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A new truncated -fractional derivative type unifying some fractional derivative types with classical properties
We introduce a truncated -fractional derivative type for
-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 -fractional
derivative, respectively. We denote this new differential operator by
, where the parameter , associated
with the order of the derivative is such that and 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 -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 -fractional heat equation and present a graphical
analysis.Comment: 16 pages, 3 figure
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