A New Generalized Definition of Fractional Derivative with Non-Singular Kernel

Khalid Hattaf

doi:10.3390/computation8020049

Abstract

This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann–Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann–Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.

Highlights

Fractional derivative is the generalization of the classical derivative of integer order
It has been recently used to study the impact of memory on the dynamics of various systems from different fields such as epidemiology [1,2], virology [3,4,5], ecology [6,7,8] and economics [9]
They proposed a new definition of fractional derivative based on Mittag–Lefler function

Summary

Introduction

Fractional derivative is the generalization of the classical derivative of integer order. In 2015, Caputo and Fabrizio [18] presented a new fractional derivative with non-singular kernel. In 2016, Atangana and Baleanu [19] remarked that the fractional derivative proposed in [18] cannot produce the original function when the order of derivative is equal to zero. To solve this problem, they proposed a new definition of fractional derivative based on Mittag–Lefler function. The main purpose of this study is to propose a new definition of fractional derivative that generalizes the above mentioned fractional derivatives with non-singular kernel for both Caputo and Riemann–Liouville types.

The New Fractional Derivative

Laplace Transform of the New Derivative

Fractional Integral Associated to the New Derivative

Application

Conclusions