Generalizations of Truncated M-Fractional Derivative Associated with (p, k)-Mittag-Leffler Function with Classical Properties

Abstract

AbstractIn the present chapter, we have generalized the truncated M-fractional derivative. This new differential operator denoted by \({ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q},\) where the parameter σ associated with the order of the derivative is such that 0 < σ < 1 and M is the notation to designate that the function to be derived involves the truncated (p, k)-Mittag-Leffler function. The operator \({ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q}\) 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, linear fractional differential equation, and present a graphical analysis.KeywordsPochhammer symbolFractional calculusMittag-Leffler functionHeat equationFractional derivativeFractional differential equations2010 MSC26A3333C4533C6033C70