Efficient conformable iterative approaches for solving nonlinear equations via comprehensive dynamical stability analysis - Asghari, Shima; Lotfi, Taher; Moccari, Mandana (The Journal of Analysis, 2025)
In this paper, we focus on the challenge of efficiently computing all roots of nonlinear equations with fractional-order derivatives, which appears in applied mathematics, physics, and engineering. To solve this problem, we propose two novel iterative schemes: one derivative-based and one derivative-free, both utilizing conformable derivatives to extend classical approaches to fractional orders. These schemes are based on the King family of fourth-order iterative methods, introduced by King in 1971. By changing the order of the fractional derivative based on an initial guess, our methods enhance computational efficiency. We comprehensively prove the convergence and the orders of convergence of the proposed methods. To ensure convergence to all roots, we introduce new convergence planes. The novelty of our study is the combination of root-finding approaches and strong convergence analysis, greatly improving the ability to solve fractional-order problems. Numerical examples demonstrate the accuracy, speed, and low computational cost of our methods, indicating their practical applicability.