Short Introduction to Fractional Calculus

Transcription

1 . Shor Inroducion o Fracional Calculus Mauro Bologna Deparameno de Física, Faculad de Ciencias Universidad de Tarapacá, Arica, Chile mbologna@ua.cl Absrac In he pas few years fracional calculus appeared as an imporan ool o deal wih anomalous diffusion processes. An anomalous diffusion process can be visualized as an an in a labyrinh where he average square of he disance covered by he an is x 2 2µ where µ is a phenomenological consan; for µ 1/2 we have he ordinary diffusion processes. A more physical approach of anomalous diffusion processes has several applicaions in many field such as diffusion in porous media or long range correlaion of DNA sequence. In his shor repor we shall presen an inroducory view of he he mahemaical aspecs of fracional calculus and is basic foundaion. 41

2 1 Elemenary properies of fracional derivaives The concep of derivaive is radiionally associaed o an ineger; given a funcion, we can derive i one, wo, hree imes and so on. I can be have an ineres o invesigae he possibiliy o derive a real number of imes a funcion. The main idea is o examine he properies of he ordinary derivaive and see where and how i is possible o generalize he conceps. As ofen happen here is no only a way o do ha; we are going o use he mos inuiive and, in a cerain sense, less rigorous way. Le us consider he general properies of he derivaive D n for n N, where n is an ineger. This operaor is, in fac, defined o have he following properies, all of which we would lie he fracional derivaive o share. The firs propery of ineres is ha of associaion D n Cf ] CD n f ] 1 where C is a consan. The second propery we would lie o incorporae ino he fracional calculus is he disribuive law D n f ± g ] D n f ] ± D n g ]. 2 The final propery is ha he operaor obeys Leibniz rule for aing he derivaive of he produc of wo funcions D n f g ] n n D n f ] D g ] n n D n g ] D f ] 3 n n! where!n! is he binomial coefficien. The above properies are cerainly reained for he nh derivaive of a monomial m wih m N, so ha D n m ] m m 1 m 2 m n + 1 m n m! m n! m n 4 for m > n. Properies 1 and 2 esablish ha he operaor D n is linear and 4 enables us o compue he n h derivaive of an analyic funcion expressed in erms of a Taylor s series. We now exend hese consideraions o fracional derivaives. Looing a Eq. 4 he mos easy hing would be o replace he ineger numbers wih real numbers. The main difficuly is how o replace he facorial funcion ha is defined for ineger numbers. Forunaely i is exiss a special funcion, he gamma funcion, ha has his propery. The gamma funcion is defined as: Γz z 1 e d. 5 42

3 The inegral 5 is defined for z > or Rez] > if z is a complex number and can be checed by elemenary inegraion ha for z ineger his funcion coincide wih he facorial; more precisely i holds: Γn + 1 n! We are ready now o define a real-indexed derivaive, or more generally, a complex-indexed derivaive D α wih α R or α C, of a monomial β, as d α d α β ] D α β ] Γ β + 1 Γ β + 1 α β α 6 where β + 1, 1,, n. I can be proved, via Eq. 3 nown as Leibniz rule, ha his equaion can be generalized o fracional derivaives as D α f g ] α α D α f ] D g ] D α g ] D f ] 7 where we use he binomial coefficien α Γ α + 1 Γ + 1 Γ α + 1. and since α is no ineger he upper limi of he sum in 7 is infinie. If one of he funcions in he produc is a consan, say g C, hen 7 reduces o D α f C] α D α f ] D C] D α f ] C 8 since only he erm survives in he series because he ineger derivaives of he consan vanish. Thus, propery 1 is reained by he generalized Leibniz rule 7. Oher properies of ordinary derivaive ha hold for he fracional derivaive can be found using Eq. 7; in paricular: D α h + g ] α α α D α ] D h + g ] D α ] D h ] + D α ] D g ] D α h ] + D α g ] 9 43

