandnumericaltechniques(MeerschaertandTadjeran,2006;Sweilametal,2007)mustbeused.Inlastdecades,fractionalcalculushasdrawnawideattentionfrommanyphysicistsandmathematicians,becauseofitsinterdisciplinaryapplicationandphysicalmeaning(OldhamandSpanier,1974;Podlubny,1999).Fractionalcalculusdealswiththegeneralizationofdifferentiationandintegrationofnon-integerorder.SeveralnumericalmethodstosolveFDEshavebeengivensuchas,homotopyperturbationmethod(Sweilametal,2007),homotopyanalysismethod(Hashimetal,2009),collocationmethod(Dohaetal,2011;Khader,2011-Khader,2012;Lietal,2012;Sweilametal,2012a)andothers(ClenshawandCurtis,1960;EvirgenandÖzdemir,2011;EvirgenandÖzdemir,2012;Sweilametal,2011;Sweilametal,2012b;Sweilametal,2012c).Representationofafunctionintermsofaseriesexpansionusingorthogonalpolynomialsisafundamentalconceptinapproximationtheoryandformthebasisofthesolutionofdifferentialequations(AbramowitzandStegun,1964;SweilamandKhader,2010).Chebyshevpolynomialsarewidelyusedinnumericalcomputation.OneoftheadvantagesofusingChebyshevpolynomialsasatoolforexpansionfunctionsisthegoodrepresentationofsmoothfunctionsbyfiniteChebyshevexpansionprovidedthatthefunction()yxisinfinitelydifferentiable.ThecoefficientsinChebyshevexpansionapproachzerofasterthananyinversepowerinnasngoestoinfinity.Inthisarticle,weintroducethenumericaltechnique,theimplicitfinitedifferencemethodtothesearchforthenumericalsolutionsoftheintroducedequations.TheFDMplaysanimportantroleinrecentresearchesinthisfield.Ithasbeenshownthatthisprocedureisapowerfultoolforsolvingvariouskindsofproblems(Smith,1965;LiandZeng,2012-LiandZeng,2013).Thistechniquereducestheproblemtoasystemofalgebraicequations.InthisworkwewillusetheNewtoniterationmethodtosolvetheresultingnon-linearsystemofalgebraicequations.ManyauthorshavepointedoutthattheFDMcanovercomethedifficultiesarisinginthecalculationofsomenumericalmethods,suchas,finiteelementmethod.ThemainaimofthepresentedpaperisconcernedwiththeapplicationoftheChebyshevcollocationmethodandfractionalfinitedifferencemethodtoobtainthenumericalsolutionofthefractionaldifferentialequationswhichgeneratedfromthenon-linearprogrammingproblems.Andstudytheconvergenceanalysisofthetwoproposedmethods.Optimizationtheoryisaimedtofindouttheoptimalsolutionofproblemswhicharedefinedmathematicallyfromamodelariseinwiderangeofscientificandengineeringdisciplines.Manymethodsandalgorithmshavebeendevelopedforthispurposesince1940.Thepenaltyfunctionmethodsareclassicalmethodsforsolvingnonlinearprogramming(NLP)problem(Luenberger,1973;SunandYuan,2006).Also,differentialequationmethodsarealternativeapproachestofindsolutionstotheseproblems.Inthistypeofmethodsandoptimizationproblemisformulatedasasystemofordinarydifferentialequationssothattheequilibriumpointofthissystemconvergestothelocalminimumoftheoptimizationproblem(Arrowetal,1958;FiaccoandMccormick,1968;Yamashita,1976).Thestructureofthispaperisarrangedinthefollowingway:Insectiontwo,weintroducesomebasicdefinitionsaboutCaputofractionalderivatives,thedefinitionoftheoptimizationproblemanditsgeneratedsystemofFDEs.Insectionthree,wederiveanapproximateformulaforfractionalderivativesusingChebyshevseriesexpansion.Insectionfour,wegivediscretizationforfractionalderivativeusingFDM.Insectionfive,numericalexamplesaregiventosolvethesystemofFDEswhichobtainedfromthenon-linearprogrammingproblemandshowtheaccuracyofthepresente