- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

International Journal of Differential Equations

Volume 2011 (2011), Article ID 852919, 9 pages

http://dx.doi.org/10.1155/2011/852919

## Solutions of the Force-Free Duffing-van der Pol Oscillator Equation

^{1}Department of Mathematics, University of Karachi, Karachi 75270, Pakistan^{2}Abdul Salam School of Mathematical Sciences, GC University, Lahore, Pakistan^{3}Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan

Received 31 May 2011; Revised 5 August 2011; Accepted 22 August 2011

Academic Editor: Yuriy Rogovchenko

Copyright © 2011 Najeeb Alam Khan 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.

#### Abstract

A new approximate method for solving the nonlinear Duffing-van der pol oscillator equation is proposed. The proposed scheme depends only on the two components of homotopy series, the Laplace transformation and, the Padé approximants. The proposed method introduces an alternative framework designed to overcome the difficulty of capturing the behavior of the solution and give a good approximation to the solution for a large time. The Runge-Kutta algorithm was used to solve the governing equation via numerical solution. Finally, to demonstrate the validity of the proposed method, the response of the oscillator, which was obtained from approximate solution, has been shown graphically and compared with that of numerical solution.

#### 1. Introduction

Considerable attention has been directed toward the solution of oscillator equations since they play crucial role in applied mathematics, physics, and engineering problems. In general, the analytical approximation to solution of a given oscillator problem is more difficult than the numerical solution approximation. Many powerful methods for solving nonlinear oscillator problems were appeared in open literature, such as variational iteration method [1–3], homotopy perturbation method [4–6], Hamiltonian method [7], Lindstedt-Poincare method [8], Variational method [9, 10], parameter-expansion method [11], max-min approach [12], iterative harmonic balance method [13] and differential transformation method [14].

Our main concern in this paper is to study the dynamics of approximate solution for the Duffing-van der pol oscillator equation of the form [15–17] With initial conditions where the overdot denotes differentiation with respect to time, , and are arbitrary parameters. Equation (1.1) is an autonomous equation which describes the propagation of voltage pulses along a neural axon. Some progress was made on the integrability of the Duffing-van der pol equation (1.1) until Chandrasekar and coworkers [15] established the complete integrability of this equation and derived a general solution for a specific choice of arbitrary parameters , given by and taking . Under the specific choice of parameters (1.3) and using a special transformation, Chandrasekar et al. were able to find the solution of this equation.

Mukherjee and colleagues [16] employed the differential transformation method to solve the Duffing-van der pol equation (1.1). Since there are some limitations in using the differential transformation method together with the fact that this method gives the solution in a very small region, developing the method for different applications is very difficult.

In the present study, we used a modified version of homotopy perturbation method which is based on two components of homotopy series. In order to improve the accuracy of the solution, we first apply the Laplace transformation, then convert the transformed series into a meromorphic function by forming the Padé approximants, and finally adopt an inverse Laplace transform to obtain an analytic solution.

Next, Runge-Kutta’s RK algorithm has been introduced to solve the governing equation (1.1). Finally, numerical examples are given to demonstrate the validity of the proposed method, and the effect of parameters on the accuracy of the method is investigated. Here, we also point out that for , , , and , (1.1) reduces to a classical van der pol equation

#### 2. Analysis of New Homotopy Perturbation Method

Let us consider the nonlinear differential equation: where is operator, is a known function, and is a sought function. Assume that operator can be written as where is the linear operator and is the nonlinear operator. Hence, (2.1) can be rewritten as follows: We define an operator as where is an embedding or homotopy parameter, , and is an initial approximation of solution of the problem in (2.4) which can be written as Clearly, the operator equations and are equivalent to the equations and , respectively. Thus, a monotonous change of parameter from zero to one corresponds to a continuous change of the trivial problem to the original problem. Operator is called a homotopy map. Next, we assume that the solution of equation can be written as a power series in embedding parameter , as follows: Now let us write (2.5) in the following form: By applying the inverse operator, , to both sides of (2.7), we have Suppose that the initial approximation of (2.1) has the form where , . are unknown coefficients and , . are specific functions on the problem. By substituting (2.6) and (2.9) into (2.8), we get Equating the coefficients of like powers of , we get following set of equations: Now, if we solve these equations in such a way that . Therefore, the approximate solution may be obtained as

#### 3. Implementation of the Method

To obtain the solution of (1.1) by NHPM, we construct the following homotopy: Applying the inverse operator, to both sides of (3.1), we obtain The solution of (1.1) is to have the following form: Substituting (3.3) in (3.2) and equating the coefficients of like powers of , we get following set of equations: Assuming , as well as solving the above equation for , leads to the following result: With vanishing , we have the following coefficients: , , Therefore, we obtain the solutions of (3.4) as The solution of (1.1) does not exhibit behavior for a large region. In order to improve the accuracy of the two-component solution, we implement the modification as follows.

Applying the Laplace transform to the series solution (3.7) yields For simplicity, let ; then, On applying Padé approximation, Recalling , we obtain Padé approximation in terms of . By using the inverse Laplace transform to the Padé approximant, we obtain the desired approximate solution of the Duffing-van der pol equation.

