tailieunhanh - A new view of the bubnov-galerkin method in the linearization context

In the study an extension of the Bubnov-Galerkin method in terms of the equivalent linearization method is presented. It is combined with sequential linearization and nonlinear procedure to yield a new method for solving nonlinear equations which can improve the accuracy when the nonlinearity is strong. For illustration the Duffing oscillator is considered to show the effectiveness of the proposed method. | Vietnam Journal of Mechanics, VAST, Vol. 34, No. 1 (2012), pp. 1 – 6 A NEW VIEW OF THE BUBNOV-GALERKIN METHOD IN THE LINEARIZATION CONTEXT N. D. Anh1 and I. Elishakoff2 Institute of Mechanics, Hanoi, Vietnam 2 Florida Atlantic University, Boca Raton, USA 1 Abstract. In the study an extension of the Bubnov-Galerkin method in terms of the equivalent linearization method is presented. It is combined with sequential linearization and nonlinear procedure to yield a new method for solving nonlinear equations which can improve the accuracy when the nonlinearity is strong. For illustration the Duffing oscillator is considered to show the effectiveness of the proposed method. Keywords: Bubnov-Galerkin, nonlinear, stochastic, Duffing. 1. INTRODUCTION Bubnov-Galerkin method is one of most popular approximate methods in many fields of applied mechanics since the method is general in scope and can be used for both conservative and nonconservative, both linear and nonlinear systems. The idea was apparently first suggested in 1913 by Bubnov [1], whereas the first paper along with elaborative examples was written in 1915 by Galerkin [2]. In 1937 Duncan [3] published the first comprehensive review of the method in the Western literature. For a given differential equation the Bubnov-Galerkin method approximates the sought solution as a linear combination of comparison functions and requires the orthogonality of the residual to each of comparison functions. In this context the Bubnov-Galerkin method is also known as a weighted residual method [4]. Although the method can be used for both linear and nonlinear systems, it is known that the accuracy of the method decreases when the nonlinearity becomes larger. Elishakoff [5] connected the Bubnov-Galerkin method with the equivalent linearization method. In this paper a representation of the Bubnov-Galerkin method in terms of the equivalent linearization method is presented and a dual approach is subsequently adopted to suggest a new .