On the study of some types of differential equations of fractional orders

Abstract

Thesubjectoffractionaldi¤erentialequationshasgainedconsiderablepopularity andimportanceduringthepastthreedecadesorso,duemainlytoitsdemonstrated applicationsinnumerousseeminglydiverseandwidespread eldsofscienceand engineering.Itdoesindeedprovideseveralpotentiallyusefultoolsforsolvingdi¤er- entialandintegralequationsaswellastheirimportanceinthemodelingofalotof physicalphenomenaassociatedtoveryrapidandveryshortchanges. Ontheotherhand,existence,uniquenessandstabilityofsolutions,represent a largepartofthequalitativetheoryofnonlinearordinaryandpartialdi¤erential equationsofnon-integerorder.Whereweareinterestedinthisthesisonthediscus- sionofqualitativeanalysisofsomekindsoffractionalordinarydi¤erentialequations andfractinalpartialdi¤erentialequations.Tothisend,weutilizethe xedpoint theoremsofBanach,SchauderandKrasnoselskiiinBanachspacesforfractionalor- dinarydi¤erentialequationsaswellasRothediscretizationmethodisusedtoshow theexistenceanduniquenessofweaksolutionforfractionaldi¤usionequationof thesecond-orderdi¤erentialVolterraoperatorinHilbertspace.Toguaranteethe e¤ectivenessandusefulnessofourobtainedresultstheoretically,someillustrative examplesaregiven.