4 and, in similar way, we can show ha: D α f a] a α D α x f x], x a. 1 Thus, we see ha he associaive propery is also rue for he fracional derivaive D α. Furher, Eqs. 8 and 9, aen ogeher, esablish ha he fracional derivaive is a linear operaor. Now le us consider he case where he index of he monomial is negaive ineger valued: β + 1, 1,, n and we operae wih he ordinary ineger derivaive. Consider he monomial funcion f m wih m a posiive definie ineger, from which we obain D n m ] 1 n m m + 1 m + n 1 m+n or using he properies of gamma funcions D n m ] 1 n Γ m + n m+n 11 Γ m wih n N. If we resric ourselves o real indices, hen again proceeding by analogy we wrie for < α < 1, D α m ] 1 α Γ m + α m+α, 12 Γ m bu we have o change he definiion of he gamma funcions when he argumen in he numeraor is a negaive ineger. This new definiion ransforms real funcions ino complex funcions and vice versa, because here is he complex facor 1 α e iαπ. We shall have occasion o use Consan funcions We define A α o be he se of consan funcions under he real indexed derivaive D α and C α is he generic consan of index α. So, for example, we consider he wo funcions: f 1/2 and f C and use he derivaive of he monomial 6 o obain: D 1/2 1/2] Γ 1/2 Γ 1 13 since Γ. Thus, a paricular funcion is effecively a consan wih regard o a cerain fracional derivaive. In he second example D 1/2 C] C Γ 1 Γ 1/2 1/2 C 14 π where we see ha a consan is no consan wih regard o fracional derivaives. These wo examples demonsrae ha here are funcions ha, under real-indexed derivaives, are addiive consans and addiive consans ha, 44

5 under real-indexed derivaives, are funcions. This funcions, ha behave as consan under fracional derivaive, can desroy he composiion propery of he index of derivaion. In fac, le f be a funcion having a power series represenaion and assume ha here exiss derivaives D µ f ], D ν f ] and D α f ] wih α µ + ν; if f does no conain funcion ha are consan for he derivaive operaor D µ and D ν hen D α f ] D µ+ν f ] D µ D ν f ]] D ν D µ f ]]. 15 In conclusion of his secion we are going o chec he inuiive idea ha D 1 is he inegraion operaor; examine he C n consan and in paricular he C 1 consan. We firs examine he operaor D 1 applied o a monomial D 1 β ] Γ β + 1 Γ β β+1 β+1 β from which we see ha effecively his is he inegral operaor. Now again using he lineariy propery of he operaor we now ha we can ae a sum of infiniesimals o obain he sandard definiion of he inegral and herefore in general we can wrie D 1 f ] 3 Applicaion o inegral calculus f τ dτ. 17 In he general siuaion he fracional derivaive of a funcion is a series. However, here are some cases where i is possible o express he resul in erms of elemenary funcions. I is no our purpose here o provide an exhausive lis of he fracional derivaive of funcions, bu i may be useful o see how such expressions are consruced from he definiions provided. An example is given by he funcion f a + b] µ 1, where applying he fracional derivaive 9 we obain D µ a + b µ 1] µ D µ ] D a + b µ 1] 18 so ha in erms of he fracional derivaive of a consan and he ineger derivaive of he funcion we have D µ a + b µ 1] µ µ Γ µ+1 µ 1 µ 2 µ 1 b a + b µ The muliplicaive facors in 19 may be expressed in erms of gamma funcions 45

6 Γ µ Γ µ µ 1 Γ µ 1 µ 1 µ 2 µ 1 Γ µ so ha D µ a + b µ 1] b µ µ b µ Γ µ Γ µ + 1 Γ µ a + bµ 1. 2 This expression may be furher simplified by using he gamma funcion relaion Γ µ Γ µ + 1 o obain, using sin π µ 1 sin πµ, π sin π µ D µ a + b µ 1] Γ µ a + b]µ 1 sin πµ π µ We sum he series using he binomial relaion µ where z b a + b 1, o obain D µ z 1 + z µ µ b a + b. 21 a + b µ 1] Γ µ a + b]µ 1 sin πµ π µ 1 b ] µ a + b aµ sin πµ Γ µ π µ a + b 22 We can use he previous resul o apply fracional derivaive wih respec o a parameer o produce he possibiliy of new ransformaions. Consider, for example, he inegral I a, b α a + b β γ 1 d 23 which is convergen for α > 1 and α + 1 /β + γ < 1 wih a, b. We rewrie 23 in erms of parameric derivaives I a, b b γ 1 α a/b + β γ 1 d b γ 1 D γ a D γ a α a/b + β ] γ 1 d b b 24 46

