A Study of the Grunwald-Letnikov Definition for Minimizing the Effects of Random Noise on Fractional Order Differential Equations

TL;DR

This paper analyzes the Grunwald-Letnikov definition for fractional derivatives, identifying its sensitivity to noise, and proposes a transformation scheme that enhances robustness in fractional order systems affected by random errors.

Contribution

It provides the first in-depth mathematical analysis of the Grunwald-Letnikov definition and introduces a novel transformation scheme to reduce noise effects in fractional systems.

Findings

The transformation scheme significantly improves noise robustness.

The scheme accurately analyzes fractional systems with high noise levels.

Experimental validation confirms the scheme's effectiveness.

Abstract

Of the many definitions for fractional order differintegral, the Grunwald-Letnikov definition is arguably the most important one. The necessity of this definition for the description and analysis of fractional order systems cannot be overstated. Unfortunately, the Fractional Order Differential Equation (FODE) describing such a systems, in its original form, highly sensitive to the effects of random noise components inevitable in a natural environment. Thus direct application of the definition in a real-life problem can yield erroneous results. In this article, we perform an in-depth mathematical analysis the Grunwald-Letnikov definition in depth and, as far as we know, we are the first to do so. Based on our analysis, we present a transformation scheme which will allow us to accurately analyze generalized fractional order systems in presence of significant quantities of random errors. Finally, by a simple experiment, we demonstrate the high degree of robustness to noise offered by the said transformation and thus validate our scheme.