An optimal and low computational cost fractional Newton-type method for solving nonlinear equations (Q2060913)

The manuscript deals with iterative methods for solving nonlinear equations \(f(x)=0\), by using the conformable fractional derivative of order \(\alpha\). The authors define the conformable fractional derivative of order \(\alpha\), for \(\alpha \in (0,1]\), present some properties of such derivative, the relation with the standard derivative and the Taylor expansion of the function \(f\) with this type of derivatives. Using the Taylor expansion, the conformable fractional Newton-type method is deduced and its convergence is proved. The order of convergence of this method is derived to be of order two. Finally, several numerical experiments confirm the theoretical results and show the advantages of the proposed method over conventional Newton-type methods designed with Caputo or Riemann-Liouville fractional derivatives.