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


  • Mohammed S. Mechee Information Technology Research and Development Center, University of Kufa, Najaf, Iraq
  • 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



α-Fractional Integral, α-power series, FrDEs


     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. 







How to Cite

Novel Definitions of α-Fractional Integral and Derivative of the Functions. (2023). Iraqi Journal of Science, 64(11), 5866-5877.

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