On fractional derivatives with generalized Mittag-Leffler kernels

Abstract

Fractional derivatives with three parameter generalized Mittag-Leffler kernels and their properties are studied. The corresponding integral operators are obtained with the help of Laplace transforms. The action of the presented fractional integrals on the Caputo and Riemann type derivatives with three parameter Mittag-Leffler kernels is analyzed. Integration by parts formulas in the sense of Riemann and Caputo are proved and then used to formulate the fractional Euler-Lagrange equations with an illustrative example. Certain nonconstant functions whose fractional derivatives are zero are determined as well.