A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law

Abstract

Abstract In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.

Keywords

• Fractional operator with the generalized Mittag–Leffler kernels
• Nonsingular kernel
• Weighted generalized Laplace transform