tailieunhanh - On the convergence of a coupling successive approximation method for solving duffing equation

In the present paper the convergence of mentioned method is proven and a condition relating coefficients of Duffing equation to provide the convergence procedure is formulated. Emphasize that the assumption of small parameters is not used in the proving. Some examples are presented to illustrate the proposed method, particularly exact solutions of some problems are compared with analytical approximate ones found by CSAM. | Volume 36 Number 3 3 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 3 (2014), pp. 185 – 200 ON THE CONVERGENCE OF A COUPLING SUCCESSIVE APPROXIMATION METHOD FOR SOLVING DUFFING EQUATION Dao Huy Bich1 , Nguyen Dang Bich2,∗ University of Science, VNU, Vietnam 2 Institute for Building Science and Technology (IBST), Hanoi, Vietnam 1 Hanoi ∗ E-mail: dangbichnguyen@ Received June 21, 2014 Abstract. General Duffing equations occur in many problems of Mechanics and Dynamics. These equations include nonlinear terms of second an third order, their coefficients are finite but not small parameters. For finding analytical approximate solutions of the general Duffing equation the coupling successive approximation method (CSAM) has been proposed by the authors. In the present paper the convergence of mentioned method is proven and a condition relating coefficients of Duffing equation to provide the convergence procedure is formulated. Emphasize that the assumption of small parameters is not used in the proving. Some examples are presented to illustrate the proposed method, particularly exact solutions of some problems are compared with analytical approximate ones found by CSAM. Keywords: General Duffing equation, coupling successive approximation method, convergence, complex valued solution, chaotic solution. 1. INTRODUCTION General Duffing equations appear in formulating and solving many problems of Mechanics and Dynamics, for example [1–7]. Different methods for finding analytical approximate solutions to nonlinear differential equations have been proposed, but the convergence are not to be established for all these methods. For the successive approximation method to nonlinear differential equation of first order [8] (p. 270) and linear differential equation of second order with functional coefficients [9] (p. 317) the convergence condition was indicated, but for nonlinear ones it is still open. Elastic solution method [10] applying to an elastic-plastic

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