A Novel Fractional Iterative Technique of Order (2 <i>λ</i> + 1) With Applications to Engineering

Fractional derivatives are widely used to model complex phenomena where classical derivatives fail to provide accurate solutions. As a result, iterative methods involving fractional derivatives have become highly significant for addressing such challenges. At present, the literature reports only two existing two‐step fractional iterative methods, highlighting a significant gap and emphasizing the need for further advancement in this area of research. In this work, we introduced a novel class of advanced fractional iterative algorithms using Caputo and Riemann‐Liouville fractional derivatives designed to enhance the efficiency of solving complex mathematical problems. Our approach presented a two‐step iterative method of order (2 λ + 1), effectively addressing situations where standard derivatives are insufficient, using fractional derivatives. This technique requires only two functions and one evaluation of the fractional derivative, making it highly efficient. We established the applicability of our method in solving nonlinear equations across diverse scientific disciplines, including chemical sciences, civil engineering, beam support system, corporate logistics, bacterial growth modeling, and bridge beam deflection analysis. Extensive numerical experiments, conducted using Maple 2022, validate the performance of the proposed algorithm, with comparisons based on absolute error and computational efficiency at each iterative step. The results demonstrate that our method outperforms existing techniques, exhibiting significantly reduced absolute errors. Furthermore, graphical error comparisons, generated using MATLAB 2018a, illustrate the superior accuracy and thus may be considered a significant addition to the existing scholarly literature.

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