Physics and applied mathematics form the basis for understanding natural phenomena using differential equations depicting the flow in porous media, the motion of viscous liquids, and the propagation of waves. These equations provide a thorough study of physical processes, enhancing the understanding of complex applications in engineering, technology, and medicine. This paper presents novel approximate solutions for the Darcy-Brinkmann-Forchheimer moment equation, the Blasius equation and the FalknerSkan equation with initial / boundary conditions by using two iterative methods: the variational iteration method and the optimal variational iteration method. The variational iteration method is effectively developed by adding a control parameter to enhance the convergence speed and prevent large-scale divergence. The influence of physical parameters on the accuracy of the solution was also analyzed, since it was noted that increasing some parameters improves accuracy, while increasing others leads to a decrease the accuracy. Also, the convergence of the proposed methods has been discussed and proved. Moreover, comparison was made with some approximate methods available in the literature were used the operational matrices methods include: Bernstein's method (BOM), Bernoulli's method (BrOM), and the shifted Legendre’s method (LOM). Furthermore, the maximum values of the residual error were computed for the proposed methods and others operational matrices methods for different cases. The results demonstrated the efficiency and accuracy of the optimal variational iteration method in solving nonlinear ordinary differential equations in comparison to other methods. All calculations in this paper were made using the Mathematica®14 software.
