Research Article | Open Access

M. Orhan Kaya, S. Altay Demirbağ, F. Özen Zengin, "Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach", *Mathematical Problems in Engineering*, vol. 2009, Article ID 450862, 9 pages, 2009. https://doi.org/10.1155/2009/450862

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

**Academic Editor:**Ekaterina Pavlovskaia

#### Abstract

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%.

#### 1. Introduction

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 [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

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

where is period of the nonlinear oscillator, . Actually, the upper limit is originally instead of . Normally, it works in most of the cases. Let us suppose that such

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

where is the frequency of the oscillator.

Inserting (2.6) into (2.5) yields

Let us define . Then (2.7) becomes

Using the Ritz method, we require

By a careful inspectation, we find that

Thus, the conditions in (2.9) reduce to

#### 3. Application

Consider the following nonlinear oscillator with discontinuity:

with initial conditions

Its variational formulation can be written as follows:

For the first approximation assume that 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:

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].

Secondly, to obtain a more accurate result, define as follows:

Notice that (3.9) satisfies the initial conditions (3.2).

Substituting (3.9) into (3.3), we obtain

The in (3.10) can be obtained as follows:

The stationary condition with respect to *A* and *B* reads

from which the relationship between oscillator frequency and amplitude can be determined.

From (3.12) we have

and the approximate period can be obtained as follows:

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 and the period 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 as follows:

Notice that (3.17) satisfies the initial conditions (3.2).

Substituting (3.17) into (3.3) gives

The stationary condition with respect to *A*, *B,* and *C* reads

Hence the approximate frequency is

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

For this nonlinear problem in (3.1), the exact period is given as follows [25]:

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

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:

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:

By using above values, the periodic function can be written for three levels of approximation as follows:

The normalized exact periodic solution 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:

**(a)**

**(b)**

**(c)**

#### 4. 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.

#### References

- J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 881–894, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,”
*Journal of Sound and Vibration*, vol. 305, no. 4-5, pp. 614–620, 2007. View at: Publisher Site | Google Scholar | MathSciNet - L.-N. Zhang and J.-H. He, “Resonance in Sirospun yarn spinning using a variational iteration method,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 1064–1066, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Öziş and A. Yıldırım, “A study of nonlinear oscillators with ${\text{u}}^{1/3}$ force by He's variational iteration method,”
*Journal of Sound and Vibration*, vol. 306, no. 1-2, pp. 372–376, 2007. View at: Publisher Site | Google Scholar - J.-H. He, “Variational iteration method-some recent results and new interpretations,”
*Journal of Computational and Applied Mathematics*, vol. 207, no. 1, pp. 3–17, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 4, pp. 699–708, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - H. Ozer, “Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 4, pp. 513–518, 2007. View at: Google Scholar - A. Beléndez, C. Pascual, M. Ortuño, T. Beléndez, and S. Gallego, “Application of a modified He's homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 2, pp. 601–610, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Beléndez, A. Hernandez, T. Beléndez, C. Neipp, and A. Marquez, “Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method,”
*Physics Letters A*, vol. 372, no. 12, pp. 2010–2016, 2008. View at: Publisher Site | Google Scholar - A. Beléndez, C. Pascual, S. Gallego, M. Ortuño, and C. Neipp, “Application of a modified He's homotopy perturbation method to obtain higher-order approximations of an ${\text{x}}^{1/3}$ force nonlinear oscillator,”
*Physics Letters A*, vol. 371, no. 5-6, pp. 421–426, 2007. View at: Publisher Site | Google Scholar - A. Beléndez, C. Pascual, T. Beléndez, and A. Hernández, “Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He's homotopy perturbation
method,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 1, pp. 416–427, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Beléndez, T. Beléndez, A. Marquez, and C. Neipp, “Application of He's homotopy perturbation method to conservative truly nonlinear oscillators,”
*Chaos, Solitons & Fractals*, vol. 37, no. 3, pp. 770–780, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,”
*Applied Mathematics and Computation*, vol. 151, no. 1, pp. 287–292, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Beléndez, A. Hernández, T. Beléndez, E. Fernández, M. L. Álvarez, and C. Neipp, “Application of He's homotopy perturbation method to the duffin-harmonic oscillator,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 1, pp. 79–88, 2007. View at: Google Scholar - T. Özis and A. Yıldırım, “A comparative study of He's homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 2, pp. 243–248, 2007. View at: Google Scholar - H.-M. Liu, “Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincare method,”
*Chaos, Solitons & Fractals*, vol. 23, no. 2, pp. 577–579, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S.-Q. Wang and J.-H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,”
*Chaos, Solitons & Fractals*, vol. 35, no. 4, pp. 688–691, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. Ö. Zengin, M. O. Kaya, and S. A. Demirbağ, “Application of parameter-expansion method to nonlinear oscillators with discontinuities,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 9, no. 3, pp. 267–270, 2008. View at: Google Scholar - S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,”
*Chaos, Solitons & Fractals*, vol. 27, no. 5, pp. 1119–1123, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Variational approach for nonlinear oscillators,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1430–1439, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “Variational principles for some nonlinear partial differential equations with variable coefficients,”
*Chaos, Solitons & Fractals*, vol. 19, no. 4, pp. 847–851, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J.-H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,”
*International Journal of Modern Physics B*, vol. 22, no. 21, pp. 3487–3578, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D.-H. Shou, “Variational approach to the nonlinear oscillator of a mass attached to a stretched wire,”
*Physica Scripta*, vol. 77, no. 4, Article ID 045006, 4 pages, 2008. View at: Publisher Site | Google Scholar - Z.-L. Tao, “The frequency-amplitude relationship for some nonlinear oscillators with discontinuity by He's variational method,”
*Physica Scripta for Experimental and Theoretical Physics*, vol. 78, no. 1, Article ID 015004, 2 pages, 2008. View at: Google Scholar | MathSciNet - B. S. Wu, W. P. Sun, and C. W. Lim, “An analytical approximate technique for a class of strongly non-linear oscillators,”
*International Journal of Non-Linear Mechanics*, vol. 41, no. 6-7, pp. 766–774, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2009 M. Orhan Kaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.