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

Nonlinear equations arise extensively in engineering and applied sciences, where efficient and reliable iterative solvers are required. This study introduces two fractional-order iterative schemes based on a common predictor–corrector structure: a Caputo-based method, NCFS${}_1$, and an Atangana–Baleanu–Caputo (ABC)-based variant, NFS${}_1^{abc}$. The proposed schemes incorporate a fractional order and two tunable parameters to improve flexibility in the iterative process. The local convergence behavior of the Caputo-based method is analyzed by means of fractional Taylor expansions, yielding an explicit error equation and convergence order, while analogous asymptotic considerations are discussed for the ABC-based variant. A dynamical-systems analysis is also performed through basins of attraction, the Convergence Area Index, and the Wada measure. Numerical experiments on application-motivated nonlinear models indicate that the proposed methods can provide faster error reduction, smaller residuals, and lower computational cost than selected existing fractional iterative schemes. These results suggest that the proposed framework is a flexible and effective approach for nonlinear root-finding problems, combining local convergence analysis with global dynamical assessment.