EXISTENCE, UNIQUENESS AND CONTROLLABILITY RESULTS FOR SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Abstract

Functionaldi erentialequationsareusedinthedailylifeofahumanbeingdirectly or indirectlylikeitisusedinMedicalSciencetocheckthegrowthofdiseases,the immuneperiod,theinfectiousperiod,etc.Inthisprocess,thespacevariableissup- pressed andthisconvertsthebiologicalmodelintoanordinarydi erentialequation havingatimevariableinanin nitedimensionalspace.Ifweareconcernedwiththe invariantpropertiesofacertainclassofproblems,thenthebestwaytostudysuch problems istoconsidertheabstractformulationsofsuchproblems.Amathemati- cal formulationoftheseproblemsisalsousedinseveralengineeringmodelslikeas semiconductor modeling,heatconduction,waveequations,genericrepression,laser optics, whalingcontrol,controltheory,climatemodels,reaction-di usionequations, coupled oscillators,populationecology,viscositymaterials,foodwebs,structured populationmodels,enzymekinetics,neuralnetworks,modernphysics,etc. Fractionalcalculusisapartofmathematicsthatinvestigatethedi erentialand integralequationshavingfractionalorderi.e.non-integerorderandthisisthe main reasonbehindthesomuchuseoffractionalcalculusatpresent.Thiscon- cerns strategiesforresolvingdi erentialequationswithfractionalderivativesofthe unknownfunction.Furthermore,di erentialequationswithcomplexordersarea part offractionalcalculus.Atpresent,fractionalcalculusisusedinseveral elds likeopticsandsignalprocessing,controltheoryofdynamicalsystems, uid ow, viscoelasticity,electricalnetworks,rheology,andaswellasinmanyother eldsof science also.Thoughtherearemanytypesoffractionalorderderivatives,Riemann- Liouville, Atangana-Baleanu,andCaputofractionalderivativesareusedcommonly.