Higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity by variational approach.(Research Article)(Report)

1. Introduction

Considerable attention has been directed toward the solution of nonlinear equations since they play crucial role in applied mathematics, physics, and engineering problems. In general, the analytical approximation to solution of a given nonlinear problem is more difficult than the numerical solution approximation. During the past decades, several types of methods are proposed to obtain approximate solution of nonlinear equations of various types. Among them are variational iteration methods [1-7], homotopy perturbation method [8-15], modified Lindstedt-Poincare method [16], parameter expansion method [17, 18],and variational methods [19-21]. The variational method is different from any other variational methods in open literature, and it is only valid for nonlinear oscillators [22]. Paper [23] is an example of use of variational approach method in nonlinear oscillator problem.

When we examine the frequency amplitude relations of some nonlinear oscillators, it is seen that paper [24] focuses on only the first-order solutions.

Variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions [20] .

In the present study, we have investigated the application of variational approach to nonlinear oscillator with discontinuity.

2. A Variational Method

Let us consider a general nonlinear oscillator in the form

u" + f (u) = 0. (2.1)

He proposed a variational principle for (2.1) as follows [20] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

where T is period of the nonlinear oscillator, [delta]F/[delta]u = f. Actually, the upper limit is originally T instead of T/4. Normally, it works in most of the cases. Let us suppose that f (u) = sgn(u) such

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

But this form is not suitable for discontinuity equation. Therefore, we propose the equation in the form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN …