Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

0.11 MB·

by  Vasily E. Tarasov

International Journal of Applied and Computational Mathematics.
2016. Vol.2. No.2. P.195-201. DOI: 10.1007/s40819-015-0054-6
Local Fractional Derivatives of
Differentiable Functions are Integer-order Derivatives or Zero
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Vasily E. Tarasov
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n
Skobeltsyn Institute of Nuclear Physics,
a
J Lomonosov Moscow State University, Moscow 119991, Russia

3 E-mail: [email protected]
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Abstract
]
A
C In this paper we prove that local fractional derivatives of differentiable functions are
. integer-order derivative or zero operator. We demonstrate thatthelocal fractional deriva-
h
t tives are limits of the left-sided Caputo fractional derivatives. The Caputo derivative of
a
fractional order α of function f(x) is defined as a fractional integration of order n−α of
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thederivative f(n)(x)of integer ordern. Therequirementof theexistence of integer-order
[
derivatives allows us to conclude that the local fractional derivative cannot be considered

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as the best method to describe nowhere differentiable functions and fractal objects. We
v
0 also prove that unviolated Leibniz rule cannot hold for derivatives of orders α 6= 1.
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MSC: 26A33 Fractional derivatives and integrals
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0 PACS: 45.10.Hjj Perturbation and fractional calculus methods
.
1 Keywords: Fractional calculus, local fractional derivative, Caputo derivative Leibniz rule,
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fractal
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:
v
i
X 1 Introduction
r
a
Fractional calculus as a theory of derivatives and integrals of non-integer orders [1, 2, 3] has
a long history [4, 5, 6]. As a mathematical tool it has a lot of applications in physics and
mechanics totodescribemediaandsystemswithnonlocalandhereditarypropertiesofofpower-
law type. During more than three hundred years of history of fractional calculus it has been
proposed different definitions of fractional derivatives by Riemann, Liouville, Riesz, Gru¨nwald,
Letnikov, Marchaud, Caputo [1, 2]. Fractional-order derivatives can be characterized by a set
of interesting unusual properties such as a violation of the usual Leibniz rule, a deformations of
the usual chain rule, a violation of the semi-group property. These unusual properties can be
used to describe nontrivial and unusual properties of complex systems and media. Therefore
1
these properties are valuable and important for applications in physics and mechanics (see
[9, 10, 11] and references therein).
Recently, so-called local fractional derivative has been proposed in [12, 13, 14, 15] by a
motivation of describing the properties of fractal objects and processes. The definition of
the local fractional derivative is obtained from the Riemann-Liouville fractional derivative by
introducing two changes such as a subtraction of finite number of terms of the Taylor series and
the taking the limit. The motivation of this definition is a studying fractional differentiability
properties of nowhere differentiable functions and fractal objects.
In this paper, we demonstrate that the definition of local fractional derivative can be pre-
sented as a limit of the Caputo fractional derivatives. We note that the Caputo fractional
derivative of order α (n − 1 < α ≤ n) of a function is defined as an action of derivative of
integer order n and then the integration of non-integer order. Therefore the function should
be differentiable n times. In this paper we also prove that local fractional derivatives of dif-
ferentiable functions are integer-order derivative or zero operator. We argued that unviolated
Leibniz rule cannot hold for local fractional derivatives of orders α 6= 1.
2 Local fractional derivative via Caputo fractional deriva-
tive
The local fractional derivative Dα of order α of a function f(x) at the point x = a is usually
defined (see Eq. 2.6 of [13]) by the equation
