Applied Mahemaical Sciences, Vol. 2, 28, no. 4, 1949-1962 From Self-Similariy o Fracional Derivaive of Non-Differeniable Funcions via Miag-Leffler Funcion Guy Jumarie Deparmen of Mahemaics Universiy of Québec a Monréal P.O. Box 8888, Downown Saion Monréal, Qc H3C 3P8, Canada jumarie.guy@uqam.ca Absrac. This shor noe focuses on he relaion beween fracional differeniabiliy, self-similariy and he Miag-Leffler funcion, and ries o clarify he origin and he very reason for inroducing fracional derivaive. There is a sriking similariy beween self-similariy and fracional differeniabiliy, and a fracional differeniable funcion can be hough of as being he sum of a consan funcion wih a self-similar funcion. Fracional derivaive is defined as he limi of fracional difference, and hen, by using Laplace s ransform, one obains an inegral represenaion which applies o non-differeniable funcions, herefore Taylor s series of fracional order for non-differeniable funcions. The discrepancy beween he modified Riemann- Liouville derivaive so obained and some sandard definiion is examined and explained. And, in quie a naural way, we conclude ha he series which defines he Miag-Leffler funcion is nohing else bu is fracional Taylor s series. Inegral wih respec o (d) appears o be quie relevan. Mahemaics Subjec Classificaion: 26A33, 41A58, 6G18 Keywords: Fracional differeniabiliy, self-similar funcion, Miag-Leffler funcion, Modified Riemann-Liouville derivaive, fracional Taylor s series, coarse-graining
195 Guy Jumarie 1. Inroducion When Mandelbro [28-3] inroduced fracional Brownian moion as he fracional derivaive [2-4.6-9,22,25-27,31-39] of a Brownian moion, he noiced ha i is also self-similar Recenly we suggesed a model of fracional derivaive which has he Miag-Leffler funcion E ( λ ) as eigenfuncion. Our purpose in he presen shor noe is o come back on he definiion of fracional derivaive, and o commen on is relaion wih self-similar funcions and he Miag-Leffler funcion, and also o provide a careful moivaion for he derivaion of he so-called modified Riemann- Liouville derivaive which we inroduced recenly o circumven some roublesome feaures involved in some by-now aken for graned classical definiions [5,1]. This shor noe is organized as follows. Saring from a qualiaive inuiive definiion of fracional differeniabiliy which is based on cusomary differenials and discards differeniable funcions, we shall bear in mind he relaion which exiss beween fracional differeniabiliy and self-similariy: clearly when a funcion is selfsimilar hen i is fracional differeniable, and he converse is parially rue, i.e., if i is fracional differeniable, hen i is he sum of a self-similar funcion wih anoher differeniable funcion (which may be a consan). As a resul, he fracional derivaive of a consan should be zero. Then by using a formal operaional calculus, one uses fracional difference o generalize derivaive of funcions, bu i appears ha his expression does no apply o consan funcions, and refers o self-similar funcions only, herefore he need o consider one a a ime consan funcion on he one hand, and self-similar funcion on he oher hand. Then we shall use he Laplace s ransform o obain he inegral represenaion of he above model, and we shall examine is discrepancy wih he socalled Capuo s definiion [5]. As a by-produc, we shall be led o inroduce fracional inegral wih respec o Wha follows focuses mainly on he fracional derivaive of non-differeniable funcions, and provides a careful derivaion of modified Riemann-Liouville derivaive. (d). 2. Desideraa for a Definiion of Fracional Derivaive 2.1 Fracional derivaive and self-similariy ( ) Definiion 2.1 A funcion x () is said o have a fracional derivaive x ( ) of order, < < 1 whenever he limi ( Δ x( ) : = x( ) x(), he symbol := means ha he lef-side is defined by he righ-one) ( ) Δx( ) x ( ) lim Δ ( Δ), (2.1) exiss and is finie.
Fracional derivaive of non-differeniable funcions 1951 Shorly, his amouns o wrie he equaliy dx = b (d). (2.2) As a direc resul of his definiion, when x () is a consan, one has ha dx ( ) =, and hus he fracional derivaive of a consan should be zero. In a like manner, when x () is differeniable, one has he differenial dx = x&( ) d, and as a resul is fracional derivaive is zero. Clearly, on he pracical sandpoin, he definiion 2.1 deals wih non-differeniable funcions only. Definiion 2.2. A funcion x () is said o be self-similar wih he Hurs parameer > whenever here exiss a posiive consan a such ha x( a) a x( ). (2.3) As a resul of (2.3), one has he equivalence x( ) x(1). (2.4) Some remarks. (i) According o (2.2), for small one has x( ) x() b, herefore x( a) x() a ( x( ) x() ), in oher words, if x () is -h differeniable, hen x( ) x() is self-similar, wih as Hurs parameer, or again when x () is locally self-similar a =. (ii) The converse is sraighforward. If x () is self-similar, hen x ( ) =, and according o (2.4) one has x( d) = dx x(1) ( d) in such a manner ha x() is locally -h differeniable a zero. (iii) According o (2.2), he fracional derivaive of a consan should be zero.. (iv) Assume ha x ( ) = f ( ) + g( ) where f () is differeniable and g () is -h differeniable, hen one has he equaliy Δx = b ( Δ) + b1δ, herefore one sill obains he limi condiion expressed by (2.1). I follows ha, exacly like a derivaive defines a funcion up o an arbirary addiive consan, a funcion of which only he fracional derivaive is known, is defined up o an addiive differeniable funcion, incluiding consan funcions. More generally, if one has he equaliy β Δ x = b ( Δ) + b1 ( Δ), < β hen x () is -h differeniable a zero. (v) And he las remark of imporance is relaed o he value of x (). In (2.3) one has x ( ) =, bu in (2.2) x () may have any value. These simple remarks show how fracional derivaive and self-similariy are deeply relaed, a such a poin ha, in a firs approach, we could have resriced ourselves o he class of self-similar funcions only.
1952 Guy Jumarie 2.2 Fracional derivaive and Miag-Leffler funcion The Miag-Leffler funcion ( y), y, is defined by he series k E ( ) = k = ( k)! E =, (2.5) wih he noaion ( k)!: = Γ(1 + k) where Γ (.) is he Euler funcion. Exacly like he exponenial e () is he soluion of he differenial equaion Dy ( ) = y( ), y() = 1, we would like o define a fracional derivaive D which would have E ( ) as eigenfuncion, ha is o say which would provide he equaliy D E ( ) = E ( ). (2.6) By his way, we would be in a posiion o expand a fracional calculus parallel o he classical one, by subsiuing E ( λ ) for e( λ ) almos everywhere. On subsiuing (2.5) ino (2.6), we hen arrive a he following se of desideraa for a suiable definiion of fracional derivaive. D ( K) =, K consa n, (2.7) k ( k)! ( k 1) D x = x. (2.8) ( k )! And well obviously his fracional derivaive should provide he cusomary derivaive when = 1. Remark ha he condiion (2.7) is no required for convenience only, bu on he conrary is quie consisen wih he remarks above, in Subsecion 2.2 3. Fracional Derivaive via Fracional Difference 3.1 Fracional derivaive for self-similar funcions Inroducion of he candidae modeling Definiion 3.1 Le f : R R, f ( ), denoe a coninuous (bu no necessarily differeniable) funcion. Le h > denoe a consan discreizaion span. On defining he forward operaor FW (h) by he equaliy FW ( h) f ( ) : = f ( + h) ; (3.1) one can wrie he finie difference Δf () of f () in he form Δ f ( ) : = ( FW 1) f ( ), (3.2) and quie in a naural way, his suggess o define he fracional difference of order, < <1, of f () by he expression [11-19]. Δ f ( ) : = ( FW 1) f ( )
