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Generalized Riemann - Liouville fractional derivatives for multifractal sets

L.Ya.KobelevDepartment of Physics, Urals State UniversityLenina Ave., 51, Ekaterinburg 620083, Russia

E-mail: [email protected]

The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractionalexponent d are functions of space and times coordinates (i.e. d = d (r (t ) , t )).

01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p.

I. INTRODUCTION

Fractional derivatives and integrals (left-sided andright-sided) Riemann - Liouville (see [ 1]- [3]) from func-tions f (t) (dened on a class of generalized functions)are

D d+

,t f (t) = 1

(n d)

d

dt

n

t

a

f (t )dt

(t t

)d

n +1 (1)

D d ,t f (t) = ( 1)n

(n d) ddt

n

b

t

f (t )dt

(t t)d n +1 (2)

where ( x) is Eulers gamma function, and a and b aresome constants from [0 , ). In these denitions, as usu-ally, n = {d} + 1 , where {d} is the integer part of d if d 0 (i.e. n 1 d < n ) and n = 0 for d < 0. Fractionalderivatives and the integrals ( 1)-(2) allow to use, insteadof usual derivatives and integrals, the integral functionalsdened on a wide class of generalized functions. It is veryuseful for the solution of a series of problems describingstochastic and chaos processes, abnormal diffusion, quan-tum theories of a eld etc. [ 4]- [10]. It is possible to con-sider appearance of integral in ( 1)-(2), from the physicalpoint of view, as the result of taking into account inu-ence of the contributions from some physical processes(characterized by the kernel ( t t ) d + n 1 1(n d)) inearlier (left-side derivative) or later (right-hand deriva-tive) times, on function f (t) that is, as the partial takinginto account the system memory about past or futuretimes. The value of fractional exponent d characterizesthe degree of the memory. Lets consider multifractal set(without self-similarity) S t consisting from innite num-ber of subsets s i (t i ), also being multifractal. Each subset

s i (t i ) is compared with fractional value (or number of values), describing its fractal (fractional) dimension (boxdimension, Hausdorff [ 11] or Renie [12] dimension etc. -see, for example, [ 13]), depending from the numbers of a subset s i (t). Let the carrier of measure of multifractalset S t be the set R n . For exposition of changes of a con-tinuous function f (t) dened on subsets si (t i ) of set S t ,it is impossible to use ordinary derivatives or Riemann -Liouville fractional derivatives ( 1), as the fractional di-mension of sets d on which f (t) is dened depends on

t i , that is on the choice of the subset si (t i ) . There is aproblem: how can the denition ( 1)-(2) be changed tofeature small (or major) changes of function f (t) denedon sets si (t i ) ? The purpose of this paper is to presentthe generalization of the Riemann - Liouville fractionalderivatives ( 1)-(2) in order to ajust them for functionsdened on multifractal sets with fractal dimension (frac-tional dimension) depending on the coordinates and time.

II. GENERALIZED FRACTIONAL DERIVATIVESAND INTEGRALS

We shall treat subsets si (t i ) as the points ti (witha continuous distribution for different multifractal sub-sets s i (t i ) ) of multifractal set S t ). Assume that the func-tion d(t i ) = d(t), describing their fractional dimension(in some cases coinciding with local fractal dimension)as function t is continuous. For the elementary gener-alization ( 1)-(2) is used physical reasons and variable tis interpreted as a time. For continuous functions f (t)

(generalized functions dened on the class of nitaryfunctions (see [ 3])), the Riemann - Liouville fractionalderivatives also are continuous. So for innitesimal in-tervals of time and the functionals ( 1)-(2) will vary oninnitesimal quantity. For continuous function d(t) thechanges thus also will be innitesimal. It allows, as theelementary generalization ( 1) suitable for describing of changes f (t) dened on multifractal subsets s(t), as wellas in (1)-(2), to summate inuence of a kernel of integral(t t ) d ( t ) n +1 1(n d(t)), depending on d(t), in allpoints of integration and, instead of ( 1)-(2), introducethe following denitions (generalized fractional deriva-tives and integrals (GFD)), taking into account also thed(t) dependence on time and vector parameter r (t) (i.e.dt dt ( r (t), t ))

D d t+ ,t f (t) = ddt

n t

a

dt f (t )

(n dt (t ))( t t )d t ( t ) n +1

(3)

D d t ,t f (t) = ( 1)n

1
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ddt

n b

t

dt f (t )

(n dt (t ))( t t)d t ( t ) n +1 (4)

In (3)-(4), as well as in ( 1)-(2), a and b stationary valuesdened on an innite axis (from to ), a < b ,n 1 dt < n , n = {dt } + 1, {dt }- the whole part of dt 0, n = 0 for dt < 0. The sole difference ( 3)-(4) from

(1)-(2) is: dt = dt (r

(t), t )- fractional dimension (furtherwill be used for it terms fractal dimension (FD) or the local fractal dimension (LFD) ) is the functionof time and coordinates, instead of stationary values in(1)-(2).