7 in order o simplify is evaluaion. We can use 22 in erms of he parameric fracional derivaive o obain from 24 γ 1 sin πγ I a, b b Γ γ D γ λ λ γ α+γβ λ + β ] 1 d π 25a where λ a/b. Maing he furher subsiuion z β in 25a we have: ] γ 1 sin πγ I a, b b Γ γ D γ λ λ γ β 1 z α+1 β +γ 1 π z + λ dz 26 and using he calculus of residues o evaluae he simple pole in he remaining inegral we have γ 1 Γ γ sin πγ I a, b b π so ha we finally obain π α+1 sin β + γ π I a, b b γ 1 Γ γ sin πγ ] Γ γ Γ α+1 β sin β + γ π Γ α+1 β α+1 β + γ ]D γ λ λ α+1 β 1] 4 The generalized exponenial funcion a b α+1 β +γ We now urn our aenion o he fracional derivaive of he exponenial funcion e, which when expressed in erms of an infinie series, yields D µ e ] ] D µ µ! Γ + 1 µ E µ 28 where we define he generalized exponenial funcion, E µ, by he series. Le us consider, for example he siuaion when he real-valued index in 28 is a negaive ineger µ 1, 2,. Saring from he definiion 28 we have for µ 1 E 1 D 1 e ] so ha reindexing he series we have E 1 j1 +1 Γ + 2 j Γ j + 1 e Of course, we can also wrie he negaively indexed generalized exponenial as he firs-order inegral 47

8 E 1 D 1 e ] e τ dτ e 1. 3 as i was expeced because of Eq. 17. Now le us consider he fracional derivaive of he negaive exponenial funcion, e. We do his by considering he fracional derivaive D µ e a ] ] D µ a a µ a µ! Γ + 1 µ aµ Eµ a 31 where a is an arbirary consan. If we choose a 1 we can use 31 o wrie D µ e ] 1 µ E µ e iπµ E µ 32 which we can furher use o define anoher generalized exponenial funcion E µ e iπµ E µ. 33 In series form we wrie his new generalized exponenial funcion as D µ e ] E µ 1 µ Γ + 1 µ. 34 Boh 33 and 34 mae i abundanly clear ha he funcion Eµ is no Eµ calculaed wih ; he new funcion differs from he old by he phase facor e iπµ. Using he propery of Eq. 1 we have: D µ e ] 1 µ D µ x e x ] x e iπµ E µ 35 jus as we obained in 32 and here Eµ is a funcion in he complex field. For real funcions i is convenien o define Eµ as Eµ calculaed wih, bu in order o do his we need o define he generalized exponenial as E µ µ Γ + 1 µ 36 where i is possible o evaluae his funcion for boh posiive and negaive values of he independen variable. In general, however, when we are dealing wih complex funcions we use he firs definiion of he generalized exponenial given by 28. For compleeness we define he inverse of he generalized exponenial funcion as he generalized logarihm, which is o say he funcion ha saisfies he relaion ln µ E µ

