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

Authors: Subrahamanyam Upadhyay; K.N. Rai

Addresses: DST – Center for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221005, India ' DST – Center for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221005, India

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.

Keywords: Legendre wavelet; collocation; Galerkin method; van der Pol equation; initial value problem; delay feedback; duffing oscillators; periodic forcing term; convergence; stability analysis; displacement; driving force.

DOI: 10.1504/IJESMS.2017.083231

International Journal of Engineering Systems Modelling and Simulation, 2017 Vol.9 No.2, pp.103 - 112

Received: 23 Apr 2015
Accepted: 21 Apr 2016

Published online: 22 Mar 2017 *

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