Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher in order o give firm undersnding in he subjec. The reder is expeced o be versed in undergrdue complex nlysis, mening h hey should lso be fmilir wih rel nlysis. In he concluding remrks, he reders fmilir wih mesure heory will find brief discussion of how o exend he opics discussed in he pper o more generl nlysis. Conens 1 Hisoricl Bckground II 2 Preliminries II 3 The Rel Cse III 4 The Complex Anlyic Mehod V 5 The Cpuo Frcionl Derivive VI 6 Properies nd Exmples VII 6.1 The Rel Riemnn-Liouville..................... VII 6.1.1 Composiion of Frcionl Derivives........... VIII 6.2 The Complex Riemnn-Liouville Definiion............ IX 6.3 The Cpuo Definiion........................ X 7 Concluding Remrks XI 8 Lis of Definiions XI 9 References XIV I
1 Hisoricl Bckground In Sepember of 1695, Leibniz wroe leer o l Hôpil regrding derivives of generl order [1]. L Hôpil wroe bck sking wh if he order is 1/2? (Ansssiou, 5). This is regrded s he sr of Frcionl Clculus. In 1832, Liouville noiced h he well-know fc D (m) (e z ) = m e z m N (where D (m) f(z) is he m h derivive of f wih respec o z) could be exended for complex numbers. Th is m N could be replced wih α C, nd we cn define D α (e z ) = α e z α C. Where D α is ody clled he frcionl derivive. There re vrious wys of defining he frcionl derivive. I will focus primrily on he rel version of he Riemnn-Liouville Frcionl Derivive, discussed in Chper 2 of Podlubny [9]. Podlubny gives few forml definiions nd heorems, so I hve wrien my own bsed on wh ws in he ex. I will lso briefly discuss The Riemnn-Liouville Frcionl Derivive in he complex cse, given by Osler (646-647) [8], s well s he Cpuo Frcionl Derivice, defined by Podlubny in chper 2.4. As his pper ims o inroduce he reder o Frcionl Clculus, following hese hree definiions here will be properies, heorems, nd exmples regrding he meril discussed. Should he reder like, lis of he vrious definiions, equions, nd heorems re provided he end of his pper, immediely before he references. I will begin wih some preliminries h will be helpful in deriving some of he resuls in he pper. 2 Preliminries This secion includes only definiions which pper in he pper h will no be defined he ime hey re menioned. Simply Conneced: Some exmples of domins h re no simply conneced domins re nnuli, puncured disks, nd puncured plnes (Gmelin, 252). The reder fmilir wih opology my be wre h being simply conneced is nlogous o hving genus. The following definiion is Compex Anlyis(Gmelin, 252-253)[4]: Definiion 2.1. Le γ() for b be closed ph in domin D. Le z 1 be he consn ph some poin in D. We sy h γ is deformble o poin if s 1, here exis closed phs γ s () for b such h γ s () depend coninuously on s nd, γ () = γ(), nd γ 1 () z 1. We sy h domin D is simply conneced if every closed ph in D is deformble o poin. Th is o sy, γ is deformble o poin if here exiss sequence of curves γ s () h depend coninuously on s nd, wih he iniil ph (h is, s = ) γ being equl o γ, nd wih he finl ph (s = 1) being equl o he consn II
ph, e.g. poin z 1 D. The Complex-Vlued Gmm Funcion: The following is from he firs chper of Frcionl Differenil Equions (Podlubny). Definiion 2.2. The Gmm Funcion, denoed Γ(z), is given by Γ(z) = z 1 e d The Gmm funcion converges on he righ hlf plne Re(z) > s shown on pge 2 of Podlubny Proposiion 2.3. Γ(z + 1) = zγ(z) Proof. Le u = z nd le dv = e. Using inegrion by prs, Γ(z + 1) = e z d = e z + e z z 1 d = Γ(z) The following definiion is quie frequen mong uhors (Podlubny, 62). In fc, mny uhors in frcionl clculus hink of he inegrl of funcion f o jus be he 1 s derivive. While his definiion