A New Generalized Truncated K–series Fractional Derivative

Abstract This paper proposes a new fractional derivative denoted by K–Fractional Truncated Derivative (K–FTD). K–FTD generalizes several existing fractional derivatives found in the literature including the conformable derivative, the truncated alternating derivative, the truncated M-fractional derivative, and the truncated M- series fractional derivative. Furthermore, it also satisfies the same properties of ordinary derivatives, such as linearity, product rule, and chain rule. In this respect, some classical results of the ordinary derivative such as Rolle’s theorem and the mean value theorem are extended. Then, the K–fractional integral is introduced, and the truncated K–fractional derivative, as well as the fractional integral of some orders α ∈ (n; n + 1], are defined. Finally, analytical solutions have been obtained for some differential and partial differential equations of fractional order α ∈ (0; 1] to demonstrate the described approach.