Novel Definitions of α-Fractional Integral and Derivative of the Functions

Mohammed S. Mechee; Zahrah I. Salman; Shaymaa Y. Alkufi

doi:10.24996/ijs.2023.64.11.32

Authors

Mohammed S. Mechee Information Technology Research and Development Center, University of Kufa, Najaf, Iraq https://orcid.org/0000-0003-2410-4851
Zahrah I. Salman Department. of Math., Faculty of Education, University of Misan, Misan, Iraq
Shaymaa Y. Alkufi Department of Math., Faculty of Computer Science and Math., University of Kufa, Najaf, Iraq

DOI:

https://doi.org/10.24996/ijs.2023.64.11.32

Keywords:

α-Fractional Integral, α-power series, FrDEs

Abstract

An α-fractional integral and derivative of real function have been introduced in new definitions and then, they compared with the existing definitions. According to the properties of these definitions, the formulas demonstrate that they are most significant and suitable in fractional integrals and derivatives. The definitions of α-fractional derivative and integral coincide with the existing definitions for the polynomials for 0 ≤ α < 1. Furthermore, if α = 1, the proposed definitions and the usual definition of integer derivative and integral are identical. Some of the properties of the new definitions are discussed and proved, as well, we have introduced some applications in the α- fractional derivatives and integrals. Moreover, α-power series and α–rule of integration by parts have been proposed and implemented in this study.