is one I m no personlly fond of, i is used by he uhors being discussed. The following definiion cn been seen s n lerne semen of he fundmenl heorem of clculus. Definiion 2.4. Le f(τ) be coninuous nd inegrble funcion. Then define he inegrl of f by 3 The Rel Cse f ( 1) () = f(τ)dτ Definiion 3.1 (Riemnn-Liouville Frcionl Derivive). Le f() be n m+1 imes differenible funcion. We sy D p f() is he p h frcionl derivive wih respec o (wih lower bound/erminl ), wih (m p < m + 1). I is given by D p f() = ( ) m+1 d ( τ) m p f(τ)dτ (m p < m + 1) d We will now see how his cn be exended o derivive of rbirry order (rher hn jus p beween m nd m + 1. Firs, I refer bck o definiion 2.3 in III
he preliminries secion (inegrion). From his, if we inegre gin, we ge f ( 2) () = = = τ1 dτ 1 f(τ)dτ f(τ)dτ τ dτ 1 ( τ)f(τ)dτ The second equliy comes from he fc h when we swich our order of inegrion, we hve o swich he bounds of inegrion in order o preserve he region being inegred over. Follnd gives brief discussion of his on pge 17 of Advnced Clculus. I cn be similrly shown h f ( 3) () = 1 2 ( τ) 2 f(τ)dτ Proceeding inducively, we rrive wh Podlubny clls he Cuchy formul f ( n) () = 1 Γ(n) ( τ) n 1 f(τ)dτ (1) This cn be clled he inegrl of order n, for fuure reference. Suppose h in he bove equion, n 1, nd le k Z, k. Then if we le D k be k ierions of inegrls, s considered bove, hen f ( k n) = 1 Γ(n) D k ( τ) n 1 f(τ)dτ Likewise, if k n, nd D k is he iered derivive operor, hen f (k n) = 1 Γ(n) Dk ( τ) n 1 f(τ)dτ (2) Therefore, we cn simply refer o (2) s generl cse of f (k n) (), wih D k being iered inegrion for k nd iered differeniion for k >. If k n <, hen (2) is o be inerpreed s iered inegrls of f(). If k n =, hen (2) represens f(), nd if k n >, hen (2) represens successive derivives of f(). We cn now define he inegrl of rbirry order. In (1), replce n wih p nd require h p >. Then we cn define D p f() = 1 Γ(p) ( τ) p 1 f(τ)dτ (3) Finlly, we will define derivives of ll orders. Le α R be number such h k α >. Then rewriing (2), we obin D k α f() = 1 Γ(α) d k d k IV ( τ) α 1 f(τ)dτ (4)
4 The Complex Anlyic Mehod The Complex nlyic mehod of definiing he Riemnn-Liouville Frcionl Derivive hs differen se-up hn he rel mehod, however he resul re much of he sme. Recll he Cuchy Inegrl Formul for he m h derivive of complex-vlued funcion f : C C on bounded domin D (Gmelin, 114): f (m) (z) = m! f(w) dw 2πi D (w z) m+1 This resul is sed s heorem proved by Gmelin, wherein f mus exend smoohly o he boundry of D. Consider wh would hppen if m is inerchnged wih non-ineger, nmely ny complex number α. The nlogy is h m! would be replced wih Γ(α + 1), nd (w z) m 1 becomes (w z) α 1. However, consider he funcions g = 1 (w z) m+1 h = 1 g = (w z)m+1 w hs of order m + 1 z =, nd so h hs pole of order m + 1 z =. Noice h before we were ble o wiggle he conour we were inegring over wihou much consequence (see Gmelin pge 81). Now, however, considering he funcions 1 η = (w z) α+1 ω = 1 η = (w z)α+1 ω hs brnch poin z = w, nd so η hs brnch poin z = w. Thus, we will define brnch cu sring he poin z = w, pssing hrough he origin, nd going ou o infiniy. Noice now h for z close o he conour, wiggling he conour my cuse us big problems. So we will ke he conour of our inegrl o be sring w =, nd enclosing z = w once in he sndrd posiive orienion, voiding (going round) ny singulriies h f my hve. Noe h his conour will no inersec he brnch cu ny poin excep w =. A picure should help clrify: V
