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Fractals, Vol. 23, No. 2 (2015) 1575001 (3 pages) c World Scientific Publishing Company DOI: 10.1142/S0218348X15750018

COMMENTS ON "RIEMANN­CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]
VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics Lomonosov Moscow State University Moscow 119991, Russia tarasov@theory.sinp.msu.ru Accepted Novemb er 25, 2014 Published April 17, 2015

Abstract
We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives. Keywords : Fractional Derivative; Leibniz Rule; Mo dified Riemann­Liouville Derivatives.

In Ref. 1, the author presents as main prop erties of suggested fractional derivatives D the equation for fractional derivative of p ower function (FD-PF) D x = ( +1) x ( - +1)
-

and the Leibniz rule D (v (x)u(x)) = (D v (x))u(x)+ v (x)(D u(x)), (2) as Eqs. (4.2) and (4.3) of Ref. 1, where D is the modified Riemann­Liouville fractional derivatives.

,

( > 0

> 0),

(1)

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Note that Eq. (1) can b e used for x > 0 since x - does not exist for - < 0 at x = 0. The relations (1) and (2) cannot b e p erformed together for fractional derivatives with orders =1. To prove this statement, we can use the functions u(x) = v (x) = x in theLeibnizrule(2). In this case, this rule is written in the form D x2 = (D x)x + x(D x). Equation (1) gives D x2 = (3) x (3 - )
2-

(3)

,

Dx =

(2) x (2 - )

1-

, (4)

Substituting Eq. (4) into Eq. (3), we obtain 2(2) (3) - = 0. (3 - ) (2 - ) (5)

If we take into account (3 - ) = (2 - )(2 - ) and (n) = (n - 1)!, then (5) can b e represented in the form 1- = 0. (6) (3 - ) As a result, we demonstrate that Eq. (1) and the Leibniz rule (2) cannot b e satisfied together for = 1. Analogously, we can use u(x) = x1 and v (x) = x2 with 1 ,2 R+ to prove that the Leibniz rule (2) holds only for = 1. In Sec. 4.2.1 of Refs. 1 and 2, there is an attempt to answer the obvious ob jection that the Leibniz rule (2) cannot hold for derivatives of orders = 1. The main assumption of this answer is that (2) holds only for non-differentiable functions u(x) and v (x). This assumption is incorrect also. Equation (2) of the Leibniz rule means that the fractional derivatives D u(x), D v (x) and D (u(x)v (x)) exist, i.e., the functions u(x) and v (x) should b e fractionally differentiable. Therefore, arbitrary non-differentiable functions cannot b e considered in the Leibniz rule (2). Using Eq. (1) (see Eq. (4.2) of Ref. 1), we can see that the author assumes that the p ower functions x ( R+ ) are fractionally differentiable. Using that p ower functions are fractionally differentiable, we can consider the Leibniz rule (2) for the p ower functions u(x) = x and v (x) = x with , = (including integer values of and ), where is the order of fractional derivative used in (2). As a result, we get by transformation similar to (3­6) that the Leibniz rule (2) holds only for = 1.

In addition, it is easy to see that nowhere in the "proof " of (2) given in Ref. 2, the assumption that u(x), v (x) are fractional differential functions but not classically differentiable is not used. Therefore, we can rep eat the same "proof " for each pair u(x), v (x) of fractional differential functions without the useless assumption that these functions are not classically differential. This allows to use p ower functions x in (2). As a result, Eqs. (1) and (2) lead to the statement that the Leibniz rule (2) cannot hold for = 1. In addition, this means that the "proof " of (2) suggested in Ref. 2 is incorrect. The violation of the Leibniz rule (2) is a characteristic prop erty of fractional-order derivatives of all typ es3 and derivatives of integer orders =1. Moreover, the fact of violation of the Leibniz rule (2) for fractional derivatives does not dep end on the class of functions (in contrast to statements in Ref. 2), if the relation (1) can b e used. A correct form of the Leibniz rule for fractional-order derivatives should b e obtained as a generalization of the Leibniz rule for integer-order derivatives (see Sec. 2.7.2 of Refs. 4 and 5). In addition, the chain rule (see Eqs. (4.4), (5.1), (5.2) of Ref. 1), which is used as the basis for formulation of the suggested generalization of differential geometry in Ref. 1, also cannot satisfy for fractional-order derivatives with = 1 (for example, see Ref. 6). Moreover, by using the Fourier transform it is easy to prove that the nonlinear coordinate transformation maps fractional order derivatives into pseudo-differential op erator of the general form that cannot b e represented as a fractional derivative. As a result, the "fractional" differential geometry of fractional differential manifold suggested in Ref. 1 and Refs. 7 and 8 is wrong.

REFERENCES
1. G. Jumarie, Riemann­Christoffel tensor in differential geometry of fractional order application to fractal space-time, Fractals 21(1) (2013) 27. 2. G. Jumarie, The Leibniz rule for fractional derivatives holds with non-differentiable functions, Math. Stat. 1(2) (2013) 50­52. 3. V. E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 18(11) (2013) 2945­2948, arXiv:1402. 7161. 4. I. Po dlubny, Fractional Differential Equations (Academic Press, San Diego, 1998).

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5. T. J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math. 18(3) (1970) 658­674. 6. C.-S. Liu, Counterexamples on Jumarie's two basic fractional calculus formulae, Commun. Nonlinear Sci. Numer. Simul. 22(1­3) (2015) 92­94.

7. G. Jumarie, An approach to differential geometry of fractional order via mo dified Riemann­Liouville derivative, Acta Math. Sin. 28(9) (2012) 1741­1768. 8. G. Jumarie, The Minkowski's space-time is consistent with differential geometry of fractional order, Phys. Lett. A 363(1­2) (2007) 5­11.

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