Solving dynamic equations by a semi-analytical method

Dynamic equations can be considered as a mathematical representation of the time-varying behavior of a system, using ordinary differential equations for state dynamics and algebraic equations for output/observation. In this paper, we consider one of the most famous dynamic equations known as the Lienard equation. In mathematics , more specifically in the study of dynamical systems and differential equations , the Lienard equation is a type of second-order ordinary differential equation. One of the applications of the Lenard equation is its use in modeling oscillatory circuits and the development of radios and vacuum tubes. Therefore, due to the wide application of the Lienard equation, it is very important to provide methods that can solve it appropriately. Here, a procedure is presented based on the homotopy perturbation method (HPM) to obtain an approximate solution for the Lienard equation. The homotopy perturbation method is a semi-analytical technique to solve nonlinear ordinary or partial differential equations . This method employs the concept of the homotopy from topology to generate a convergent series solution for nonlinear systems. The HPM is defined as a mathematical technique that assumes the solution of a nonlinear differential equation can be expressed as a power series in a homotopy parameter. In the proposed method, the solution is considered as the sum of an infinite series which converges rapidly to the accurate solutions. Using homotopy technique in topology, a homotopy is constructed with an embedding parameter which is considered as a small parameter. Two numerical examples are given to apply the homotopy perturbation method for solving the Lienard equation. Also, the obtained approximate solutions have been compared with their known theoretical solutions and results obtained via the variational iteration method (VIM). The results show the efficiency and capability of HPM in solving dynamic equations.