A Modified Fractional Newton’s Solver

Abstract

Fractional calculus extends the conventional concepts of derivatives and integrals to non-integer orders, providing a robust mathematical framework for modeling complex systems characterized by memory and hereditary properties. This study enhances the convergence rate of the Caputo-based Newton’s solver for solving one-dimensional nonlinear equations. By modifying the order to 1+η, we provide a thorough analysis of the convergence order and present numerical simulations that demonstrate the improved efficiency of the proposed modified fractional Newton’s solver. The numerical simulations indicate significant advancements over traditional and existing fractional Newton-type approaches.