Multipoint fractional iterative methods with (2α+1)th-order of convergence for solving nonlinear problems

Abstract

In the recent literature, some fractional one-point Newton-type methods have been proposed in order to find roots of nonlinear equations using fractional derivatives. In this paper, we introduce a new fractional Newton-type method with order of convergence α + 1 and compare it with the existing fractional Newton method with order 2a. Moreover, we also introduce a multipoint fractional Traub-type method with order 2α+1 and compare its performance with that of its first step. Some numerical tests and analysis of the dependence on the initial estimations are made for each case, including a comparison with classical Newton (α = 1 of the first step of the class) and classical Traub's scheme (α = 1 of fractional proposed multipoint method). In this comparison, some cases are found where classical Newton and Traub's methods do not converge and the proposed methods do, among other advantages.