The definition of the fractional derivative was discussed in the last chapter. These

Transcription

1 Chapter 3 Local Fractional Derivatives 3.1 Motivation The definition of the fractional derivative was discussed in the last chapter. These derivatives differ in some aspects from integer order derivatives. In order to see this, one may note, from equation (2.4), that except when (51 is a positive integer, the q^^ derivative is nonlocal as it depends on the lower limit 'a'. The same feature is also shown by other definitions. However, we wish to study local scaling properties and hence we need to modify this definition accordingly. Secondly from equation (2.23) it is clear that the fractional derivative of a constant function is not zero. Therefore adding a constant to a function alters the value of the fractional derivative. This is an undesirable property of the fractional derivatives to study fractional differentiability. While constructing local fractional derivative operator, we have to correct for these two features. This forces one to choose the lower limit as well as the additive constant 47

2 before hand. The most natural choices are as follows. 1) We subtract, from the function, the value of the function at the point where we want to study the local scaling property. This makes the value of the function zero at that point, canceling the effect of any constant term. 2) Natural choice of a lower limit will again be that point itself, where we intend to examine the local scaling. In the following section we formally introduce the concept of local fractional derivative. 3.2 Definition Definition 4 If, for a function f : [0,1] -^ IR, the limit l)'/to) = lim'"'^'"'-(<"». 0<,<1 (3.1) ^->y d{x y)i exists and is finite, then we say that the local fractional derivative (LFD) of order q (denoted by E)'^f{y)), at y, exists. This defines the LFD for 0 < g < 1. It was first introduced in [1], and later generalized [2] to include all positive values of q as follows. Definition 5 //, for a function f : [0,1] > JR, the limit ^^' ^^y [d{x-y)]i exists and is finite, where N is the largest integer for which N^ derivative of f(x) at y exists and is finite, then we say that the local fractional derivative (LFD) of order q {N < q < N + 1), at X = y, exists. 48

3 We subtract the Taylor series term in the above definition for the same reason as one subtracts f{y) in the definition 4. We do this to supress any regular behavior that may mask the local singularity. Definition 6 The critical order a, at y, of a function f is a{y) = sup{q\d'''f{y), q < q, exists}. Sometimes it is essential to distinguish between limits, and hence the critical order, taken from above and below. In that case we define ^^^^'^ = i?^ [d±i.~y)]^ ^^-'^ We will assume E)'^ E)\ unless mentioned otherwise. As an example consider the function f{x) = Ixl'', a > 0. The critical order of this function from above at origin is a when a is noninteger and is oo when a is an integer. The local fractional derivative that we have defined above reduces to the usual derivatives of integer order when <? is a positive integer. In order to see this point for g = 1 we consider equation (3.1). Since the Riemann-Liouville fractional derivative on RHS of equation (3.1) reduces to ordinary first derivative when q \ (This follows from the relation d d'^fjx) d^+^f{x) dx [d{x - y)y ~ [d{x - y)]«+i and setting q = 0. For details, see the book by Oldham and Spanier [3], page 50. See also ref. [5]), we get ID^fiy) = lim^impm. (3.4) ^->-y ax 49

4 Now since f{y) is constant its derivative is zero. Therefore after taking the limit we get ID'fiy) = ^. (3.5) For N < q < N + I the definition of local fractional derivative is «,;( =,j:mz^j^,mtziyi,3., '^^' ^^y [d{x-y)]i ^ ' Now, li q n (i.e. A^ = n 1 in view of the above range of q), we get ^"[/(^) - E.-0 TT^i^ - y)'] DV(y) = lim ^^ ^ ^'-oro+d^ ^^ ^ ^^ Again since y is a constant idv(y) = ^ ^. (3.8) Therefore, for g = n, local fractional derivative reduces to nth order derivative. From this it is clear that in our construction local fractional derivatives generalize the usual derivatives to fractional order keeping the local nature of the derivative operator intact, in contrast to other definitions of fractional derivatives. The local nature of the operation of derivation is crucial at many places, for instance, in studying differentiable structure of complicated manifolds, studying evolution of physical systems locally, etc. The virtue of such a local quantity will be evident in the following section where we show that the local fractional derivative appears naturally in the fractional Taylor expansion. This will imply that the LFDs are not introduced in an ad hoc manner merely to satisfy the two conditions mentioned in the beginning, but they have their own importance. 50

5 3.3 Local fractional Taylor expansion An interesting consequence of the above definitions is that the LFDs appear naturally in the fractional Taylor expansion. Derivation of local fractional Taylor expansion We follow the usual procedure to derive Taylor expansion with a remainder [10]. In order to derive local fractional Taylor expansion, let It is clear that ld'f{y) = F{yAq) (3-10) Now, for 0 < 9 < 1, [d{x-y)\ ^[d{x-y)\i provided the last term exists. Thus = ^^[F{y.t;q)j{x-y-tr'dt]r 1 f-ydf{y,t;q){x-y-t)^,-. mvjlsl^szlzlljt (3.12) r(g) Jo dt q f[x)-f[y) = ^flj(^-^)', _ l r - ^ d F i y J ^ (3.13) r(<?+i)yo dt ^ '^ ' ^ ^ 51

