Benyamin Ebrahimpour - Dept. of Mechanical Engineering, Babol Noshirvani University of Technology, Mazandaran, Iran
Seyed Ali MirMohammad Sadeghi - Dept. of Mechanical Engineering, Babol Noshirvani University of Technology, Mazandaran, Iran
Nima Bakhshinezhad - Dept. of Mechanical Engineering, Babol Noshirvani University of Technology, Mazandaran, Iran

Van der Pol (VDP) oscillator is an essential topic in applied dynamics to describe complex dynamical behaviors, chaotic motions, etc. In this study, a novel method is proposed towards the analytical solution of the VDP equation-as a quintessential example of nonlinear ordinary differential equations. The presented technique takes advantage of generic trial functions and their derivatives substituted into the target differential equation. Next, an optimization algorithm is tailored to minimize residual mean. Initial/boundary conditions, on the other hand, are dealt with as the second objective of the optimization problem. The proposed method was compared to two conventional analytical methods. The judgment on their accuracy was made based on the Runge-Kutta numerical method. Accordingly, it was observed that the solutions of the proposed method were in tight agreement with the numerical method. Furthermore, it was observed that the newly developed method circumvented the computational burden encountered when using the rival techniques. The obtained results champion outperformance of this method compared with the rival analytical methods in terms of accuracy, simplicity, and robustness.

Analytical solution, Differential equation, Van der Pol oscillator, Inverse method, Optimization