No Nonlocality. No Fractional Derivative

TL;DR

The paper argues that many fractional derivatives do not truly exhibit nonlocality and can be reduced to integer-order derivatives, challenging their interpretation as non-integer order derivatives.

Contribution

It introduces a principle of nonlocality for fractional derivatives and demonstrates that several fractional derivatives lack genuine nonlocality, reducing to integer-order derivatives.

Findings

Several fractional derivatives cannot be considered non-integer order derivatives.

The principle of nonlocality is used to classify fractional derivatives.

Many fractional derivatives are equivalent to finite-order integer derivatives.

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.