(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this study, we investigate two fundamental fractal-fractional (FF) models: the competitive dynamics among Egyptian banks and the Brusselator system. For the banking model, optimal control strategies are proposed to mitigate profit downturns during crises, such as the COVID-19 pandemic, through a system of four fractional differential equations. Recognizing the slow convergence of traditional numerical methods, an efficient integration technique is developed to simulate both models with enhanced accuracy and computational efficiency. The simulation results reveal the dynamic behaviors of the studied systems for various FF-operator values, confirming the robustness and precision of the proposed approach when compared with the classical fourth-order Runge–Kutta method (RK4M). The presented technique offers a simple yet powerful framework for modeling and analyzing complex FF-based dynamical systems.