About Convergence and Order of Convergence of Some Fractional Derivatives

Abstract

In this paper we obtain some convergence results for Riemann-Liouville, Caputo, and Caputo–Fabrizio fractional operators when the order of differentiation approaches one. We consider the errors given by D 1−α f − f ′ p for p=1 and p = ∞ and we prove that for bothm the Caputo and Caputo Fabrizio operators, the order of convergence is a positive real r ∈ (0,1). Finally, we compare the speed of convergence between Caputo and Caputo–Fabrizio operators obtaining that they are related by the Digamma function.