Fractional derivatives of the generalized Mittag-Leffler functions

Abstract

Abstract In this paper, we derive the compositions of the fractional derivatives with the Shukla function, a four-parameter Mittag-Leffler function. We investigate and compare the difference between the Riemann–Liouville and Caputo derivatives of the generalized Mittag-Leffler functions and obtain the reason causing the difference and expand the fractional derivatives of the generalized Mittag-Leffler functions. Two illustrative examples and the related numerical results are provided to demonstrate the validity.

Keywords

• Mittag-Leffler function
• Riemann–Liouville derivative
• Caputo derivative