Finlly, since we cn wrie (w z) α 1 ( α 1)(log (w z)) = e nd log is mulivlued funcion, we will ke he rel pr of he logrihm when w z >. We rrive he following definiion: Definiion 4.1. Le f(z) = z p g(z), where g(z) is nlyic on simply conneced domin D Ω C : D, nd le Re(p) > 1. Then we define he Frcionl Derivive of order α of f(z) (denoed D α z f(z)) s for α 1, 2, 3,.... f (α) (z) = D α z f(z) = Γ(α + 1) 2πi z + f(w) dw (5) (w z) α+1 5 The Cpuo Frcionl Derivive Podlubny wries his book on Frcionl Differenil equions, nd s he describes i, he Riemnn-Liouville is no he bes definiion o ke when solving such problems. In pplicion (such s viscoelsiciy nd herediry solid mechnics), i is beer o use differen definiion, such s he Cpuo definiion (Podlubny, 78). The Cpuo pproch mkes iniil condiions for differenil equions nicer, while he Riemnn-Liouville definiion is beer from pure mh pproch. I is no my im in his pper o discuss differenil equions VI
or he pplicions of frcionl clculus, rher I hough h he reder would find i useful o see differen wy of pproching Frcionl Clculus. We define he α h Cpuo Frcionl Derivive of f() wih respec o, C D α f(), s C D α 1 f() Γ(α n) 6 Properies nd Exmples 6.1 The Rel Riemnn-Liouville Proposiion 6.1. Suppose f() is C 1 for. Then lim p D p f() = f() Proof. We will use inegrion by prs wih u = f(τ) nd dv = ( τ) p 1. Then du = f (τ)dτ nd v = p( τ) p. By proposiion (Γ(z + 1) = zγ(z)), Thus, 1 Γ(p) = p Γ(p + 1) D p f() = ( )p f() 1 + Γ(p + 1) Γ(p + 1) ( τ) p f (τ)dτ Tking he limi on eiher side nd pssing he limi under he inegrl (Podlubny does no check for uniform convergence here, 66), we obin lim p D p f() = f() + f (τ)dτ = f() If we weken our ssumpion h f() C 1 for o f() C for, he resul sill holds, bu n epsilon del proof is needed. Proposiion 6.2. If f() C for, hen D p (D q f() ) = D p q f() (6) The proof of his is given on Podlubny, pge 67. I will prove more generl resul, bu will use his proposiion in he proof. Podlubny does no jusify swpping limiing operions in his proofs. Theorem 6.3. Suppose p > nd >. Then D p (D p f() ) = f() VII
Proof. Consider he cse where p = n N. Then D n (D n f() ) = dn d n Swpping he limiing operions, we obin d d f(τ)dτ = f() ( τ) n 1 f(τ)dτ Now consider he cse where k 1 p < k. Applying he bove proposiion, D k f() = D (k p) (D p f() ) According o Podlubny (69), his implies h D p (D p f() ) = dk d k [ = dk d k [ D (k p) D p (D p f() )] f() ] = f() Exmple Le ν R, ν > 1, nd le f() = ( ) ν Suppose n 1 p < n. By definiion of he Riemnn-Liouville Derivive, D p f() = ( dn d n D (n p) ) f() If we le α = n p nd subsiue in (3), hen we obin D α f() = 1 Γ(α) B(α, ν + 1)( )ν+α = Where B(x, y) is he be funcion. Thus, Γ(ν + 1) ( )ν+α Γ(ν + α + 1) D p f() = 1 Γ( p) B( p, ν + 1)( )ν p = Γ(ν + 1) ( )ν p Γ(ν p + 1) A similr exmple will be discussed in he complex secion. 6.1.1 Composiion of Frcionl Derivives Wih Ineger-Order Derivives Proposiion 6.4. d n d n ( D p f()) = D n+p f() VIII
The moivion for his propery, rher hn deiled proof, will be shown. The full discussion is on Podlubny 73, nd uses resuls h were no discussed in his pper. Using (4), we hve h d n ( d n D k α ) f() = 1 Γ(α) Wiring p = k α, we obin d n d n+k d n+k ( τ) α 1 f(τ)dτ = D (n+k) α d n ( D p f()) = D (n+p) α f() f() The oher direcion requires lile bi more work, nd gin is voided due o he exen of resuls used o prove i. Wih Frcionl Derivives The following proposiion will be sed wihou proof, s Podlubny gin using resuls from erlier in his book, resuls which I m no covering. The propery is quie hndy, however. Proposiion 6.5. Suppose f() is k imes differenible, where k = mx{m, n}, nd m 1 p < m nd n 1 q < n. If f (j) () = for j = 1,..., k, hen he following is rue: D p ( D q f()) = D p ( D q f()) = D p+q f() In generl, hese wo operors do no commue, nd his proposiion will be discussed more when discussing he Cpuo Derivive s properies. 