资源描述:
《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