Closed-Form Solutions for a Dynamical System Using Optimal Parametric Iteration Method

The vibrations of the dynamical system play an important role in biological processes, electrical engineering, and mechanical structures. In this work, we focus on the behaviors of dynamical systems, such as periodical or damped oscillations and asymptotic behaviors. Theorems for explicitly integrability of the dynamical system are established. The effect of the physical parameters 0<a≤1, d≥0 is semi-analytically analyzed by means of the Optimal Parametric Iteration Method (OPIM). We pointed out some cases when the investigated system admits only one first integral or two first integrals. These cases are reduced to a second-order nonlinear differential equations, which are solved by OPIM. The OPIM solutions are highlighted qualitatively by figures and quantitatively by tables, respectively, and are in good agreement with corresponding numerical ones. The accuracy of the obtained results are emphasized by comparison with the iterative solutions, via the classical iterative method and new optimal iterative method, respectively. Other advantages of the applied method are pointed out.