No nonlocality. No fractional derivative

Abstract

Abstract:The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small neighborhood of the considered point. Therefore differential equation, which is considered for this point and contains a finite number of integer-order derivatives, cannot describe nonlocality in space and time. This allows us to propose a principle of nonlocality for fractional derivatives. We state that if the differential equation with fractional derivative can be presented as a differential equation with a finite number of integer-order derivatives, then this fractional derivative cannot be considered as a derivative of non-integer order. This means that all results obtained for this type of fractional derivatives can be derived by using differential operators with integer orders. To illustrate the application of the nonlocality principle, we prove that the conformable fractional derivative, the M-fractional derivative, the alternative fractional derivative, the local fractional derivative and the Caputo-Fabrizio fractional derivatives with exponential kernels cannot be considered as fractional derivatives of non-integer orders.

A Comment on a Controversial Issue: A Generalized Fractional Derivative Cannot Have a Regular Kernel

“…Several paradoxes involving the GFD with regular kernels have been pointed out in [21,14,22,5]. In [15] it has been shown that the models involving the GFD with regular kernels poorly reflect the real world data.…”

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations

“…• there is no memory term and the approximation of the solution is obtained by using just local information; • the numerical approximation, which is obtained by means of linear interpolant polynomials, is claimed to converge to the exact solution with order 3 under the simple assumption that f (t, y(t)) is bounded. It is a well-known fact that fractional derivatives are nonlocal operators and, as discussed in some recent papers [15,16], nonlocality is an essential, distinctive and inviolable feature of fractional-order operators.…”

A sigmoidal fractional derivative for regularization

“…Theorem 2.11 (Fractional Gradient Descent). Let f be as in (13). Then, the left-sigmoidal fractional-order gradient method (13) converges to the true critical point t * .…”