Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

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%.


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 ismore 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 .

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

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 : where T is period of the nonlinear oscillator, ∂F/∂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 sgn u

2.4
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 where ω is the frequency of the oscillator.

2.8
Using the Ritz method, we require

2.11
Thus, the conditions in 2.9 reduce to

Application
Consider the following nonlinear oscillator with discontinuity: Its variational formulation can be written as follows: For the first approximation assume that u t is in the following form: Substitute this first approximation into 3.3 : The stationary condition with respect to A reads which leads to the result and the approximate period can be obtained as follows: Secondly, to obtain a more accurate result, define u as follows: u A cos ωt B cos ωt − cos 3ωt . 3.9 Notice that 3.9 satisfies the initial conditions 3.2 .

3.10
The J A, B, ω in 3.10 can be obtained as follows:

3.11
The stationary condition with respect to A and B reads In this study, we obtained the relative error as 1.6% for the first-order approximation while the other researchers 2, 16 obtained the relative error as 1.8%.The reason for the difference in the relative error is that the other researchers take less precision in the decimal numbers during calculations.In 9 , the frequency ω 1.107452/ √ A and the period T 2app A 5.67440 √ A were found for the same problem by second-order approximation and the relative error was calculated as %0.30.Equation 3.1 was approximately solved in 25 using an improved harmonic balance method that incorporates salient features of both Newtons's method and the harmonic balance method.In 25 , the following results for the first and second-order approximations were obtained: To obtain a more accurate result, define u as follows: u A cos ωt B cos ωt − cos 3ωt C cos 3ωt − cos 5ωt .

3.17
Notice that 3.17 satisfies the initial conditions 3.2 .Substituting 3.17 into 3.3 gives

3.18
The stationary condition with respect to A, B, and C reads For this nonlinear problem in 3.1 , the exact period is given as follows 25 :

3.22
The period values and these relative errors obtained in this method for nonlinear oscillator with discontinuity are the following:

3.23
Mathematical Problems in Engineering Equation 3.1 was approximately solved in 25 using an improved harmonic balance method that incorporates salient features of both Newtons's method and the harmonic balance method.In 25 , the following result for the third-order approximations was obtained: T WSL3 A 5.650976 √ A, relative error 0.10%.3.24 Equation 3.1 was approximately solved in 9 using a homotopy perturbation method.In 9 , the following result for the third-order approximations was obtained:

3.26
The normalized exact periodic solution u ex /A has been obtained by numerically integrating 3.1 and 3.2 and compared with approximate solutions 3.26 in Figure 1.Here nondimensional time h is defined as follows: 3.27

Conclusions
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.

Figure 1 :
Figure 1: Comparison of approximate normalized functions dashed line with exact normalized functions continuous line for a first approximation b second approximation c third approximation.
This solution agrees with Liu's solution obtained by He's modified Lindsted-Poincaré method 16 , Rafei et al.'s solution obtained by He's variational iteration method 2 , Wu et al.'s solution obtained by the low-order harmonic balance method 25 , and A. Belendéz et al.'s solution obtained by He's homotopy perturbation method 9 .