An optimal and low computational cost fractional Newton-type method for solving nonlinear equations

Abstract

In recent papers, some fractional Newton-type methods have been proposed by using the Riemann–Liouville and Caputo fractional derivatives in their iterative schemes, with order 2α or 1+α. In this manuscript, we introduce the Conformable fractional Newton-type method by using the so-called fractional derivative. The convergence analysis is made, proving its quadratic convergence, and the numerical results confirm the theory and improve the results obtained by classical Newton’s method. Unlike previous fractional Newton-type methods, this one involves a low computational cost, and the order of convergence is at least quadratic.