^{1}

^{2}

^{2}

^{1}

^{2}

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

Considerableattention 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 approximationto solution of a given nonlinear problem ismoredifficult 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 [

When we examine the frequency amplitude relations of some nonlinear oscillators, it is seen that paper [

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 [

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

Let us consider a general nonlinear oscillator in the form

He proposed a variational principle for (

therefore

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

Assume that its solution can be expressed as

Inserting (

Let us define

Using the Ritz method, we require

By a careful inspectation, we find that

Thus, the conditions in (

Consider the following nonlinear oscillator with discontinuity:

Its variational formulation can be written as follows:

For the first approximation assume that

Substitute this first approximation into (

The stationary condition with respect to

This solution agrees with Liu’s solution obtained by He’s modified Lindsted-Poincaré method [

Secondly, to obtain a more accurate result, define

Notice that (

Substituting (

The stationary condition with respect to

From (

In this study, we obtained the relative error as 1.6% for the first-order approximation while the other researchers [

Equation (

To obtain a more accurate result, define

Notice that (

Substituting (

The stationary condition with respect to

Hence the approximate frequency is

Therefore, approximate period of the nonlinear oscillator can be obtained as follows:

For this nonlinear problem in (

The period values and these relative errors obtained in this method for nonlinear oscillator with discontinuity are the following:

Equation (

Equation (

By using above values, the periodic function

The normalized exact periodic solution

Comparison of approximate normalized functions (dashed line) with exact normalized functions (continuous line) for (a) first approximation (b) second approximation (c) third approximation.

He’s variational approach is modified for nonlinear oscillator with discontinuities. The method has been applied to obtain three levels of approximation of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(u). We reached 1.6%, 0.31%, and 0.1% relative errors for the first, second, and third approximate periods, respectively. One can obtain higher-order accuracy by extending the idea given in this paper.