Local Fractional Derivatives of Differentiable Functions Are Integer

Home , Fractional calculus

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2 Local fractional derivative via Caputo fractional derivative

The local fractional derivative Dα of order α of a function f(x) at the point x = a is usually deﬁned (see Eq. 26 of [13]) by the equation

− n 1 (k) α RL α f (a) k (D f)(a) := lim a Dx f(x) − (x − a) , (n − 1 <α ≤ n) (1) x→a k! k ! X=0 RL α where a Dx is the left-sided Riemann-Liouville fractional derivative of order α (see Eqs. (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 RL α The Riemann-Liouville fractional derivative a Dx is nonlocal operator. For 0 <α ≤ 1, equations (1) and (2) have the forms

α RL α (D f)(a) := lim a Dx (f(x) − f(a)), (0 <α ≤ 1), (3) x→a

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

2 above Riemann-Liouville fractional derivative (see Eq. (2.4.1) and (2.4.3) of [2]) by

n− 1 f (k)(a) ( C Dαf)(x) := RLDα f(x) − (x − a)k , (n − 1 <α ≤ n). (5) a x a x k! k ! X=0 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

α C α (D f)(a) := lim ( a Dx f)(x), (n − 1 <α ≤ n). (6) x→a As a result, we can see that local fractional derivative is a limit of the Caputo fractional derivative.

Proposition. The local fractional derivative (1) of order α, where n − 1 < α ≤ n and n ∈ N, at the point x = x0 can be represented by the equation 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) C Dαf(x) := , (n − 1 <α ≤ n). (8) a x Γ(n − α) (x − z)α−n+1 dzn Za

C α The Caputo fractional derivatives a Dx can be deﬁned for functions belonging to the space 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) C Dα(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:

C α a Dx C =0.

For k =0, 1, 2, ..., n − 1, we have C α k a Dx (x − a) =0.

3 It is important to emphasize that the Caputo fractional derivative is represented by equation (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]):

C α RL n−α (n) ( a Dx f)(x)= a Ix f (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 functions.

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(x0) of x0 (i.e. f(x) has a derivative of order n ∈ N in a neighborhood of x0), then the function can be represented by the Taylor series n f (k)(x ) f(x)= 0 (x − x )k + o((x − x )n). (9) k! 0 0 k X=0 If f(x) is a diﬀerentiable function (i.e.n=1), then

(1) f(x)= f(x0)+ f (x − x0)+ o((x − x0)). (10) For n-diﬀerentiable functions, we can rewrite the local fractional derivative of order n − 1 < α ≤ n in the form − n 1 (k) α RL α f (x0) k (D f)(x0) = lim x0 Dx f(x) − (x − x0) = x→x0 k! k ! X=0 (n) RL α f (x0) n n = lim x0 Dx (x − x0) + o((x − x0) ) = x→x0 n! (n) f (x0) RL α n n = lim x0 Dx (x − x0) + o((x − x0) ) . (11) n! x→x0 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

4 we obtain

(n) α f (x0) Γ(n + 1) n−α n−α (D f)(x0)= lim (x − x0) + o((x − x0) ) = n! x→x0 Γ(n +1 − α) (n) f (x0) n−α n−α = lim (x − x0) + o((x − x0) ) . (13) Γ(n +1 − α) x→x0 As a result, we get 0 n − 1 < α < n, α (D f)(x0)= (14)  (n)  f (x0) α = n, N where n − 1 <α ≤ n and 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 X=0 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, α (D f)(x0)= (16)  ′  f (x0) α =1. 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 operator 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).

5 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 k X=0 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) 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 X=0 RL α RL α RL α Substitution of equation (19) for ( a D fg)(x), ( a D f)(x) and ( a D g)(x) into equations (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 functions that are not fractional-differentiable since the derivatives Dαf(x), Dαg(x) and Dα(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. C α (3) The Caputo fractional derivatives a Dx can be defined for functions belonging to the space 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]. If f(x) ∈ ACn[a, b], then this functions is n-differentiable almost everywhere on [a, b]. The Leibniz rule (17) holds on a set of differentiable functions only for α = 1.

6 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

We prove that local fractional derivatives of differentiable functions are integer-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 the classical and fractional calculus to nowhere differentiable functions and fractal 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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