6.2 The Complex Riemnn-Liouville Definiion For he complex cse, I will jus show one exmple of he frcionl derivive. I is discussed on pge 647 on Osler, lhough I emp o give more descripion hn he does. He skips quie few seps in sing his exmple, nd hus I m inerpreing some of his resuls in-beween seps. Exmple Le f(z) = z p for Re(p) >. Noe h in he cse where α = N N. Then D N z f(z) = p! (N p)! zp N In he cse where N > p, we mus invoke he gmm funcion. In (5), prmerize w in erms of s: w = zs for s 1. Then dw = zds nd we re sked wih evluion of D α z f(z) = Γ(α + 1) 2πi 1 + = zp α Γ(α + 1) 2πi (zs) p z (sz z) 1 + α+1 ds s p ds (s 1) α+1 IX
Osler hen prescribes conour h runs from o 1 ε long he rel xis, rverses he circle s 1 = ε, nd hen runs bck o he origin on he rel xis. I ssume on he wy bck long he rel xis here ws phse shif cused by rversing he circle, s Osler rrives he following expression z p α Γ(α + 1) 1 [1 e 2πi(α+1) s p ] ds 2πi (s 1) α+1 Osler hen uses properies bou he Gmm nd Be funcion (no explicily), nd simplifies he bove expression o z p α Γ(p + 1) Γ(p α + 1) Compre his o he resul in he rel cse. 6.3 The Cpuo Definiion Here, I will discuss some simple properies of he Cpuo Frcionl Derivive simply by conrsing i o he Riemnn-Liouville Frcionl Derivive. They re sed s fcul resuls rher hn precise properies. Firsly, he Cpuo definiion sisfies he propery h he Cpuo Derivive of consn is. This is fmilir o rdiionl clculus. However, in he Riemnn-Liouville definiion, if we ke K o be consn, ssuming we hve finie lower bound(or erminl) on he inegrl (ssume i is ), D α K = K α Γ(1 α) Podlubny noes h i is somewh common o le =, s his preserves he propery from rdiionl clculus h he derivive of consn is (Podlubny, 8). Secondly, recll proposiion 6.5 bove. The more generl cse is D α ( D m f()) = D m ( D α f()) = D α+m f() m N, n 1 < α < n which is only sisfied if f (s) () = for s =,..., m. condiion wih he Cpuo definiion, However, he sme C D α ( C D m f() ) = C D m ( C D α f() ) = C D α+m f() m N, n 1 < α < n is sisfied if f (s) () = for s = n, n + 1,..., m Hence he Cpuo inegrl cn be nicer in pplicions nd formuls. X
7 Concluding Remrks Mny of he ppers nd books considered in wriing his pper were eiher oo big or oo smll. By his I men he uhor eiher compleely ignored mesure heory, nd did no discuss how frcionl clculus reled o Husdorff mesure nd oher such opics, or he uhor did cover hese hings, bu such ppers nd journls were inended for reders well versed in Lebesgue inegrion, mesure heory, nd occsionlly more dvnced opics. Thus, I will give n exremely brief discussion of Lebesgue mesure, doped from Follnd pg. 27-28 [3]. Then I will briefly explin Husdorff mesure nd i s loose-pplicions o frcionl clculus (s I don hve ime o go ino furher deil). Definiion 7.1. Suppose T is iled se such h i is composed of finie number of recngles R k wih disjoin ineriors. Th is, T = K k=1 R k. Then he Lebesgue mesure m(t ) is he sum of he re s of he R k s. The Lebesgue mesure of compc se K is m(k) = sup{m(t ) : T is iled se ndk T } While he Lebesgue mesure of n open se U is given by m(u) = inf{m(t ) : T is iled se nd T U} A subse S of R 2 is clled Lebesgue mesurble if for compc K S nd open S U, sup{m(k)} = inf{m(u)} in which cse we denoe he Lebesgue mesure of S by m(s), which is equl o boh of hese vlues. Husdorff mesure is slighly more difficul o define, bu i is more generl exension of Lebesgue mesure, nd cn be defined by king he inf defined by of sum of dimeers of smll coverings of se, where he dimeer of se is he supermom of of he disnce beween ny wo poins in he se [7]. While his my seem