#### 4. Numerical Solutions

In order to verify the procedure of the method, we consider the following particular cases and comparison will be made with RK4 method as well as [16].

*Numerical Experiment 1*

Consider the Duffing-van der pol equation [16] by taking , :
With initial conditions,
The approximate analytical solution of (4.1) with conditions (4.2) can be obtained by applying the procedure mentioned in previous section as
In Table 1, the results of proposed method are compared to DTM and the fourth-order Runge-Kutta method. For comparison, the displacements of the oscillator corresponding to the four different methods are depicted in Figures 1(a)–1(d) for the same values of the parameters. It is clearly seen from Figure 1(d) that the DTM solution converges in a small region.

*Numerical Experiment 2*

Consider the Duffing-van der pol equation taking , , , and :
With initial conditions,
The solution of (4.1) with conditions (4.5) exhibits the periodic behavior that is the characteristic of the oscillatory system. A comparison between the approximate solution and the solution that is obtained by the fourth-order Runge-Kutta method in Figures 2(a) and 2(b) shows that it converges in a wider region. Figures 3(a) and 3(b) represent approximate shape of the Duffing-van der pol limit cycle.

#### 5. Closing Remarks

In this work, the modified NHPM has been employed to analyze the force-free Duffing-van der pol oscillator with strong cubic nonlinearity. The results obtained from this method have been compared with those obtained from numerical method using RK algorithm and [15, 16]. This comparison shows excellent agreement between these methods. The presented scheme provides concise and straightforward solution to approach reliable results, and it overcomes the difficulties that have been arisen in conventional methods. Unlike the ADM, VIM, and HPM [17–20], the modified HPM [21] is free from the need to use the Adomian polynomials, the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals. The present method is very simple method, leading to high accuracy of the obtained results.

#### Acknowledgments

The author N. A. Khan is thankful and grateful to the Dean of Faculty of Sciences, University of Karachi, Karachi, Pakistan, for supporting and facilitating this research work. The author M. Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan; Department of Mathematics, NED University of Engineering & Technology, Karachi, Pakistan, and also Higher Education Commission of Pakistan for generous support and facilitating this research work.

#### References

- 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 · View at Google Scholar · View at MathSciNet - Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 7, no. 1, pp. 27–34, 2006. View at Publisher · View at Google Scholar · View at Scopus - T. Öziş and A. Yildirim, “A study of nonlinear oscillators with ${u}^{1/3}$ force by He's variational iteration method,”
*Journal of Sound and Vibration*, vol. 306, no. 1-2, pp. 372–376, 2007. View at Publisher · View at Google Scholar · View at Scopus - 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*, vol. 10, no. 2, pp. 601–610, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at 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 Publisher · View at Google Scholar - N. A. Khan, M. Jamil, and A. Ara, “Multiple-parameter Hamiltonian approach for higher accurate approximations of a nonlinear oscillator with discontinuity,”
*International Journal of Differential Equations*, vol. 2011, Article ID 649748, 7 pages, 2011. View at Publisher · 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 · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. H. He, “Variational approach for nonlinear oscillators,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1430–1439, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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, 2008. View at Publisher · View at Google Scholar · View at Scopus - F. Ö. Zengin, M. O. Kaya, and S. A. Demirbaǧ, “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 Publisher · View at Google Scholar · View at Scopus - J. H. He, “Max-min approach to nonlinear oscillators,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 9, no. 2, pp. 207–210, 2008. View at Publisher · View at Google Scholar · View at Scopus - Z. Guo and A. Y. T. Leung, “The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3163–3169, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. E. Ebaid, “A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 1, pp. 528–536, 2011. View at Publisher · View at Google Scholar - V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, “New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems,”
*Journal of Physics: A*, vol. 37, no. 16, pp. 4527–4534, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Mukherjee, B. Roy, and S. Dutta, “Solution of the Duffing-van der pol oscillator equation by a differential transform method,”
*Physica Scripta*, vol. 83, Article ID 015006, 2011. View at Publisher · View at Google Scholar - N. A. Khan, A. Ara, S. A. Ali, and A. Mahmood, “Analytical study of Navier-Stokes equation with fractional orders using He's homotopy perturbation and variational iteration methods,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 9, pp. 1127–1134, 2009. View at Publisher · View at Google Scholar · View at Scopus - N. A. Khan, A. Ara, and A. Mahmood, “Approximate solution of time-fractional chemical engineering equations: a comparative study,”
*International Journal of Chemical Reactor Engineering*, vol. 8, article A19, 2010. View at Google Scholar · View at Scopus - N. A. Khan, M. Jamil, and A. Ara, “An efficient approach for solving the Riccati equation with fractional orders,”
*Computers and Mathematics with Applications*, vol. 61, pp. 2683–2689, 2011. View at Publisher · View at Google Scholar - M. Madani and M. Fathizadeh, “Homotopy perturbation algorithm using Laplace transformation,”
*Nonlinear Science Letters: A*, vol. 1, pp. 263–267, 2010. View at Google Scholar - N. A. Khan, M. Jamil, A. Ara, and N.-U. Khan, “On efficient method for system of fractional differential equations,”
*Advances in Difference Equations*, Article ID 303472, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH