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1、FractionalCalculus.FractionalCalculus.2014年4月19日1FractionalCalculus1.1TheBasicIdea..Theorem.1.A(FundamentalTheoremofClassicalCalculus)Letf:[a,b]→Rbeacontinuousfunction,andletF:[a,b]→Rbede?nedby∫xF(x):=f(t)dt.aThen,Fisdi?erentiableandF′=f..2FractionalCalculus.De?nition.1.1.(a)B
2、yD,wedenotetheoperatorthatmapsadi?erentiablefunctionontoitsderivatives,i.e.Df(x):=f′(x).(b)ByJa,wedenotetheoperatorthatmapsafunctionf,assumedtobe(Riemann)integrableonthecompactinterval[a,b],ontoitsprimitivecenteredata,i.e.∫xJaf(x):=f(t)dtafor.a≤x≤b.3FractionalCalculus.De?nitio
3、n.(c)Forn∈NweusethesymbolsDnandJantodenotethen-folditeratesofDandJa,respectively,i.e.wesetD1:=D,Ja1:=Ja,.andDn=DDn?1andJan:=JaJan?1forn≥2.4FractionalCalculus.Lemma.1.1.LetfbeRiemannintegrableon[a,b].Then,fora≤x≤bandn∈N,wehave∫xn1n?1Jaf(x)=(x?t)f(t)dt..(n?1)!a.Lemma.1.2.Letm,n∈
4、N,suchthatm>n,andletfbeafunctionhavingacontinuousnthderivativesontheinterval[a,b].Then,Dnf=DmJm?nf..a5FractionalCalculus.proofoflemma1.2..ByDnJanf=f,wehavef=Dm?nJam?nf.ApplyingtheoperatorDntobothsidesofthisrelationandusingthefactthat.DnDm?n=Dm,thestatementfollows..De?nition.1.
5、2Thefunction
