Non-local fractional calculus from different viewpoint generated by truncated [formula omitted]-derivative

Abstract

The core of this paper is to propose more favored novel non-local fractional integral and derivative operators with three parameters by using the iteration manner on truncated M-derivative involving truncated Mittag-Leffler function. This is related closely to the fact that, in a certain sense, to iterate a convenient local derivative for real phenomena under investigation is one of the best ways for obtaining more comprehensive fractional integration and differentiation. In order to gain a deeper grip, we address some significant theorems, reflected accomplishedly the features of the standard fractional calculus, about these aidful operators encompassing some other types of fractional operators. The presented new operators in Riemann–Liouville and Caputo sense are utilized to solve Sturm–Liouville problem subjected to the initial conditions. As a useful method, Laplace transform of the aforementioned fractional integral and derivative is introduced to solve anomalous dynamics of various complex systems. Moreover, we carry out comparison analysis by plotting governing Sturm–Liouville equation in order to grasp the advantages of these new fractional operators performing much better on account of the additional parameter γ.