Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation

Abstract

In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases.