In this paper, the alternating group explicit (AGE) iterative method is applied to a nonlinear fourth-order PDE describing the flow of an incompressible fluid. This equation is a Ladyzhenskaya equation. The AGE method is shown to be extremely powerful and flexible and affords its users many advantages. Computational results are obtained to demonstrate the applicability of the method on some problems with known solutions. This paper demonstrates that the AGE method can be implemented to approximate solutions efficiently to the Navier-Stokes equations and the Ladyzhenskaya equations. Problems with a known solution are considered to test the method and to compare the computed results with the exact values. Streamfunction contours and some plots are displayed showing the main features of the solution.