Numerical approximation of nonlinear fractional dynamical systems for coastal wave modeling: rigorous analysis of convergence and computational efficiency

In the present work, we study a family of nonlinear fractional dynamical systems and propose a numerical procedure based on the power series iterative method (PSIM). The proposed framework is based on the Atangana-Baleanu-Caputo (ABC) fractional derivative which is able to capture memory and nonlocal effects in dynamic processes. The method transforms the original fractional system into an equivalent integral representation and generates approximate solutions through an efficient iterative scheme. We also discuss the analytical properties of the proposed method such as convergence to the exact solution under appropriate conditions and stability to small perturbations. To illustrate its practical utility, we apply the method to the fractional Whitham-Broer-Kaup system (WBKS) for the purpose of coastal wave modeling, analyzing wave propagation, surface elevation dynamics and memory effects in shallow-water environments. The obtained results show that PSIM provides accurate approximations with fast convergence, thus serving as an efficient and reliable computational tool for nonlinear fractional dynamical systems and their applications in coastal hazard assessment.