9 5 Generalized rigonomeric funcions Now ha we have a generalizaion of he complex exponenial funcion, i should, of course, be possible o consruc a generalizaion of he Euler relaion, ha being, E i µ cos µ + i sin µ. 38 From he real par of 38 we obain he equaion for he generalized cosine funcion cos µ 1 2 E i µ + Eµ i 39 and from he imaginary par of 38 we obain he equaion for he generalized sine funcion sin µ 1 2i E i µ Eµ i Figure 1: The dashed line is sin α x wih α.3, he doed line is sin α x wih α.3 and he coninuous line is he ordinary rigonomeric funcion sin x. Clearly afer a period he hree funcion assume he same values. We can hen exend hese definiions even furher and consruc he generalized angen funcion as well an µ sin µ cos µ. 41 We can also express he generalized sine and generalized cosine funcions in series form using he series definiion of he generalized exponenial. The generalized cosine funcion is given by 49

10 cos µ 1 2 E i µ + Eµ i and he generalized sine funcion is given by sin µ 1 2i µ e i µπ/2 + e i µπ/2 Γ + 1 µ 2 µ cos µ π/2] 42 Γ + 1 µ E i µ Eµ i µ e i µπ/2 e i µπ/2 Γ + 1 µ 2i µ sin µ π/2]. 43 Γ + 1 µ From 42 and 43 we can see ha for ineger µ he generalized rigonomeric series sin µ and cos µ become he ordinary rigonomeric funcions sin and cos. I is useful o sudy he derivaives of he generalized rigonomeric funcions in order o undersand how hese periodic funcions differ from hose in he sandard form. Consider he firs-order ime derivaive of he generalized cosine funcion D cos µ ] µ µ 1 Γ + 1 µ where by reindexing he series, j + 1, we can wrie D cos µ ] j 1 cos µ π/2] µ 1 cos µ π/2] 44 Γ µ j µ cos j + 1 µ π/2]. 45 Γ j + 1 µ Separaing he j 1 erm from he series and using he rigonomeric ideniy cos j + 1 µ π/2 sin j µ π/2 yields D cos µ ] sin µ + cos µπ/2 Γ µ µ+1 46 where we have used 43 o replace he series. We see ha he formal relaion resuling from he derivaive of he generalized cosine differs from ha of he 5

11 derivaive of he cosine by a erm ha decays as an inverse power law in he independen variable. Thus, as, he formal relaion for he wo derivaives approach one anoher: lim D cos µ ] sin. 47 The inverse power-law form of he erm in 46 is quie suggesive, since he memory in dynamical processes ha mae i impossible o join he microscopic and macroscopic descripions of complex phenomena are exacly of his inverse power-law form. Le us now examine he derivaive of he generalized sine funcion D sin µ ] µ µ 1 Γ + 1 µ where by reindexing he series, j + 1, we can wrie D sin µ ] j 1 sin µ π/2] µ 1 sin µ π/2] 48 Γ µ j µ sin j + 1 µ π/2]. 49 Γ j + 1 µ Separaing he j 1 erm from he series and using he rigonomeric ideniy sin j + 1 µ π/2 cos j µ π/2 yields D sin µ ] cos µ sin µπ/2 Γ µ µ+1 5 where we have used 42 o replace he series. We see ha he formal relaion resuling from he derivaive of he generalized sine differs from he derivaive of he sine by a erm ha decays as an inverse power law in he independen variable, jus as i did for he generalized cosine. Thus, as he formal relaions for he wo derivaives approach one anoher: lim D sin µ ] D sin ] cos. 51 Here again, he inverse power-law form of he erm in 5 is quie suggesive. We have examined wha happens o a generalized rigonomeric funcion when we ae an ordinary derivaive. Now le us examine wha happens o an ordinary rigonomeric funcion when we ae a fracional derivaive. Consider he fracional derivaive of he sine funcion D µ sin ] 1 2i 1 2i D µ e i ] D µ e i ] e iµπ/2 Eµ i e iµπ/2 Eµ i 51

