Abstract

Fractional differential equations have emerged as powerful mathematical tools for modeling complex dynamical systems exhibiting memory and hereditary characteristics across engineering, physics, and applied sciences. However, obtaining accurate and computationally efficient numerical solutions for nonlinear fractional systems remains a significant challenge due to slow convergence and instability associated with conventional analytical–numerical techniques. In this study, an Adaptive Residual-Corrected Accelerated Adomian Decomposition Method (ARC-AA-ADM) is proposed for the numerical solution of nonlinear fractional differential equations formulated in the Caputo sense. The proposed framework enhances the classical Adomian Decomposition Method by incorporating a residual-based adaptive iteration mechanism, enabling automatic truncation control and improved numerical stability. Furthermore, a convergence acceleration strategy based on rational approximation techniques is introduced to significantly reduce series convergence time while maintaining solution accuracy. A residual correction procedure is subsequently employed to minimize approximation errors through fractional integral compensation, leading to refined solution reconstruction with fewer decomposition terms. Theoretical analysis establishes the stability and convergence of the proposed iterative operator using fixed-point arguments in Banach spaces. The effectiveness and robustness of ARC-AA-ADM are validated through several benchmark fractional models, where numerical results demonstrate substantial reductions in residual error and faster convergence compared with classical ADM and existing semi-analytical methods. The obtained results confirm that the proposed method provides an efficient, accurate, and computationally reliable framework for solving a broad class of nonlinear fractional differential equations, thereby offering a significant advancement over traditional decomposition-based numerical schemes.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 1
Year: 2023

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