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Commun Nonlinear Sci Numer Simulat 18 (2013) 2945-2948

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Commun Nonlinear Sci Numer Simulat
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No violation of the Leibniz rule. No fractional derivative
Vasily E. Tarasov
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

article

info

abstract
We demonstrate that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. We prove that all fractional derivatives Da , which satisfy the Leibniz rule Da ?fg î Ì ?Da f îg ? f ?Da g î, should have the integer order a Ì 1, i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule. ã 2013 Elsevier B.V. All rights reserved.

Article history: Received 20 March 2013 Accepted 1 April 2013 Available online 13 April 2013 Keywords: Fractional derivative Leibniz rule

1. Introduction Fractional derivatives of non-integer orders [1,2] have wide applications in physics and mechanics [4-13]. The tools of fractional derivatives and integrals allow us to investigate the behavior of objects and systems that are characterized by power-law non-locality, power-law long-term memory or fractal properties. There are different definitions of fractional derivatives such as Riemann-Liouville, Riesz, Caputo, Gr?nwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov and others [1,2]. Unfortunately all these fractional derivatives have a lot of unusual properties. The well-known Leibniz rule Da ?fg î Ì ?Da f îg ? f ?Da g î is not satisfied for differentiation of non-integer orders [1]. For example, we have the infinite series

Da ?fg î Ì

1 X kÌ 0

C?a ? 1î ?D C?k ? 1îC?a Ð k ? 1î

aÐk

f î?Dk g î

? 1î

for analytic functions on Ía; b (see Theorem 15.1 in [1]), where Da is the Riemann-Liouville derivative, Dk is derivative of integer order k. Note that the sum is infinite and contains integrals of fractional order for k > Ía? 1. Formula (1) first appeared in the paper by Liouville [3] in 1832. The unusual properties lead to some difficulties in application of fractional derivatives in physics and mechanics. There are some attempts to define new type of fractional derivative such that the Leibniz rule holds (for example, see [15-17]). In this paper we proof that a violation of the Leibniz rule is one of the main characteristic properties of fractional derivatives. We state that linear operator Da that can be defined on C 2 ?U î, where U & R1 , such that it satisfied the Leibniz rule cannot have a non-integer order a. In other words, a fractional derivative that satisfies the Leibniz rule is not fractional. It should have integer order. 2. Hadamard's theorem We denote by C m ?U î a space of functions f ?xî, which are m times continuously differentiable on U & R1 . Let D Ì d=dx : C m ?U î ! C mÐ1 ?U î be a usual derivative of first order with respect to coordinate x.
1 x

Tel.: +7 495 939 5989; fax: +7 495 939 0397.
E-mail address: tarasov@theory.sinp.msu.ru 1007-5704/$ - see front matter ã 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.04.001


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V.E. Tarasov / Commun Nonlinear Sci Numer Simulat 18 (2013) 2945-2948

It is well-known the following Hadamard's theorem [14]. Hadamard's Theorem. Any function f ?xî 2 C 1 ?U î in a neighborhood U of a point x0 can be represented in the form

f ?xî Ì f ?x0 î??x Ð x0 îg ?xî;
where g ?xî 2 C ?U î. Proof. Let us consider the function
m

?2î

F ?t î Ì f ?x0 ??x Ð x0 îtî:
Then F ?0î Ì f ?x0 î and F ?1î Ì f ?xî. The Newton-Leibniz formula gives

?3î Z
0

F ?1îÐ F ?0î Ì

Z
0

1

dt ?D1 F î?t î Ì t

1

dt ?D1 f î?x0 ??x Ð x0 îtî?x Ð x0 î Ì ?x Ð x0 î x

Z
0

1

dt ?D1 f î?x0 ??x Ð x0 ît î: x

?4î

We define the function

g ?xî Ì

Z
0

1

dt ?D1 f î?x0 ??x Ð x0 ît î: x

?5î

As the result, we have proved representation (2). h

3. Algebraic approach to fractional derivatives We consider fractional derivatives Da of non-integer orders a by using an algebraic approach. Special forms of fractional derivatives are not important for our consideration. We take into account the property of linearity and the Leibniz rule only. For the operator Da we will consider the following conditions. (1) R-linearity:

Da ?c1 f ?xî? c2 g ?xîî Ì c1 ?Da f ?xîî ? c2 ?Da g ?xîî; x x x
where c1 and c2 are real numbers. Note that all known fractional derivatives are linear [1,2]. (2) The Leibniz rule:

?6î

Da ?f ?xî g ?xîî Ì ?Da f ?xîî g ?xî? f ?xî?Da g ?xîî: x x x

?7î

(3) If the linear operator satisfies the Leibniz rule, then the action on the unit (and on a constant function) is equal to zero:

Da 1 Ì 0: x
Let us proof the following theorem.