n−1 f(k)(a)
(Dαf)(a) := lim RLDα f(x)− (x−a)k , (n−1 < α ≤ n) (1)
x→a a x k! !
k=0
X
where RLDα is the left-sided Riemann-Liouville fractional derivative of order α (see Eqs.
a x
(2.1.15) and (2.1.1) of [2]) that is defined by
1 dn x f(z)dz
RLDαf(x) := . (n−1 < α ≤ n) (2)
a x Γ(n−α)dxn (x−z)α−n+1
Za
The Riemann-Liouville fractional derivative RLDα is nonlocal operator. For 0 < α ≤ 1, equa-
a x
tions (1) and (2) have the forms
(Dαf)(a) := lim RLDα(f(x)−f(a)), (0 < α ≤ 1), (3)
x→a a x
1 d x f(z)dz
RLDαf(x) := , (0 < α ≤ 1). (4)
a x Γ(1−α)dx (x−z)α
Za
In the fractional calculus is well-known the Caputo fractional derivatives [2, 3] suggested in
[16, 17, 18, 19]. The left-sided Caputo fractional derivative of order α can be defined via the
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above Riemann-Liouville fractional derivative (see Eq. (2.4.1) and (2.4.3) of [2]) by
n−1 f(k)(a)
(CDαf)(x) := RLDα f(x)− (x−a)k , (n−1 < α ≤ n). (5)
a x a x k!
!
k=0
X
As a result, the local fractional derivative (1) of order α at the point x = a can be defined via
the above Caputo fractional derivative by the equation
(Dαf)(a) := lim(CDαf)(x), (n−1 < α ≤ n). (6)
x→a a x
As a result, we can see that local fractional derivative is a limit of the Caputo fractional deriva-
tive.
Proposition.
The local fractional derivative (1) of order α, where n− 1 < α ≤ n and n ∈ N, at the point
x = x can be represented by the equation
0
1 x dz dnf(z)
(Dαf)(a) := lim , (n−1 < α ≤ n). (7)
Γ(n−α) x→a (x−z)α−n+1 dzn
Za
Proof. This statement is direct corollary of Theorem 2.1 of [2] (see Eq. (2.4.15) of [2]),
where the left-sided Caputo fractional derivative is represented in the form
1 x dz dnf(z)
CDαf(x) := , (n−1 < α ≤ n). (8)
a x Γ(n−α) (x−z)α−n+1 dzn
Za
The Caputo fractional derivatives CDα can be defined for functions belonging to the space
a x
ACn[a,b] of absolutely continuous functions [2]. If f(x) ∈ ACn[a,b], then the Caputo fractional
derivatives exist almost everywhere on [a,b].
It can be directly derived that the left-sided Caputo fractional derivative of the power
function (x−a)β is
Γ(β +1)
CDα(x−a)β = (x−a)β−α,
a x Γ(β −α+1)
where β > −1 and α > 0. In particular, if β = 0 and α > 0, then the Caputo fractional
derivative of a constant C is equal to zero:
CDαC = 0.
a x
For k = 0,1,2,...,n−1, we have
CDα(x−a)k = 0.
a x
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Itisimportanttoemphasize thattheCaputo fractionalderivative isrepresented byequation
(8) as a sequence of two operations, namely, first taking the ordinary derivative of integer order
n and then the integration of fractional order n−α, (see Eq. 2.4.17 of [2]):
(CDαf)(x) = RLIαf(n)(x).
a x a x
The Riemann-Liouville fractional derivative is represented by the inverse sequence of the same
operations.
3 Local fractional derivative is derivative of integer-order
Let us prove the following statement for the local fractional derivatives of n-differentiable func-
tions.
Proposition.
Local fractional derivative of order α, where n−1 < α ≤ n, n ∈ N, on a set of n-differentiable
functions is a derivative of order n ∈ N or zero operator.
Proof.
If f(x) is a n-differentiable function in a neighborhood U(x ) of x (i.e. f(x) has a derivative
0 0
of order n ∈ N in a neighborhood of x ), then the function can be represented by the Taylor
0
series
n f(k)(x )
f(x) = 0 (x−x )k +o((x−x )n). (9)
k! 0 0
k=0
X
If f(x) is a differentiable function (i.e.n=1), then
f(x) = f(x )+f(1)(x−x )+o((x−x )). (10)
0 0 0
For n-differentiable functions, we can rewrite the local fractional derivative of order n − 1 <
α ≤ n in the form
n−1 f(k)(x )
(Dαf)(x ) = lim RLDα f(x)− 0 (x−x )k =
0 x→x0 x0 x k! 0 !
k=0
X
f(n)(x )
= lim RLDα 0 (x−x )n +o((x−x )n) =
x→x0 x0 x n! 0 0
(cid:18) (cid:19)
f(n)(x )
= 0 lim RLDα (x−x )n +o((x−x )n) . (11)
n! x→x0 x0 x 0 0
(cid:16) (cid:17)
Using Γ(n+1) = n! and the equation (see Property 2.1 of [2])
Γ(β +1)
RLDα(x−x )β = (x−x )β−α, (β > −1), (12)
x0 x 0 Γ(β +1−α) 0
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we obtain
f(n)(x ) Γ(n+1)
(Dαf)(x ) = 0 lim (x−x )n−α +o((x−x )n−α) =
0 n! x→x0 Γ(n+1−α) 0 0
(cid:16) (cid:17)
f(n)(x )
= 0 lim (x−x )n−α +o((x−x )n−α) . (13)
Γ(n+1−α) x→x0 0 0
(cid:0) (cid:1)
As a result, we get
0 n−1 < α < n,
(Dαf)(x ) = (14)
0