Similar to ( 3)-(4), it is possible to dene the GFD,(coinciding for integer values of fractional dimensiond r ( r , t ) with derivatives with respect to vector variable r )D d bfr+ , r f (r , t ) respect to vector r (t) variables (spatial co-ordinates). We pay attention, that denitions ( 3)-(4)area special case of Hadamard derivatives [ 14].

III. FRACTIONAL DERIVATIVES FORD (R (T ), T ) 1

For FD which have very small differences from of in-teger values it is possible approximate to change theGFD by the usual derivatives and integrals. For an es-tablishment of connection of GFD with orderly deriva-tives we shall see ( 3), for example, for a case d(r (t), t ) =1 + (r (t), t ), 1, d < 1, (if utilize the theorem of themean value of integral) as

D 1 + ,t f (t) = t

t

0

f (t )d ((t ))( i )1 ( t )

=

= t

[ f (t cp (t))t

0

d ( )1 ( t )

] (5)

where f = 1f and tmed - some value of . As 0 it ispossible to estimate values of integral in ( 5) for minimumand maximal values of ( > 0)

t

0

d 1 min ( t )

= t min

min,

t

0

d 1 max ( t )

= t max

max(6)

For selection from integrals ( 6) the trms which are inde-

pendent from (because of f ) we use decompositiont = 1 + ln t + ... We obtain

D 1 + ,t f (t) f (t)

t +

f (t cp (t)t

ln t +f (t cp (t)

t(7)

For major times t = t0 + ( t t0),t t0 t0 the ap-proximate representation GFD ( 7) through usual deriva-tives will accept a view (if neglect by additions of order

f /t 0 ,(t t0)/t 0 , to use the designation ln t0 = and if itis accounted, that cp t because of the basic contribu-tion to integral ( 5) is stipulated by small )

D 1 + ,t f (t) f (t)

t +

f (t)t

(8)

In (8) play a role of a stationary value of parameter

of regularization and if change (in f = f ) on quantity 1 , GFD ( 8) is not depends practically on this

parameter.We shall give below, another method of deduction the

relation ( 8) using an expansion 1 in a power seriesof under sign of integral and again for d(r (t), t ) withpoorly difference from the whole value (but d > 1). Letsfractional dimension d(r (t), t ) is equal unity with smallvalue (d(r (t), t ) = 1 + (r (t), t ), 1) and expandFD in ( 3) in a power series on by a rule ( t ) =1 ln(t ) + .... Restricted expansion FD by the rsttwo members of a series, we obtain for left-side fractionalderivative (for a = 0)

D 1+ + ,t f (t) = 2

t 2

t

0

f ( )(1 )( t ) ( r ( ) , )

d

t

( 1

(1 )f (t))

2

t 2

t

0

f ( )d (1 )[(t ) i ]

,

0 (9)

Integral in ( 9) is considered as a generalized function withdetermined regularizasion and after regularization of in-tegral the parameter of regularization is picked by arequirement of the best coincidence of approximate and

exact results of integral calculation the members of a rstorder at (it is necessary to take its real part, the pa-rameter is necessary to put zero after calculations). Af-ter an integration by parts and also using of a relation1/x = P (1/x ) or another regularizations we shall re-ceive (if take into account that integrals ( 1)-(2) are realvalues and dene the fractal addendum to derivatives asa coefficients at the imaginary parts of integrals)

D 1+ + ,t f (t) = t

[ 1

(1 )f (t)]

t

[ (t)f (t)(1 (t))

] (10)

were dened by selection of regularization .The selec-tion of a sign in ( 6) is dened by a selection of the reg-ularization. From ( 10) the opportunity follows (at thesmall fractional additives to FD of time) to use for de-scribing of changes of functions dened on multifractalsets of time by means of using the renormalized ordinaryderivatives. At the same time, the dependence FD of the time from coordinates and time is concerned. Letsconsider fractional dimension d for case when d smaller of unity ( d = 1+ ,d < 1). For this case fractional derivative(see (3) for n = 1) looks like

