A new algorithm for solving Van der Pol equation based on piecewise spectral Adomian decomposition ‎method‎

Subject Areas : International Journal of Industrial Mathematics

S. GH Hosseini ¹ , E. Babolian ² , S. Abbasbandy ³

1 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
2 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
3 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.

Received: 2016-02-27 Accepted : 2016-02-27 Published : 2016-08-01

Keywords: Van der Pol ‎equation, ‎S‎pectral Adomian decomposi, Piecewise method, ‎Runge-Kutta ‎method,

Abstract :

‎‎In this article‎, ‎a new method is introduced to give approximate solution to Van der Pol equation‎. ‎The proposed method is based on the combination of two different methods‎, ‎the spectral Adomian decomposition method (SADM) and piecewise method‎, ‎called the piecewise Adomian decomposition method (PSADM)‎. ‎The numerical results obtained from the proposed method show that this method is an effective‎, accurate and powerful tool for solving Van der Pol equation and‎, ‎the comparison show that the proposed technique is in good agreement with the numerical results obtained using Runge-Kutta method‎. ‎The advantage of piecewise spectral Adomian decomposition method over piecewise Adomian decomposition method is that it does not need to calculate complex integrals‎. ‎Another merit of this method is that‎, ‎unlike the spectral method‎, ‎it does not require the solution of any linear or nonlinear system of equations‎. Furthermore‎, ‎the proposed method is easy to implement and computationally very ‎attractive.