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We introduce a truncated $M$-fractional derivative type for$\alpha$-differentiable functions that generalizes four other fractionalderivatives types recently introduced by Khalil et al., Katugampola and Sousaet al., the so-called conformable fractional derivative, alternative fractionalderivative, generalized alternative fractional derivative and $M$-fractionalderivative, respectively. We denote this new differential operator by$_{i}\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associatedwith the order of the derivative is such that $ 0 <\alpha<1 $, $\beta>0$ and $M $ is the notation to designate that the function to be derived involves thetruncated Mittag-Leffler function with one parameter.  The definition of this truncated $M$-fractional derivative type satisfies theproperties of the integer-order calculus. We also present, the respectivefractional integral from which emerges, as a natural consequence, the result,which can be interpreted as an inverse property. Finally, we obtain theanalytical solution of the $M$-fractional heat equation and present a graphicalanalysis.

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