Fracional derivaive of non-differeniable funcions 1953 k = ( 1) f [ + ( k) h], (3.3) k = k wih he noaion ρ ( FW ) f ( ) = f ( + ρh). Wih his definiion, in quie a naural way, we are led o define he fracional derivaive of order of f () as he limi ( ) Δ f ( x) f ss ( x) = lim. (3.4) h h Inconsisency of his model wih he expeced derivaive of a consan According o he remark (iii) in he subsecion (2.1) ogeher wih he equaion (2.7), we have seen ha he fracional derivaive of a consan should be zero, and here wih he expression (3.4), we have o examine wha happens abou.to his end, in order o cope wih problem of serial convergence which so appears, we shall work wih he Laplace ransform. ( ) ( ) According o (3.4), he Laplace s ransform L{ f ss ( ) } of f ss ( ) is equal o ( ) L{ f ( ) } s Fˆ ss = ( s) (3.5) where F ˆ ( s ) denoes he Laplace s ransform of f (). This expression can be derived as follows. Firsly, as a preliminary resul, we remark ha one has he following equaliy, sh sh { f + h) } = e Fˆ ( s) e h s ( e f ( ) d, L which provides, for small h, L{ f ( + h) } sh ˆ sh e F( s) h e f () (3.6) This being he case, aking he Laplace ransform of (3.3) yields hs { } ˆ hs k ( k ) hs L Δ f ( ) = e F( s) he f () + ( 1) e Fˆ ( s) k = 1 k sh ( ) ˆ hs = 1 e F( s) he f (). For a small h, we hen have { } ˆ 1 L h hs Δ f ( ) h (1 1 + sh) F( s) h e f (), (3.7) and by making h end o zero, we obain he resul. Furher remarks and commens (i) According o (3.7), for a consan K differen from zero, we would have ( ) 1 { K } = s K L ss, in oher words, wih his definiion, he fracional derivaive of a consan would no be zero, wha conradics one of our desideraa above (see subsecion 2.1 and 2.2). 1 This feaure is due o he presence of h in he coefficien of f () in he expression (3.7). (ii) Remark ha he classical formula for derivaive is L f ( ) = sfˆ ( s) f (, (3.8) { } )
1954 Guy Jumarie and as in evidence i is no equivalen o (3.5) when = 1. And his gives rise o he quesion as o wheher here is some inconsisency or misake somewhere? (iii) Firs of all, he consisency beween (3.5) and (3.6) is complee when f ( ) =, which is he condiion saisfied by self-similar funcions. In oher words, he definiion (3.4) can be considered as being quie saisfacory for self-similar funcions. (iv) Moreover, according o he subsecion 2.1, when we resric ourselves o he class of funcions which saisfy he condiion f ( ) =, hen we can claim he following: if he funcion is fracional differeniable, hen i is self-similar; and conversely, if i is self-similar, hen i is fracional differeniable. (v) To cope wih he problem which is so occurring wih he consan erm f (), we shall sae ha he fracional derivaive of a consan is zero, and we are so led o he following definiion. 3.1 Fracional derivaive for funcions wih nonzero iniial condiions Definiion 3.1 Le f : R R, f ( ), denoe a coninuous (bu no necessarily ~ differeniable) funcion, and define f ( ) : = f ( ) f (). Le h > denoe a consan discreizaion span. The fracional derivaive of order of f () is defined as he limi ~ ( ) Δ f ( ) f ( ) : = lim. (3.9) h h An alernaive is he following: Definiion 3.2. Assume ha g () is a self-similar funcion wih he Hurs parameer H, < H < 1, hen is fracional derivaive of order H is defined by he expression H ( H ) Δ g( ) g ( ) = lim h H h According o his definiion 3.1, he fracional derivaive of a consan K is ~ auomaically zero, since hen one has K =. On has he following ( ) Lemma 3.1 According o he Definiion 3.1, he Laplace ransform L{ f ( ) } is provided by he equaliy ( ) { ( )} ˆ 1 L f = s F( s) s f (). (3.1) ~ Proof. I is sufficien o apply he equaliy (3.5) o he funcion f ( ). - Remark ha (3.1) provides (3.8) when = 1. 4. Derivaion of he Inegral Represenaion 4.1 Towards he definiion One has he Laplace s ransform
Fracional derivaive of non-differeniable funcions 1955 { 1 Γ 1 ( 1) } = s, < < 1 L, and by using he convoluion heorem, in quie a direc way, he equaliy (3.1) suggess o define he fracional derivaive by he inverse Laplace ransform { s Fˆ 1 ( s) } = 1 1 L ( τ ) f ( τ ) dτ. Γ( 1) Unforunaely such a definiion comes wih some problems on he convergence of he inegral, and o cope wih his difficuly, we may re-wrie (3.1) eiher in he form ( ) 1 { ( )} ˆ 1 L f = s sf( s) s f (), (4.1) or as ( ) 1 { ( )} ˆ 1 L f = s s F( s) s f (). (4.2) 1 The equaion (4.1) involves he erm s ( sfˆ ( s) ) and hus assumes explicily ha f () is differeniable. In oher words, (4.1) refers o funcions which are boh differeniable and fracional differeniable wih < < 1. In conras, (4.2) does no make any assumpion on he differeniabiliy