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Physics Letters A 379 (2015) 1071­1072

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Physics Letters A
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Comments on "The Minkowski's space­time is consistent with differential geometry of fractional order" [Phys. Lett. A 363 (2007) 5­11]
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

article

info

abstract
We prove that Eq. (3.6) for fractional-order derivative of power function and the usual Leibniz rule (3.7) in the form D ( v (x) u (x)) = (D v (x)) u (x) + v (x) (D u (x)), which are based relations of the considered paper, cannot hold together for fractional derivatives of order = 1. We also prove that the usual Leibniz rule with = 1 cannot hold for sets of differentiable and non-differentiable functions. © 2015 Elsevier B.V. All rights reserved.

Article history: Received 5 November 2014 Received in revised form 2 February 2015 Accepted 2 February 2015 Available online 7 February 2015 Communicated by C.R. Doering Keywords: Fractional derivative Leibniz rule Modified Riemann­Liouville derivatives

In paper [1], G. Jumarie uses as main properties of fractionalorder derivatives D the equation for fractional derivative of power function

2

(3 - )

-

2

(2 - )

=0

(5 )

D x =

( + 1) x - ( > 0), ( - + 1)

(1)

for x R+ . Let us take into account Then Eq. (5) gives

(3 - ) = (2 - ) (2 - ).

and the Leibniz rule

1- = 0. (3 - ) (2)

(6)

D ( v (x) u (x)) = (D v (x)) u (x) + v (x)(D u (x)),
as Eqs. (3.6) and (3.7) of [1], where D Liouville fractional derivatives. Let us prove that Eqs. (1) and (2) cannot for fractional-order derivatives with order statement, we can use the functions u (x) = v niz rule (2). In this case, this rule is written in

is the modified Riemann­

performed together = 1. To prove this (x) = x in the Leibthe form

D x2 = (D x) x + x (D x).
Eq. (1) gives

(3)

D x2 =

2

(3 - )

x2- ,

D x =

1

(2 - )

x1- ,

(4)

where (n + 1) = n! is taken into account. Substituting of Eq. (4) into Eq. (3), we obtain the condition

DOI of original article: http://dx.doi.org/10.1016/j.physleta.2006.10.105. E-mail address: tarasov@theory.sinp.msu.ru. http://dx.doi.org/10.1016/j.physleta.2015.02.005 0375-9601/© 2015 Elsevier B.V. All rights reserved.

As a result, we prove that Eq. (1) with x R+ and the Leibniz rule (2) cannot be satisfied together for = 1. To prove this proposition, we also can use u (x) = x1 and v (x) = x2 with arbitrary positive real values of 1 , 2 and x R+ to prove that the Leibniz rule (2) holds only for = 1. Note that our proof does not depend on the definition of fractional derivative and the type of the derivative, i.e. this statement can be used for all types of fractional derivatives including the modified Riemann­Liouville derivatives. The statement [5] that the Leibniz rule in the form (2) holds for non-differentiable functions is also incorrect. Let us give the proof. Step 1. If the Leibniz rule (2) is considered then functions u (x) and v (x) should be fractional differentiable. In order to use the left and right sides of the Leibniz rule (2), the functions u (x) and v (x) should be differentiable in a fractional sense, i.e. fractional derivatives D u (x), D v (x) and D (u (x) v (x)) should exist. Therefore arbitrary non-differentiable functions cannot be considered in the Leibniz rule (2). In paper [2] these functions are called " -differentiable" or/and fractional differentiable. Therefore the Leibniz rule (2) with fractional derivatives should be considered for fractional-differentiable functions.


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V.E. Tarasov / Physics Letters A 379 (2015) 1071­1072

Step 2. Power functions are fractional differentiable. In paper [2] (see remarks on Definition 2.5), we can see that u (x) is -differentiable at x = 0, if and only if, when u (x) - u (0) is locally self-similar with the parameter . It is easy to see that u (x) = v (x) = x are self-similar with the parameter . Then these functions are -differentiable. In addition, Eq. (1) (see Eq. (3.6) of [1]) means that fractional derivatives of power functions exist. Step 3. Using that power functions are fractional differentiable, we can repeat our proof. Using u (x) = v (x) = x for x R+ , we obtain that the Leibniz rule (2) holds only if the condition

(2 + 1) - 2

2

( + 1) = 0

(7)

