A Unified Caputo–ABC Fractional Framework for High-Order Iterative Methods in Nonlinear Equations

Nonlinear equations arise extensively in engineering and applied sciences. This study introduces a family of Caputo and Atangana–Baleanu–Caputo (ABC) fractional order iterative methods for solving nonlinear problems. The proposed schemes are designed to enhance convergence behavior and improve robustness compared to existing fractional Newton-type methods. Local convergence is analyzed using fractional Taylor expansions, establishing the order of convergence and associated error equations. In addition, a dynamical systems perspective is adopted to investigate global convergence properties through basin of attraction analysis, including fractal structures and the Wada measure. Numerical experiments on application-inspired nonlinear models demonstrate that the proposed methods achieve faster error reduction, lower residuals, and improved computational efficiency compared to existing schemes. These results indicate that the proposed framework provides an effective and flexible approach for solving nonlinear equations, combining accuracy, stability, and dynamical insight.