Robust Fractional Quantum Two-Step Schemes with Enhanced Stability for Nonlinear Equations

Fractional quantum calculus provides a powerful mathematical framework for incorporating memory and scaling effects into numerical models. However, classical iterative methods for nonlinear equations often suffer from limited stability, sensitivity to initial guesses, and restricted convergence domains, particularly in highly nonlinear settings. In this work, we introduce a new Caputo fractional–quantum iterative scheme, denoted by MSBq:α, formulated as a parameterized two-step method based on a Caputo-type fractional quantum derivative. The proposed framework incorporates additional structural parameters that regulate the iterative dynamics and enable enhanced control over convergence behavior and stability properties. To assess the performance of the method, we employ tools from complex dynamical systems, including stability analysis and fractal basin investigations in the complex plane. These analyses provide insight into how the fractional and quantum parameters influence the geometry of attraction domains and the global convergence behavior of the scheme. Numerical experiments on representative nonlinear problems arising in engineering and biomedical applications demonstrate improved robustness with respect to initial guesses, reduced residual errors, and competitive computational efficiency compared with existing iterative methods. Overall, the results indicate that the proposed fractional–quantum framework offers an effective and versatile approach for the numerical solution of challenging nonlinear equations.