can be used. This condition holds only for = 1. For example, if = 1/2, then condition (7) has the form 1 - /2 = 0. Remark 1. Note that if u (x) and v (x) is -differentiable at x = 0, then u (x) and v (x) are locally self-similar with the parameter , and the product u (x) v (x) is locally self-similar with the parameter 2 , i.e. the product is not -differentiable functions since it is 2 -differentiable. Therefore the Leibniz rule cannot be formulated for class of -differentiable functions with fixed order , which is equal to order of the used fractional derivative. Then we can consider fractional differentiable functions with orders , R+ . This means that we can consider the power functions u (x) = x and v (x) = x with , = (including integer values of and ), where is the order of fractional derivative. Remark 2. Using Remark 1, we can consider the Leibniz rule (2) for fractional differentiable functions with orders , , which may differ from the order of derivative in this rule, then we can repeat our proof. Using the Leibniz rule (2) with fractional derivative of order for product of -differentiable and -differentiable functions, we can repeat our proof for power functions u (x) = x and v (x) = x with , R+ (including , N) to get the statement that the Leibniz rule (2) holds only for = 1. Remark 3. To consider fractional differentiable functions u (x) and v (x) at x = x0 = 0 with , R+ , we can use for example u (x) = v (x) = x - x0 and then use (1) for x - x0 and (x - x0 )2 = x2 - 2x0 x + x2 . 0 As a result, we prove the following statements: 1. The Leibniz rule (2) for fractional derivatives of orders = 1 is not satisfied on a set of differentiable functions. The Leibniz rule (2) holds on a set of differentiable functions only for = 1. 2. The Leibniz rule (2) for fractional derivatives of orders = 1 is not satisfied on a set of fractional-differentiable functions. Eq. (1) for fractional-order derivative of power function and the Leibniz rule (2) on a set of fractional-differentiable functions hold together only for = 1. 3. The Leibniz rule (2) cannot be used for non-differentiable functions that are not fractional-differentiable. Let us note some additional questionable steps and mistakes in papers [1­3,5]: (1) In [5], author tried to answer to the obvious objection [4] that Leibniz rule cannot hold in the form (2) for fractional derivatives = 1. The key of the answer is that (2) holds only for functions u (x) and v (x), which are fractional differentiable but not classically differentiable. It is remarkable to note that nowhere in the "proof" of (2) given in [5], the assumption for u , v of being non-classically differentiable is not used. Therefore, using exactly the same "proof", we can get the statement that the Leibniz rule (2) holds for each pair u , v of fractional differentiable functions without the useless assumption that these functions are not classically differentiable. As a result, Eqs. (1) and (2) give that the Leibniz rule (2) holds only for = 1.

(2) In Eq. (2.6) of [1], we can see that the fractional derivative f ( ) for > 1 is defined by f ( ) (x) := ( f n (x))( -n) , where n is integer part of and f (n) is the ordinary classical derivative of integer order n, i.e. f (x) is classically differentiable. Therefore the suggested Taylor's formula (see Eq. (3.1) and Corollary 3.1 of [1]) hold if f (n) exists for all n N, i.e. if the function f (x) is smooth and f (x) classically differentiable for all n N. As a result, the suggested Taylor's formula can be used for classically differentiable functions. (3) In [5], a generalization of the Hadamard's theorem is considered. From Eq. (2) of [5] with f C and the fractional differentiability of (x - x0 ) , it follows that also f C m . Therefore we have that f C implies f C m for m N. This reveals incorrectness in the suggested generalization of Hadamard's theorem and the correspondent Taylor's series in the Hadamard's form. We should emphasize that the violation of the Leibniz rule (2) is a characteristic property of fractional-order derivatives of all types [4] and derivatives of integer orders = 1. Moreover the violation of the Leibniz rule (2) for fractional-order derivatives does not depend on the class of functions (in contrast to statement in [5]), if the relation (1) can be used. A correct form of the Leibniz rule for fractional-order derivatives should be obtained as a generalization of the Leibniz rule for integer-order derivatives (for example, see Section 2.7.2 of [6]). We should note that the chain rule in the form of Eq. (4.4) of [5], which is used as basis for formulation of the suggested generalization of differential geometry of fractional order, also cannot satisfied for fractional derivatives with orders = 1. It should be noted that fractional calculus as a theory of integrals and derivatives of non-integer real (or complex) order has a long history from 30 September 1695, and it go back to many great mathematicians such as Leibniz, Liouville, Riemann, Abel, Letnikov, Riesz, Weyl. There are different correct definitions of fractional derivatives such as Riemann­Liouville, Riesz, Caputo, GrÝnwald­ Letnikov, Marchaud, Weyl, Sonin­Letnikov (for example, see [7,8], where the correct properties of these fractional derivatives are described). The theory of fractional-order derivatives and integrals has a wide application in mechanics and physics (for example, see papers and books cited in [9]). In my opinion, unusual properties of fractional-order derivatives such as a violation of the usual Leibniz rule, a deformations of the usual chain rule a violation of the semi-group property allow us to describe new unusual properties of complex media and physical systems. References
[1] G. Jumarie, The Minkowski's space­time is consistent with differential geometry of fractional order, Phys. Lett. A 363 (1­2) (2007) 5­11. [2] G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann­Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22 (3) (2009) 378­385. [3] G. Jumarie, Modified Riemann­Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (9­10) (2006) 1367­1376. [4] V.E. Tarasov, No violation of the Leibniz rule. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 18 (11) (2013) 2945­2948, arXiv:1402.7161. [5] G. Jumarie, The Leibniz rule for fractional derivatives holds with nondifferentiable functions, Math. Stat. 1 (2) (2013) 50­52. [6] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1998, 340 p. [7] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993, 1006 p. [8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, 353 p. [9] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2011, 504 p.