AN ADAPTED ANALYTICAL SOLUTION TO THE MULHOLLAND EQUATION: MODIFIED DIRECT ITERATION PROCEDURE

The Mulholland equation is a third-order ordinary differential equation characterized by its nonlinearity. It particularly represents a central restoring force. Mulholland equations have many real applications in various fields, such as modern control theory, phase plane analysis, stability analysis, bifurcation analysis, etc. This paper employed the modified direct iteration method to solve the Mulholland equation, analytically incorporating all its terms (sometimes in approximated forms) during each iterative step. The analytical solutions are compared with existing results. The analytical solutions also showed remarkable precision when compared with numerical outcomes. It also becomes clear that compared to other approaches already in use, the modified direct iteration method is substantially easier to use, more accurate, efficient, and uncomplicated. Moreover, the fourth approximated frequency exhibited only a 0.022 percentage error. The suggested method can be widely applied to different engineering issues, while it is primarily demonstrated in nonlinear models with strong nonlinear factors.