Exploring Fractional q-Kinetic Equations via Generalized q-Mittag-Leffler Type Functions: Applications and Analysis

In this study, the q-calculus is employed to introduce a novel generalization of the Mittag-Leffler function. In the following, we investigate a novel q-exponential function with five parameters, resulting in the generalized q-Mittag-Leffler function.  Some q-integral representations and fractional q-derivative (Caputo  and Hilfer) for this q-Mittag-Leffler type function are derived. Moreover, we obtain the solutions to the q-fractional kinetic equations includes this function by applying q-Laplace and q-Sumudu transforms defined using fractional q-calculus operators of the Riemann-Liouville (R-L) type. A few particular scenarios to illustrate the use of our primary finding. Further, we state some significant and special cases of our main results. Finally, we present the obtained solutions in the form of numerical graphs using MATLAB ۱۶.

Keywords:

q-Mittag-Leffler function , q-Laplace transform , q-Beta function , q-Gamma function , Fractional q-calculus operator