分数阶微积分.pdf

《分数阶微积分.pdf》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、FractionalCalculus.FractionalCalculus.2014年4月19日1FractionalCalculus1.1TheBasicIdea..Theorem.1.A(FundamentalTheoremofClassicalCalculus)Letf:[a,b]→Rbeacontinuousfunction,andletF:[a,b]→Rbedefinedby∫xF(x):=f(t)dt.aThen,FisdifferentiableandF′=f..2FractionalCalculus.Definition.1.1.(a)B

2、yD,wedenotetheoperatorthatmapsadifferentiablefunctionontoitsderivatives,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.Definitio

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..Definition.1.

5、2Thefunction:(0,∞)→R,definedby∫∞(x):=tx−1e−tdt,0.iscalledEuler’sfunctionorEuler’sintegralofthesecondkind.6FractionalCalculus.Theorem..1.3Forn∈N,wehave(n−1)!=(n)..Proof..用归纳法证明。∫∞n=1,(1)=e−1dt=1.0假设n=k−1时成立，那么n=k时,∫∞(k−1)=tk−2e−tdt.0(k)=∫∞tk−1e−1dt=−tk−1e−t

6、∞+∫∞(k−1)tk−2e−

7、tdt=(k−000∫∞1)tk−2e−tdt=(k−1)(k−1)=(k−1)(k−2)!=(k−1)!..07FractionalCalculus.Definition.1.3.Let0<µ<1,k∈Nand1≤p.Lp[a,b]:={f:[a,b]→∫bR;fismeasurableon[a,b]and

8、f(x)

9、pdt<∞},aL∞[a,b]:={f:[a,b]→R;fismeasurableandessentiallyboundedon[a,b]},H[a,b]:={f:[a,b]→R;∃c>0∀x,y∈[a,b]:

10、f(x)−f(y)

11、

12、≤.c

13、x−y

14、},8FractionalCalculus.Definition.Ck[a,b]:={f:[a,b]→R;fhasacontinuouskthderivative},C[a,b]:=C0[a,b],.H0[a,b]:=C[a,b].9FractionalCalculus.Definition.1.4ByH∗orH∗[a,b]wedenotethesetoffunctionsf:[a,b]→RwiththepropertythatthereexistssomeL>0suchthat

15、f(x+h)−f(x)

16、≤L

17、h

18、ln

19、h

20、−1.w

21、here

22、h

23、≤1/2andx,x+h∈[a,b]..Theorem.1.B(FundamentalTheoreminLe