A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties

We introduce a truncated M-fractional derivative type for α-differentiable functions that generalizes four other fractional derivatives types recently introduced by Khalil et al. , Katugampola and Sousa et al. , the so-called conformable fractional derivative, alternative fractional derivative, generalized alternative fractional derivative and M-fractional derivative, respectively. We denote this new differential operator by ₈D₌^α, β, where the parameter α, associated with the order of the derivative is such that 0 0 and M is the notation to designate that the function to be derived involves the truncated Mittag-Leffler function with one parameter. The definition of this truncated M-fractional derivative type satisfies the properties of the integer-order calculus. We also present, the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the M-fractional heat equation and present a graphical analysis.