Significant Behavior of Generalized Fractional Operators with Different Versions of Mittag-Leffler Functions and its Applications

Abstract

Special functions are considering as the initiative step for the development of modern fractional calculus, and moreover to resolve many issues, which are facing to the applied industries. In this thesis, we consider generalized fractional operators, and discuss its composition with extenion and generalization of Mittag-Leffler type functions. The product of Wiman function and Prabhakar function, and establish its composition with different type of left-and right-sided fractional operators. Furthermore, we construct generalized k-fractional operators involving various forms of the k-Mittag-Leffler functions.

Keywords: Fractional calculus; Generalized fractional operators; Mittag–Leffler functions; k-Mittag–Leffler functions; Wiman function; Prabhakar function; Left-sided fractional operators; Right-sided fractional operators; Special functions; k-fractional operators.