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D 1 + ,t f (t) = t

t

0

f ( )d (( ))( t i )1 ( )

(11)

Taking into account, that for ( 11) selection , by virtueof denition ( 3)-(4)), is prohibited, for including in ( 11)value D1 + ,t f (t) at = 0 before a right member in ( 11)(applicable only for > 0) it is necessary to take into

account a addendum from ( 9) with = 0, i.e. f ( t )

t .We receive (if use a rule of a regularization that was hadused before for d > 1 i.e. Reg(1/x ) (x) and relation(1 + ) = ())

1

t 1 f (t)

t

f (t) t

[ (r (t), t )f (t)(1 + (r (t), t ))

f (t)] (12)

The approximate representation GFD by ordinaryderivatives (relations ( 8),( 10),( 12)) if use different meth-ods are very similar, so any of them may be used in fol-low calculations. The above mentioned approximate con-nections of generalized fractional derivatives ( 3)-(4) de-ned on the multifractal sets with fractional dimensiond( r (t), t ) (if d(r (t), t ) poorly distinguished from unity)with ordinary derivatives may be ex-tend for the caseswith arbitrary n:d(r (t), t ) = n + (r (t), t ), || 1. Abovementioned reasoning make possible to show, for a cases 1 (but not close to integer values), that the represen-tations of generalized fractional derivative by means of derivatives of integer order will contain integer deriva-tives of arbitrary high orders. Lets consider a sym-metrical generalized fractional derivatives Dd ,t f (t) andD d+ ,t f (t):

D dt f (t) = 0 .5(Dd+ ,t + D

d ,t )f (t) (13)

The symmetry of GFD allows to take into account theinuence on event that happens in the given instant fea-tured by function f (t) both past, and future (by frac-tional integration and differentiation on time). For frac-tional integration and differentiation at coordinates thesymmetrical GFD takes into account inuence the eventwith given coordinate of all points of space

D dr f (t) = 0 .5(D d+ , r + Dd , r )f (t)

At small difference of dimensions of time (or space)from unity D 1+ + ,t D

1+ ,t and so on.

IV. CONNECTION WITH COVARIANTDERIVATIVES

Let dene ( 10) as

D 1+ + ,t f (t) A t

f Bf (14)

where

A(r (t), t ) = (1 ) 1 + a (15)

B ( r (t), t ) = (1 ) 2(1 + a) t

at

, (a = 1)

(16)

The relation ( 16) reminds the covariant derivatives, fre-

quently meeting in physics. It is possible to show, that ata various selection of a mathematical nature of functionf (vector, tensor etc.) and relevant selection of function, GFD ( 10) really coincides with covariant derivatives(see [15], [16]).

V. EQUATIONS WITH GENERALIZEDFRACTIONAL DERIVATIVES

The equations with GFD are possible to connect withnatural sciences (in particular, physics) when the fractaldimensions dt and dr are connected with describing mul-

tifractal structure of a surfaces of solid bodies, structureof chaos, structure of time and space (see, for example,[5], [7], [8], [11], [16]. In some cases GFD are related toequations with FD that depends from functions (or func-tionals) the same to which GFD was applied. It givesin the interesting nonlinear fractional integral-differentialequations with GFD

F (D d t ( f ( t ))+ ,t )f (t) = 0 (17)

where F - function or functional from GFD. A new class of the equations in fractional integral-differential function-als represent the equations such as ( 13). Their examina-tion, apparently, is an interesting problem and representsa new approach to describe problems of chaos.

VI. CONCLUSION

The generalized Riemann-Liouville fractional deriva-tives dened in the paper allow to describe dynamicsand changes of functions dened on multifractal sets,in which every element of sets is characterized by itsown fractional dimension (depending on coordinates andtime). At small differences of fractional dimensions fromtopological dimensions, generalized fractional derivativesare represented through expressions similar to covariant

derivatives used in physics.

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[10] G.M.Zaslavsky, Chaos, 4,25, (1994)[11] Hausdorf F., Math. Ann. 79 (1919), P.157-179[12] Renyi A. Introduction to information theory, Appendix

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[17] L.Ya.Kobelev Multifractal time, covariant derivatives and gauge invariance Urals State University, Dep. vVINITI 07.09.99, No.2791-B99

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