Résumé:
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.
