2017/05/10 by J. Vanterler da C. Sousa, Sousa, J. Vanterler da C., E. Capelas de Oliveira +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.1705.07181
openalex publication_date 2017/05/10 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28
We introduce a new derivative, the so-called truncated V-fractional derivative for α-differentiable functions through the six parameters truncated Mittag-Leffler function, which generalizes different fractional derivatives, recently introduced: conformable fractional derivatives, alternative fractional derivative, truncated alternative fractional derivative, M-fractional derivative and truncated M-fractional derivative. This new truncated V-fractional derivative satisfies properties of the entire order calculus, among them: linearity, product rule, quotient rule, function composition, and chain rule. Also, as in the case of the Caputo derivative, the derivative of a constant is zero. Since the six parameters Mittag-Leffler function is a generalization of Mittag-Leffler functions of one, two, three, four, and five parameters, we can extend some of the classic results of the entire order calculus, namely: Rolle's theorem, the mean value theorem and its extension. In addition, we present the theorem involving the law of exponents for derivatives and we calculated the truncated V-fractional derivative of the two parameters Mittag-Leffler function. Finally, we present the V-fractional integral from which, as a natural consequence, new results appear as applications. Specifically, we generalize inverse property, the fundamental theorem of calculus, a theorem associated with classical integration by parts, and the mean value theorem for integrals. Also, we calculate the V-fractional integral of the two parameters Mittag-Leffler function. Further, through the truncated V-fractional derivative and the V-fractional integral, we obtain a relation with the fractional derivative and integral in the Riemann-Liouville sense, in the case 0