A new modified definition of Caputo–Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM)

Abstract

In this paper, we present a new definition of fractional-order derivative with a smooth kernel based on the Caputo–Fabrizio fractional-order operator which takes into account some problems related with the conventional Caputo–Fabrizio factional-order derivative definition. The Modified-Caputo–Fabrizio fractional-order derivative here introduced presents some advantages when some approximated analytical methods are applied to solve non-linear fractional differential equations. We consider two approximated analytical methods to find analytical solutions for this novel operator; the homotopy analysis method (HAM) and the multi step homotopy analysis method (MHAM). The results obtained suggest that the introduction of the Modified-Caputo–Fabrizio fractional-order derivative can be applied in the future to many different scenarios in fractional dynamics.