Development of Two-Step Fractional Iterative Technique of Convergence Order(2µ + 1) with Applications

Fractional iterative techniques possess the capability to model complex dynamic systems with greater accuracy, playing a crucial role in the advancement of numerical analysis. This study presents a novel class of advanced fractional iterative algorithms, developed to improve the efficiency and precision of solving challenging mathematical problems. Riemann-Liouville and Caputo fractional derivatives, with order (2µ) or (µ + 1), have been used in several recent publications to suggest single step fractional Newton-type approaches. In this article, we provide a twostep conformable fractional Newton-type approach with a (2µ + 1) convergence order employing the same derivatives. Convergence is analyzed, demonstrating its (2µ + 1) convergence. We conducted extensive analysis to evaluate the performance of our algorithm, focusing on absolute error and function evaluations at each iteration. The test functions used in these experiments span a wide range of applications in chemical sciences, civil engineering, and bacterial growth. The numerical outcomes support the theory and enhance the findings. The dynamical portraits reveal that the M SF R method is a highly sensitive iterative approach, particularly effective for capturing complex dynamics.