bsrc, Husdorff mesure cn be used o define such spces s R α for < α 1. Husdorff mesure lso llows one who is ineresed in frcl geomery o mke more precise semens[5]. Once spces such s R α re se up, one cn lk bou mpping funcions ino R α, nd develop more dvnced heory of frcionl clculus h my beer resemble undergrd nlysis. This kind of rigorous exension cn led o inequliies h some migh consider o be hrd nlysis, such s inegrl inequliies [6]. As sed he sr of he pper, he ide of frcionl clculus les in concep des bck o he le 16 s, so i is no surprise h hs mny exensions nd cn be widely used. 8 Lis of Definiions Definiion. Le γ() for b be closed ph in domin D. Le z 1 be he consn ph some poin in D. We sy h γ) is deformble o poin XI
if s 1, here exis closed phs γ s () for b such h γ s () depend coninuously on s nd, γ () = γ(), nd γ 1 () z 1. We sy h domin D is simply conneced if every closed ph in D is deformble o poin. Definiion. The Gmm Funcion, denoed Γ(z), is given by Γ(z) = z 1 e d Proposiion. Γ(z + 1) = zγ(z) Definiion. Le f(τ) be coninuous nd inegrble funcion. Then define he inegrl of f by f ( 1) () = f(τ)dτ Definiion (Riemnn-Liouville Frcionl Derivive). Le f() be m+1 imes differenible funcion. The le D p f() be he p h frcionl derivive wih respec o (wih lower bound ). frcionl derivive of of D p f() = ( ) m+1 d ( τ) m p f(τ)dτ (m p < m + 1) d f ( n) () = 1 Γ(n) f (k n) = 1 Γ(n) Dk D p f() = 1 Γ(p) D k α f() = 1 Γ(α) d k d k ( τ) n 1 f(τ)dτ (1) ( τ) n 1 f(τ)dτ (2) ( τ) p 1 f(τ)dτ (3) ( τ) α 1 f(τ)dτ (4) Definiion. Le f(z) = z p g(z), where g(z) is nlyic on simply conneced domin D Ω C : D, nd le Re(p) > 1. Then we define he Frcionl Derivive of order α of f(z) (denoed Dz α f(z)) s for α 1, 2, 3,.... f (α) (z) = D α z f(z) = Γ(α + 1) 2πi z + f(w) dw (5) (w z) α+1 Proposiion. Suppose f() is C 1 for. Then lim p D p f() = f() XII
Proposiion. If f() C for, hen D p Theorem. Suppose p > nd >. Then (D p f() ) = f() Proposiion. d n (D q f() ) = D p q f() (6) D p d n ( D p f()) = D n+p f() Proposiion. 6.5 Suppose f() is k imes differenible, where k = mx{m, n}, nd m 1 p < m nd n 1 q < n. If f (j) () = for j = 1,..., k, hen he following is rue: D p ( D q f()) = D p ( D q f()) = D p+q f() Definiion. Suppose T is iled se such h i is composed of finie number of recngles R k wih disjoin ineriors. Th is, T = K k=1 R k. Then he Lebesgue mesure m(t ) is he sum of he re s of he R k s. XIII
9 References 1. Ansssiou, G. (29):. Frcionl differeniion inequliies. New York; London: Springer. Pge 5 2. Diehelm, K. (21). The nlysis of frcionl differenil equions: An pplicion-oriened exposiion using differenil operors of cpuo ype. Lecure Noes in Mhemics, 24, 1-262. 3. Follnd, G. (22). Advnced clculus. Upper Sddle River, NJ: Prenice Hll. Pges 17, 27-28. 4. Gmelin, T. (21). Complex nlysis (Undergrdue exs in mhemics). New York: Springer. Chper IV.4, pges 81, 252-253. 5. Ling, Y., & Su, S. (216). Frcl dimensions of frcionl inegrl of coninuous funcions. Ac Mhemic Sinic, English Series, 32(12), 1494-15. 6. Liu, Q., & Sun, W. (217). A Hilber-ype frcl inegrl inequliy nd is pplicions. Journl of Inequliies nd Applicions, 217(1), Pges 1-8. 7. Mkrov, B., & Podkoryov, Anolii. (213). Rel nlysis : Mesures, inegrls nd pplicions (Universiex). London ; New York: Springer. Chper 2.6. 8. Osler, T.J.(1971). Frcionl Derivives nd Leibniz Rule. The Americn Mhemicl Monhly, 78(6), 646-647. 9. Podlubny, I. (1999). Frcionl differenil equions : An inroducion o frcionl derivives, frcionl differenil equions, o mehods of heir soluion nd some of heir pplicions (Mhemics in science nd engineering ; v. 198). Sn Diego: Acdemic Press. Chpers 1.1, 2.3, nd 2.4. XIV