of f () : sricly speaking, i would apply o a funcion which is coninuous bu no necessarily differeniable. To summarize, (4.2) applies o self-similar funcions whils (4.1) does no. For he sake of generalizaion, we selec (4.2) and as a resul we so arrive a he following definiion 4.2 Modified fracional Riemann-Liouville derivaive (via inegral). Definiion 4.1 (Riemann-Liouville definiion revisied). Refer o he funcion f(x) of he Definiion 3.1. (i) Assume ha f () is a consan K. Then is fracional derivaive of order is K D K =,, (4.3) Γ(1 ) =, >. (4.4) (ii) When f () is no a consan, hen one will se f ( ) = f () + ( f ( ) f ()), and is fracional derivaive will be defined by he expression ( ) f ( ) = D f () + D ( f ( ) f ()) in which, for negaive, one has 1 1 Dx ( f ( ) f ()) : = ( ) ( ), Γ( ) τ f τ dτ <, (4.5) whils for posiive, one will se D ( f ( ) f ()) = D f ( x) = ( ( ( )) 1), < < 1 f,
1956 Guy Jumarie When n < n + 1, one will se d = 1 ( τ ) Γ(1 ) d ) ( n) ( n) ( f ( ) ), n < n + 1, n 1 ( f ( τ ) f ( ) dτ. (4.6) ( ) f ( ) : =. (4.7) We shall refer o his fracional derivaive as o he modified Riemann Liouville derivaive, and i is of order o poin ou ha i is sricly equivalen o he definiion 3.1, via he equaion (4.1).. Wih his definiion 4.1, he Laplace ransform L{ } ( ) 1 { f ( x) } = s L{ f ( x) } s f (), < < 1 of he fracional derivaive is L. (4.8) 4.3 Furher remarks and commens As i is well known, many auhors prefer o selec he equaion (4.1) which yields he derivaive ( ) 1 ( ) = ( ) ( ), < < 1 Γ(1 ) f c τ f τ dτ. (4.9) Neverheless, his definiion may appear o be somewha quesionable, due o various roublesome feaures (We beg he reader o see here only an academic conenion and nohing more, please!) (i) Here one uses he derivaive of order one o calculae a derivaive of order lower han he uni. Usually one works in he opposie way. The derivaive of order, < <1, would be used o calculae he firs order derivaive. Here, a he exreme, when =, f & () is used o define f (). Or again, i looks like if we were using he second order derivaive o define he firs one. This is raher disurbing from a physical sandpoin. (ii) The use of his definiion is more especially quesionable in he modeling of physical phenomena involving coarse-graining space, in which, loosely speaking, he poin has a hickness, in such a manner ha coninuousness may be no quie relevan. (iii) Anoher poin is ha (4.9) assumes explicily ha x () is differeniable. Bu ( ) hen, wha happens when i is no he case? Sricly speaking, f ( ) may exis whils f & () does no. The ypical example of such a case is provided by he Gaussian whie noise. Irrespecive of any sochasic framework, if we consider a realizaion of his process as a non-random funcion, hen we canno apply (4.9). To summarize, we shall say ha he fracional derivaive (4.9) applies o differeniable funcions only, whils he modified Riemann-Liouville derivaive deals wih funcions which may be differeniable.
Fracional derivaive of non-differeniable funcions 1957 5. On he Relaion Beween Fracional Derivaive And Derivaive 5.1 Fracional derivaive and fracional ani-derivaive In usual calculus, he equaliy dx = g( ), x() = (5.1) d is equivalen o x( ) = g( τ ) dτ. (5.2) The righ-side erm of (5.2) is considered as an ani-derivaive, in he sense ha i is defined by he equaliy d g( ) d g( ) d τ τ =. (5.3) On applying he same poin of view o fracional derivaive, we shall say ha he fracional differenial equaliy dy = g( )( d), y() =, < < 1 (5.4) is equivalen o he fracional inegral y( ) = g( τ )( dτ ), (5.5) and here, he righ-side erm of (5.5) would be he ani-deraive of order defined by he ideniy d g( τ )( dτ ) = g( )( d). (5.6) 5.2 Applicaion o he pair (derivaive, fracional derivaive) Assume ha he derivaive x& () is known whils x () is unknown. A suiable definiion of fracional derivaive should, will saisfy he equaliy 1 ( ) D x& ( ) = x ( ) (5.7) which provides 1 ( ) x& ( ) = D x ( ). (5.8) According o he definiion 2.1, (5.8) can be re-wrien in he form ( ) 1 dx = x(d) &, herefore (see (5.4),(5.5)) 5.3 On he definiion of fracional inegral x ( ) ( ) 1 ( ) = x () + x& ( d). (5.9) On combining he fracional Taylor s series (see he secion 6 below wih he definiion 4.1, we have proposed he definiion [14,16,19]
