On the Development and Stability Analysis of Two Step Fractional Iterative Scheme with Applications

Abstract

Fractional iterative techniques are pivotal for modeling complex dynamic systems with enhanced accuracy. Despite their significance, scarcity of two step fractional iterative methods presents critical research gap. To fill this void, we propose a novel class of conformable two step Newton type fractional algorithm based on Caputo derivative, designed to achieve higher computational efficiency and precision. Unlike existing single step fractional Newton type methods, our approach attains a convergence order of (2℘ + 1). This claim is thoroughly proved through theoretical analysis. The algorithm is computationally efficient requiring only two function evaluations and one fractional derivative calculation per iteration. Extensive numerical experiments conducted in Maple 2022, using test functions from various fields such as quarter-car suspension modeling, hanging object, spring-mass system analogue, bookshelf deflection analysis and acidity determination in chemical systems. Absolute error per iteration and elapsed time are key metrics that indicate its superiority over existing methods. Furthermore, a comprehensive fractal analysis using Mobius transformations for rational functions with fixed and critical point analysis deepens the understanding of dynamical properties and stability of the proposed algorithm. Parametric planes generated by MATLAB R2014a confirm enhanced stability. The results highlight algorithm’s broad applicability in solving nonlinear equations across scientific and engineering domains, establishing it as a valuable contribution to numerical analysis.