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

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    We introduce a truncated MM-fractional derivative type for α\alpha-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 MM-fractional derivative, respectively. We denote this new differential operator by iDMα,β_{i}\mathscr{D}_{M}^{\alpha,\beta }, where the parameter α\alpha, associated with the order of the derivative is such that 00 0 0 and M 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 MM-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 MM-fractional heat equation and present a graphical analysis.Comment: 16 pages, 3 figure
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