1958 Guy Jumarie 1 g ( τ )( dτ ) : = ( ) ( ), < < 1 τ g τ dτ, (5.1) which relaes fracional inegral (i.e. inegral wih respec o (d) ) and Riemann inegral. Le Y () denoe he Heaviside sep funcion, i.e. Y ( ) 1 when and Y ( ) = elsewhere; and define he impulse funcion U h (), U h ( ) : = Y ( ) Y ( h). (5.11) Wih his noaion, (5.1) provides he equaliy N 1 g( τ )( dτ ) lim U ( τ jh) g( τ )( dτ ) = h j= which could serve as a poin of deparure o an approach via analysis. Remark ha h N 1 ( j+ 1) h g( τ )( dτ ) lim U ( τ jh) g( τ )( dτ ) h j= jh h, (5.12) 6. On he Compleeness of he Approach In he equaion (2.1), we have defined he fracional derivaive by means of he quoien Δ x /( Δ). In he equaion (3.9) we explicily defined fracional derivaive by using a fracional difference. For he compleeness of he approach, i is compulsory o exhibi he relaion beween (2.1) and (3.9). This can be done as follows. (Sep 1) Wih he above definiion of fracional derivaive, i is easy o show ha he soluion of he fracional differenial equaion ( ) y ( ) = λy( ), y() = y, (6.1) is y ( ) = ye ( λ ). (6.2) (Sep 2) A simple calculaion provides he formal operaional equaliy Dh FW ( h) = D FW ( h), (6.3) which, considered as a fracional differenial equaion wih respec o FW (h), yields he formal expression (see (6.1),(6.2)) FW ( h) = E ( h D ). (6.4) (Sep 3) Expliciing (6.4) provides he fracional Taylor s series k h ( k ) x( h) x( ) + + = x ( ). (6.5) k = 1 ( k)! (Sep 4) For a small h, we shall consider he erm k = 1 only o obain he fracional Rolle formula 1 ( ) dx( ) = (!) x ( )( d) (6.6)
Fracional derivaive of non-differeniable funcions 1959 which is exacly (2.1). Remark ha if we define he Miag-Leffler funcion as he soluion of he fracional differenial equaion (6.1), hen he series (2.5) appears o be exacly is fracional Taylor s series. 7. Concluding Remarks Prospecs and commens We believe ha many auhors who deal wih fracional derivaive have no in mind, or have forgoen ha here are deep relaions wih self-similariy, wha could be of help o significanly apprehend he quesion. A self-similar funcion has a fracional derivaive and he converse is almos exac, and we should have his feaure in mind when we use fracional derivaive in he modeling of physical phenomena. A funcion which is coninuous everywhere bu nowhere differeniable canno be replicaed. As a resul, i necessarily exhibis random-like characerisics, and his is he reason why fracional sochasic processes and fracals appear in he lieraure as companion opics of analysis [28-3]. On a modeling sandpoin, he rend of our hough is ha when a funcion is differeniable, he knowledge of is fracional derivaive will no, should no conribue significanly any more o is pracical meaning. Shorly, he very reason of using fracional derivaive would be he analysis of non-differeniable funcions defined in coarse-graining space. The fracional Taylor s series so obained is very aracive, and allows us o expand a fracional analysis quie parallel o he sandard one, and mainly o define inegral wih respec o (d).by his way we have been in a posiion o go a sep furher and o inroduce a fracional probabiliy measure in he form μ ( x ) = p( x)( dx) which provides a suppor o some space and ime fracal Fokker-Planck equaions which have been proposed in he lieraure [21], and exhibis some similariies wih quanum probabiliy. Kolwankar s approach As a las remark, i is of order o menion ha he equaion (5.6) has been independenly obained by Kolwankar [23, 24] who refers o i as o local Taylor s formula. The framework is no he same, and his auhor uses an approach which is quie differen from ours, by using funcions defined on fracal ses o arrive a nondiffereniable funcions. In some sense, Kolwankar explains he fracal naure of he non-differeniabiliy, whils we describe i only. Kolwankar s work is heoreical whils our modeling is merely formal. Neverheless, i is ineresing o noice ha he wo approaches provide he same Rolle-Taylor s formula.
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