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

Fractional quantum calculus provides a flexible mathematical framework for incorporating memory and scaling effects into numerical models. However, classical iterative methods for nonlinear equations often suffer from limited stability, strong dependence on initial guesses, and restricted convergence domains, particularly for highly nonlinear problems. In this work, we introduce a new Caputo fractional–quantum iterative scheme, denoted by MSB$_{mathfrak{q}:alpha}$, 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 provide enhanced control over convergence behavior and stability properties.
To investigate the performance of the proposed scheme, we employ tools from complex dynamical systems, including stability analysis and fractal basin investigations in the complex plane. These analyses illustrate how the fractional and quantum parameters influence the geometry of attraction domains and the global convergence behavior of the method. Numerical experiments on representative nonlinear test problems motivated by engineering and biomedical applications demonstrate improved robustness with respect to initial guesses, reduced residual errors, and competitive computational efficiency compared with classical iterative solvers.
Overall, the results indicate that the proposed fractional–quantum framework provides an effective and flexible approach for the numerical solution of challenging nonlinear equations.