On a fractional derivative operator with a singular kernel: definition, properties and numerical simulation

Abstract

This paper is concerned with proposing a novel nonlocal fractional derivative operator with a singular kernel. We considered a fractional integral operator as a single integral of convolution type combined with a Mittag-Leffler kernel of Prabhakar type. The proposed singular fractional derivative operator is formulated as a proper inverse of the considered integral operator. We provided some useful features and relationships of the proposed derivative and introduced comparisons with the Caputo derivative which can be utilized for potential applications. Next, we presented numerical solutions for some nonlinear fractional order models incorporating the proposed derivative using a numerical algorithm developed in this paper. As a case study, we discussed the dynamic behavior of a fractional logistic model with the proposed derivative.