Optimal homotopy asymptotic method with Caputo fractional derivatives: a new approach for solving time-fractional Navier-Stokes equation

Fractional calculus has emerged as a powerful mathematical framework for modeling and solving complex phenomena characterized by noninteger order derivatives. In this research, we delve into the utilization and efficacy of the fractional-order optimal homotopy asymptotic method (OHAM) in solving time-fractional Navier–Stokes equations, a class of equations widely encountered in fluid dynamics and engineering. OHAM combines asymptotic series and homotopy perturbation techniques and offers a promising avenue for tackling fractional-order problems with tunable convergence properties. This research unveils OHAM’s effectiveness through a comprehensive analysis of its outcomes. Our investigation begins with a meticulous examination of two illustrative examples. In the first case, OHAM’s solution aligns perfectly with the exact solution, showcasing its remarkable precision. As we delve deeper into the method by increasing the order of approximation, OHAM’s accuracy further improves. This outcome underlines OHAM’s reliability and potential as a dependable approach for addressing time-fractional dynamics in fluid mechanics.