The Gr\"unwald-Letnikov fractional-order derivative with fixed memory length

TL;DR

This paper proposes a modified Gr"unwald-Letnikov fractional derivative with fixed memory length that preserves periodicity, enhancing the applicability of fractional derivatives to periodic real-world phenomena.

Contribution

It introduces a simple modification to the Gr"unwald-Letnikov fractional derivative by fixing the memory length, which preserves periodicity in fractional derivatives.

Findings

The modified derivative preserves periodicity of functions.

It extends the applicability of fractional derivatives to periodic phenomena.

The approach is practical for real-world periodic systems.

Abstract

Contrary to integer order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period, as a consequence of this property the time-invariant fractional order system does not have any non-constant periodic solution unless the lower terminal of the derivative is plus or minus infinite, which is not practical. This property limits the applicability areas of fractional derivatives and makes it unfavorable, for a wide range of periodic real phenomena. Therefore enlarging the applicability of fractional system to such real area is an important research topic. In this paper we give a solution for the above problem by imposing a simple modification on the Gr\"unwald-Letnikov definition of fractional derivative, this modification consists of fixing the memory length and varying the lower terminal of the derivative. It is shown that the new proposed definition of fractional derivative preserves the periodicity.