f(n)(x ) α = n,
 0
where n−1 < α ≤ n and n ∈ N, and we use Γ(n+1−n) = Γ(1) = 1.

The proved statement means that the local fractional derivatives cannot be considered as
fractional derivatives on sets of finite (or infinite) numbers of times differentiable functions.
As a corollary of the proved statment we can say that generalizations of the Leibniz rule
for local fractional derivatives Dα of order α > 0 for n-differentiable functions f(x) and g(x)
should give the relation
n
Γ(α+1)
Dα(f(x)g(x)) = (Dα−kf(x))(Dkg(x)), (α = n ∈ N) (15)
Γ(α−k +1)Γ(k +1)
k=0
X
for integer orders α = n ∈ N or it should give the equality 0 = 0 for noninteger orders α,
n−1 < α ≤ n.
For 0 < α ≤ 1, equation (14) has the form
0 0 < α < 1,
(Dαf)(x ) = (16)
0

f′(x ) α = 1.
 0
Therefore we get the statement that hasbeen proved in [7] as Theorem ”No violation of the
Leibniz rule. No fractional derivative”: If the unviolated Leibniz rule
Dα(f(x)g(x)) = (Dαf(x))g(x)+f(x)(Dαg(x)), (17)
holds for the linear operator Dα and the n-differentiable functions f(x) and g(x), then this op-
erator is the derivative of integer (first) order, that can be represented as Dα = a(x)D1, where
a(x) are functions on R.
In paper [7] it was proved that violation of relation (17) is a characteristic property of all
types of fractional-order derivatives with α > 0 (and it is obvious for derivatives of integer-
orders n ∈ N greater than one).
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A correct form of the Leibniz rule for local fractional derivatives should be obtained from
the fractional generalization of the Leibniz rule for the Riemann-Liouville derivatives that has
the form of the infinite series. The symmetrized expression of this rule is
∞
k
RLDα(fg)(x) = (RLDα−kf)(x)(Dkg(x))+(RLDα−kg)(x)(Dkf(x)) , (18)
a x α a x a x
Xk=0(cid:18) (cid:19)(cid:16) (cid:17)
where the f(x) and g(x) are analytic functions on the interval [a,b] (Theorem 15.1 of [1]),
Dkf(x) = f(k)(x), and
k Γ(α+1)
= .
α 2Γ(α−k +1)Γ(k+1)
(cid:18) (cid:19)
It should be noted that the sum of (18) is infinite and it contains integrals of fractional order
for the values k > [α]+1. Equation (18) has been proposed by Liouville [21] in 1832. Note that
extensions of the Leibniz rule to fractional derivatives are suggested by Osler [22, 23, 24, 25].
The left-sided Riemann-Liouville fractional derivative of order α (n − 1 < α ≤ n) of a
function f(x) (see Eqs. (2.4.6) of [2]) can be presented via the Caputo fractional derivative by
n−1 f(k)(a)
(RLDαf)(x) = (CDαf)(x)+ (x−a)k−α, (n−1 < α ≤ n). (19)
a x a x k!
k=0
X
Substitution of equation (19) for (RLDαfg)(x), (RLDαf)(x) and (RLDαg)(x) into equations
a a a
(18) gives a generalized Leibniz rule for the Caputo fractional derivatives. The extension of the
Leibniz rule for the Caputo fractional derivatives also has been suggested by Theorem 3.17 of
the Diethelm’s book [20].
Let us give some remarks about Leibniz rule (17) for local fractional derivatives on a set of
non-differentiable functions. It should be noted that statement [26, 27] that the Leibniz rule
