The Q-Analogues Of Nonsingular Fractional Operators With Mittag-Leffler And Exponential Kernels

This paper is devoted to introducing a new q-fractional calculus in the framework of Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) operators. First, an appropriate q-Mittag-Leffler function is defined, and then q-analogues of fractional derivatives of Atangana-Baleanu-Riemann (ABR) and Atangana-Baleanu-Caputo (ABC) are derived. Next, the q-analogues of proper fractional integrals in the AB sense are proved. Several important properties of these definitions are investigated by using the q-Laplace transform. Additionally, a suitable q-exponential function is defined, and the q-analogues of CF fractional derivatives with their inverse operators are introduced. The higher-order extension of the q-analogues of AB and CF fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of q-ABC calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.