12 so ha using he Euler relaions for boh he exponenial and generalized exponenial and combining erms we obain D µ sin ] sin µπ/2 cos µ + cos µπ/2 sin µ. 52 Equaion 52 is reminiscen of he rigonomeric expansion of sin + µπ/2. In he same way we ae he fracional derivaive of he cosine funcion D µ co ] D µ e i ] + D µ e i ] e iµπ/2 Eµ i + e iµπ/2 Eµ i so again using he Euler relaions and combining erms we obain D µ co ] cos µπ/2 cos µ sin µπ/2 sin µ. 53 Equaion 53 is reminiscen of he rigonomeric expansion of cos + µπ/2. A geomerical inerpreaion of he derivaive relaions in 52 and 53 can be obained by inroducing he roaion marix cos µπ/2 sin µπ/2 R sin µπ/2 cos µπ/2 and he vecor so ha we can wrie v µ sinµ cos µ D µ v ] Rv µ. 54 Boh he above fracional derivaives are included in he roaion equaion given by 54 since we now ha he generalized funcions reduce o heir ordinary counerpars when µ v sin cos The exisence of 54 allows us o infer ha he vecors v µ and w have he same lengh, where w D µ v ], since he lengh of he roaion marix is uniy. A similar ind of analysis can be done for he generalizaion of he hyperbolic sines and cosines.. 52

13 6 Cerain fracional inegrals We now consider how o consruc he definie inegrals of cerain funcions using he properies of he fracional derivaives discussed in he previous secions using negaive values of he µ index and he series represenaions of he generalized funcions. Consider an inegral of he form I α τ α e τ dτ 55 where α > 1. Using he fracional calculus formalism we wrie I α D 1 α e ] 56 and applying he generalized Leibniz rule 7 o 56 we obain I α 1 Γ α + 1 D 1 α ] D e ] 1 1 +α+1 Γ + α + 2 e. 57 Inroducing he series definiion of he generalized exponenial ino 57 we have I α Γ α + 1 e E α+1 58 for he inegral 55. We can exend 58 o α < 1 wih α 1, 2,. Equaion 58 can be used o deermine he inegral for he gamma funcion lim I α Γ α since E α+1 e as. In general we use he same logic o obain a τ α e ±τ dτ Γ α + 1 e ± E α+1 + c 6 where c is a consan dependen on he lower limi of he inegral. We can now use 6 o again obain he derivaive of he generalized exponenial. We now consider he inegrals of rigonomeric funcions of monomials such as I α cos τ α dτ. 61 ha appear in several field of physics. For example for α 2, he inegral 61 is basically he Fresnel cosine inegral Cτ, well nown in opics. Maing he change of variables y τ α so ha dy ατ α 1 dτ and 61 becomes 53

14 I α 1 α and using 6 we obain dyy 1 1 α 1 cos y Re α dyy 1 α 1 e iy ] 62 ] Γ 1/α cos τ α dτ Re αi 1/α e iy E iy 1/α Γ 1/α cos π/2α. 63 α In a similar way we obain for he inegral of he sine funcion ] Γ 1/α sin τ α dτ Im αi 1/α e iy E iy 1/α Γ 1/α sin π/2α 64 α where we have used he propery ha for he generalized exponenial funcion becomes he ordinary exponenial funcion. 7 Concluding Remars This shor review showed as he fracional calculus is a very helpful ool o perform calculaion specifically dealing wih power law1, 2]. Despie o he mahemaical examples presened, many physical applicaion can be faced3]; le us, for example, consider he ordinary diffusion equaion, P x, 2 P x, ha leads o gaussian processes. A possible generalizaion of he diffusion equaion is he fracional diffusion equaion P x, α P x, where he second derivaive is subsiued by a non ineger order of derivaion α. The soluion of his new diffusion equaion fracional diffusion equaion leads o Levy processes ha are considered as possible source of anomalous diffusion processes. Finally a brief and absoluely no exhausive bibliography is repored. References 1] K. S. Miller and B. Ross, An Inroducion o he Fracional Calculus and Fracional Differenial Equaions, John Wiley, New Yor ] B. J. Wes, M. Bologna and P. Grigolini, Physics of Fracal Operaors, Springer-Verlag 23. 3] R. Hilfer, Applicaion of Fracional Calculus in Physics, World Scienific, Singapore