in the unviolated form (17) holds for non-differentiable functions is incorrect [8]. Let us give
some explanations below.
(1) It should be noted that the Leibniz rule (17) cannot be used for non-differentiable func-
tionsthatarenotfractional-differentiablesincethederivativesDαf(x),Dαg(x)andDα(f(x)g(x))
should exist.
(2) It is easy to see that nowhere in the proofs proposed in [26, 27], the requirement that the
functions f(x) and g(x) are not classically differentiable is not used. Therefore, using the same
proofs, we can get the statement that the Leibniz rule (17) holds for fractional-differentiable
functions f(x), g(x) without the useless assumption that these functions are not classically
differentiable.
(3) The Caputo fractional derivatives CDα can be defined for functions belonging to the
a x
space ACn[a,b] of absolutely continuous functions [2]. If f(x) ∈ ACn[a,b], then the Caputo
fractionalderivativesexistalmosteverywhere on[a,b]. Iff(x) ∈ ACn[a,b], thenthisfunctionsis
n-differentiable almost everywhere on[a,b]. TheLeibnizrule(17) holdsonaset ofdifferentiable
functions only for α = 1.
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As a result, we get that the Leibniz rule (17) with local fractional derivative for f(x),g(x) ∈
ACn[a,b] holds only for α = 1 almost everywhere on [a,b]. In the countable number of points
the Leibniz rule is not performed since the derivatives Dαf(x), Dαg(x) and Dα(f(x)g(x)) does
not exist. Note that the Leibniz rule for fractional derivatives of orders α = n > 1 (n ∈ N)
cannot have the form (17). It should be get the rule in the form (15).
4 Conclusion
Weprove that localfractionalderivatives ofdifferentiable functionsareinteger-order derivatives
or zero operator. We show that definition of local fractional derivative of non-integer order α
(n − 1 < α ≤ n) can be represented by limit of the Caputo fractional derivative of order
α. The Caputo derivative of fractional order α of a function f(x) is defined as an action of
derivative of integer order n and then the integration of non-integer order n−α. Therefore the
function f(x) should be classically differentiable n times, and derivative f(n)(x) of integer order
n should exist. As a result, the local fractional derivative cannot be considered as the best tool
for description of nowhere differentiable functions and fractal objects despite the motivation to
consider these derivatives. We also argued that unviolated Leibniz rule cannot hold for local
fractional derivatives of orders α 6= 1.
A problem of strict formulation of ”fractal” analogs of derivatives and integrals by extension
of theclassical andfractionalcalculus tonowhere differentiable functionsandfractal sets is very
difficult. We can assume that the most promising approaches to fractional calculus on fractals
can be based on methods of ”non-integer dimensional spaces” (see [28, 29] and references
therein) or ”analysis on fractals” [30, 31, 32, 33].
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