Modeling and Stability Analysis of Time-Dependent Free-Fall Motion in Random Environments

This paper examines the stability of a fractional-order model that describes the free-fall motion of a football in changing environmental conditions. Traditional models often overlook memory effects and nonlocal influences like air resistance, humidity, and turbulence. To tackle this issue, we enhance the classical Newtonian model by using the Hilfer fractional derivative, which includes memory and hereditary properties. We analyze the resulting equation for Ulam–Hyers–Rassias stability within the context of random Banach spaces, using distribution-valued norms and t–norms to manage uncertainty and oscillatory behavior. We apply fixed-point theory to establish sufficient conditions for stability and uniqueness of solutions. Numerical simulations and error analyses with special functions show the effectiveness and reliability of the proposed model, demonstrating better accuracy and stability compared to traditional methods.