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

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.

A New Fractional Model for Cancer Therapy with M1 Oncolytic Virus

“…e results obtained in this study are based on the fractional derivative in sense of Caputo with singular kernel. It will be more interesting to model the dynamics of oncolytic M1 virotherapy by using the new generalized fractional derivative with nonsingular kernel [10]. Moreover, we will extend our model presented in (1) by taking into account other biological factors such as diffusion [11,12] and immunity [13,14].…”

On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel

“…Due to the importance of weighted fractional derivatives to solve several types of integral equations with elegant ways, Al-Refai [3] introduced the weighted Atangana-Baleanu fractional operators and he studied their properties. A generalized version of all previous fractional derivative operators with non-singular kernel was recently proposed in [4]. e objective of this study is to establish some properties and formulas of the new generalized fractional derivative introduced in [4] in order to extend several results presented in recent works.…”

“…A generalized version of all previous fractional derivative operators with non-singular kernel was recently proposed in [4]. e objective of this study is to establish some properties and formulas of the new generalized fractional derivative introduced in [4] in order to extend several results presented in recent works. To achieve this goal, the remainder of this article is outlined as follows.…”

Analysis of a Fractional Reaction-Diffusion HBV Model with Cure of Infected Cells

“…In the recent years, fractional calculus has attracted the attention of many researchers. Hattaf [1] proposed a new fractional derivative with nonsingular kernel which generalizes many forms existing in the literature such as the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. Furthermore, there are some new methods used to solve numerically fractional models considered to explain deeper investigations of real-world problems [2][3][4][5][6].…”

“…e rest of the paper is organized as follows. e following section is devoted to the calculations of the basic reproduction number and steady states of model (1). e global dynamics of the FPDE model is analyzed in Section 3.…”