?8î

Theorem (``No violation of the Leibniz rule. No fractional derivative''). If an operator Da can be applied to functions from x C 2 ?U î, where U & R1 be a neighborhood of the point x0 , and conditions (6) and (7) are satisfied, then the operator Da is the x derivative D1 of integer (first) order, i.e. it can be represented in the form x

Da Ì a?xî D1 ; x x
where a?xî are functions on R . Proof
1

?9î

(1) Using Hadamard's theorem for the function g ?xî in the decomposition (2), the function f ?xî for x 2 U can be represented in the form

f ?xî Ì f ?x0 î??x Ð x0 îg ?x0 î??x Ð x0 î2 g 2 ?xî;
where g 2 ?xî 2 C 2 ?U î, and U & R1 is a neighborhood of the point x0 . Applying to equality (10) the operator D1 and use D1 f ?x0 î Ì 0, we get x x

?10î

?D1 f î?xî Ì g ?x0 î? 2?x Ð x0 î g 2 ?xî??x Ð x0 î2 ?D1 g 2 î?xî: x x
Then

?11î

?D1 f î?x0 î Ì g ?x0 î: x


V.E. Tarasov / Commun Nonlinear Sci Numer Simulat 18 (2013) 2945-2948

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As a result, we have

f ?xî Ì f ?x0 î??x Ð x0 î?D1 f î?x0 î??x Ð x0 î2 g 2 ?xî: x
(2) Applying to equality (12) the operator Da , we get x

?12î

?Da f î?xî Ì Da f ?x0 î?Da ?x Ð x0 î?D1 f î?x0 î ?Da ?x Ð x0 î2 g 2 ?xî : x x x x x
The Leibniz rule gives

?13î

?Da f î?xî Ì Da f ?x0 î? a?xî?D1 f î?x0 î??x Ð x0 î?Da ?D1 f î?x0 î? 2a?xî?x Ð x0 î g 2 ?xî??x Ð x0 î2 ?Da g 2 î?xî; x x x x x x
where we use the notation

?14î

a?xî Ì ?Da ?x Ð x0 îî?xî: x
Then

?15î

?Da f î?x0 î Ì Da f ?x0 î? a?x0 î?D1 f î?x0 î: x x x
As a result, we have

?16î

Da Ì a?xî D1 ? b?xî; x x
where we define the function

?17î

b?xî Ì Da 1 x
and we use the R-linearity in the form

?18î

Da f ?x0 î Ì f ?x0 î?Da 1î: x x
(3) Using that Da 1 Ì 0 for linear operator, which satisfies the Leibniz rule, we get b?xî Ì 0; Da x0 Ì x0 Da 1 Ì 0 and x x x

Da Ì a?xî D1 ; x x
where a?xî Ì Da x. x As the result, we prove (9). h

?19î

Remark. In general, the property (8) is not satisfied for all type of fractional derivatives. For example, we have

Da 1 Ì x

1

C?1 Ð aî

x

Ða

for Riemann-Liouville fractional derivative [2]. Note that Da x is not equal to one in general. For example, x

Da x Ì x

1

C?2 Ð aî

x

1Ð a

for Riemann-Liouville fractional derivative [2]. Note that this theorem can be proved for multivariable case. The theorem state that fractional derivative that satisfies the Leibniz rule coincides with differentiation of the order equal to one, i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule. Unfortunately the Leibniz rule is suggested for some new fractional derivatives (the modified Riemann-Liouville derivative that is suggested by Jumarie [15,16], and local fractional derivative in the form that is suggested by Yang [17] and some other derivatives). Linear operators Da that satisfy the Leibniz rule cannot be considered as fractional derivatives of non-integer orders. Fracx tional derivatives should be subject to a rule that is a generalization of the classical Leibniz rule

Dn ?fg î Ì x

n X ?D kÌ 0

nÐk

f î?Dk g î

to the case of differentiation and integration of fractional order (see Section 15. ``The generalized Leibniz rule'' of [1] and references in it). It can be assumed with a high degree of reliability that the generalization of the Leibniz rule for all types of fractional derivatives should be represented by an infinite series in general. The history of the generalizations of the Leibniz rule for fractional derivatives, which is began from the paper [3] in 1868, is described in Section 17 (Bibliographical Remarks and Additional Information to Chapter 3.) in [2].


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