On generalized fractional integral with multivariate Mittag-Leffler function and its applications

The fractional calculus (FC) has been extensively studied by researchers due to its vast applications in sciences in the last few years. In fractional calculus, multivariate Mittag–Leffler functions are considered the powerful extension of the classical Mittag–Leffler functions. This paper defines the generalized fractional integral operator with multivariate Mittag–Leffler (M-L) function. We prove certain basic properties of the proposed operators, such as an expansion of an infinite series of Riemann–Liouville integrals, Laplace transform (LT), semigroup property, composition with Riemann–Liouville integrals. Also, we present the fractional differential operators and their properties. The application of the proposed operators like the fractional kinetic differential and the time-fractional heat equation are also discussed.