6 I.e. /(x) = f{y) + ^ ^ ( ^ - y)" + U^. y) (3.i4) 1(9 + 1) where Rg{x^y) is a remainder given by Equation (3.14) is a fractional Taylor expansion of f{x) involving only the lowest and the second leading terms. Using the general definition of LFD and following similar steps one arrives at the fractional Taylor expansion for A^ < 9 < A'^ + 1 (provided id' exists), given by, w here /w = :4f^(-yr + f(^(-w' + fi,(x,.) (3.16) We note that the local fractional derivative (not just fractional derivative) as defined above provides the coefficient A in the approximation of f{x) by the function f{y) + A{x - yyit{q + 1), for 0 < 9 < 1, in the vicinity of y. We further note that the terms on the RHS of eqn(3.13) are nontrivial and finite only in the case q = a. Osier in ref.[ll] has constructed fractional Taylor series using usual (not local in the sense above) fractional derivatives. His results are, however, applicable to analytic functions and cannot be used for non-differentiable scaling functions directly. Furthermore, Osier's formulation involves terms with negative q also and hence is not suitable for approximating schemes. 52

7 Let us consider the function f{x) = x'^, x,a > 0. Then D'^f{0) = T{a + 1) and using equation (3.16) at j/ = 0 we get J{x) = x" since the remainder term turns out to be zero. Geometrical interpretation of LFD It is well known [5] that one can not attach any geometrical interpretation to the conventional fractional derivatives as one does for ordinary derivatives. Whereas, the local fractional Taylor expansion of section 3.3 suggests a possibility of such an interpretation for LFDs. In order to see this note that when q is set equal to unity in the equation (3.14) one gets the equation of the tangent. It may be recalled that all the curves passing through a point y and having same tangent form an equivalence class (which is modeled by a linear behavior). Analogously all the functions (curves) with the same critical order a and the same E)" will form an equivalence class modeled by x ' (If / differs from x by a logarithmic correction then terms on RHS of eqn(3.13) do not make sense like in the ordinary calculus). This is how one may generalize the geometric interpretation of derivatives in terms of 'tangents'. This observation is useful when one wants to approximate an irregular function by a piecewise smooth (scaling) function. 53

8 3.4 Generalization to higher dimensional functions The definition of the Local fractional derivative can be generalized [4] for higher dimensional functions in the following manner. Consider a function / : IR^ > R. We define ^(y,0 = /(y + vo-/(y), veir", teir. (3.18) Then the directional-lfd of / at y of order g, 0 < g < 1, in the direction v is given (provided it exists) by ldim=^^^^\i=o (3.19) where the RHS involves the usual fractional derivative of equation (2.4). The directional LFDs along the unit vector e, will be called i'^ partial-lfd. 3.5 Some remarks 1. We would like to point out that there is a multiplicity of definitions of fractional derivatives. The use of Riemann-Liouville definition, and other equivalent definitions such as Grunwald's definition are suitable for our purpose. The other definitions of fractional derivatives which do not allow control over both the limits, such as WAj'eL's definition or definition using Fourier transforms, are not suitable since it would not be possible to retrieve local nature of differentiability property which is essential for study of local behavior. Also, the important 54

9 difference between our work and the work of [8, 9] is that while we are trying to study the local scaling behavior these works apply to asymptotic scaling properties. 2. It is interesting to note that the same definition of LFD can be used for negative values of the critical order between -1 and 0. For this range of critical orders iv = 1 and the sum in equation (3.2) is empty. As a result the expression for LFD becomes ^' ^(2/) - l i m - ^ ^ ^ (3.20) ^^y [d[x - y)y An equivalence between the critical order and the Holder exponent, for positive values of critical order, will be proved in chapter 3. Here we would like to point out that the negative Holder exponents do arise in real physical situation of turbulent velocity field (see [12, 13] and references therein). 3. Another way of generalizing the LFD to the values of critical order beyond 1 would have been to write it as ^.;(,),H OT^)-/""(=/)),3,21) ^ ' ^^y [(/(x-y)]? ^ ' But the existence of A^"* derivative of / at x may not be guaranteed in general. Such a situation may arise in the ca^e of multifractal functions to be treated in chapter 5. 00

10 References [1] K. M. Kolwankar and A. D. Gangal, Chaos (1996). [2] K. M. Kolwankar and A. D. Gangal, Pramana - J. Phys (1997). [3] K. B. Oldham and J. Spanier The Fractional Calculus (Academic Press, New York, 1974). [4] K. M. Kolwankar and A. D. Gangal, in the proceedings of the Conference 'Fractals in Engineering', Archanon, [5] K. S. Miller and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations ( John Wiley, New York, 1993). [6] Hilfer R., Phys. Scr (1991). [7] Hilfer R., Phys. Rev. Lett (1992). [8] Giona M. and Roman H. E., J. Phys. A: Math Gen (1992). [9] G15ckle W. G. and Nonnenmacher T. F.. J. Stat. Phys (1993). 56

11 [10] R. Courant and F. John, Introduction to Calculus and Analysis (John Wiley, New York, 1965) Vol 1. [11] Osier T. J., SIAM J. Math. Anal (1971). [12] I. Eyink J. Stat. Phys. 78, 353 (1995). [13] S. Jaffard, To appear in SIAM J. of Math. Anal.. 0/