A generalization of truncated M-fractional derivative and applications to fractional differential equations

TLDR: In this paper, Sousa and de Oliveira proposed a new truncated M-fractional derivative type unifying some fractional derivative types with classical properties and obtained the analytical solutions of some M-series fractional differential equations.

Abstract: In this paper, our aim is to generalize the truncated M-fractional derivative which was recently introduced [Sousa and de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), 83–96, 2018]. To do that, we used generalized M-series, which has a more general form than Mittag-Leffler and hypergeometric functions. We called this generalization as truncated M-series fractional derivative. This new derivative generalizes several fractional derivatives and satisfies important properties of the integerorder derivatives. Finally, we obtain the analytical solutions of some M-series fractional differential equations.