On the new properties of Caputo–Fabrizio operator and its application in deriving shifted Legendre operational matrix

Abstract

In this paper, we study the recently introduced Caputo and Fabrizio operator, which this new operator was derived by replacing the singular kernel in the classical Caputo derivative with the regular kernel. We introduce some useful properties based on the definition by Caputo and Fabrizio for a general order n<α<n+1, n∈N. Here, we extend the associated integral of Caputo–Fabrizio sense to n<α<n+1, n∈N. We also find the general formula for the Caputo–Fabrizio operator of (t−a)β. Then, we derive Legendre operational matrix based on this new operator and together with Tau method, we use it to solve the differential equations defined in the Caputo–Fabrizio sense. As far as we know, the operational matrix method has yet been derived or attempted for solving the differential equations in Caputo–Fabrizio sense, while it has been successfully used to solve fractional calculus problems involving the classical Caputo sense. Some numerical examples are given to display the simplicity and accuracy of the proposed technique.