Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 569098 | https://doi.org/10.1155/2012/569098

Mohammed Ali, Marwan Alquran, Mahmoud Mohammad, "Solitonic Solutions for Homogeneous KdV Systems by Homotopy Analysis Method", Journal of Applied Mathematics, vol. 2012, Article ID 569098, 10 pages, 2012. https://doi.org/10.1155/2012/569098

Solitonic Solutions for Homogeneous KdV Systems by Homotopy Analysis Method

Academic Editor: J. C. Butcher
Received14 Jul 2012
Accepted28 Aug 2012
Published04 Oct 2012

Abstract

We study two-component evolutionary systems of a homogeneous KdV equations of second and third order. The homotopy analysis method (HAM) is used for analytical treatment of these systems. The auxiliary parameter h of HAM is freely chosen from the stability region of the h-curve obtained for each proposed system.

1. Introduction

Applications in physics are modeled by nonlinear systems. Very few nonlinear systems have closed form solutions, therefore, many researchers stress their goals to search numerical solutions. Homotopy analysis method (HAM), first proposed by Liao [1], is an elegant method which has proved its effectiveness and efficiency in solving many types of nonlinear equations [2, 3]. Liao in his book [4] proved that HAM is a generalization of some previously used techniques such as the d-expansion method, artificial small parameter method [5], and Adomian decomposition method. Moreover, unlike previous analytic techniques, the HAM provides a convenient way to adjust and control the region and rate of convergence [6]. Recently, new interested applications of the homotopy analysis have been introduced by Abbasbandy and coauthors [7, 8]. Also, in [9] HAM is used to study the effects of thermocapillarity and thermal radiation on flow and heat transfer in a thin liquid film.

In this work, we consider a two-component evolutionary system of a homogeneous KdV equations of third order type (I) and (II) given, respectively, by Also, we study a two-component evolutionary system of a homogeneous KdV equations of second order given by In the literature many other direct mathematical methods such as the sine-cosine method, rational sine-cosine method, and extended tanh-coth method [1014] have been implemented in obtaining different solitonic solutions to these systems. Our main goal in this paper is to see how much accuracy we may gain by applying HAM to such evolutionary systems.

In what follows, we highlight the main features of the homotopy analysis method. More details and examples can be found in [7, 8, 15] and the references therein.

2. Survey of Homotopy Analysis Method

To illustrate the basic ideas of this method, we consider the following nonlinear system of differential equations: where are nonlinear operators, is an independent variable, are unknown functions. By means of generalizing the traditional homotopy method, Liao construct the zeroth-order deformation equation as follows: where , is an embedding parameter, is a general differential linear operator, are initial guesses of , are unknown functions, and and are auxiliary parameter and auxiliary function, respectively. It is important note that, one has great freedom to choose auxiliary objects such as and in HAM; this freedom plays an important role in establishing the key stone of validity and flexibility of HAM as shown in this paper. obviously, when and , both thus as increasing from to , the solutions of change from the initial guesses to the solutions . Expanding in Taylor series with respect to , one has where if the auxiliary linear operators, the initial guesses, the auxiliary parameter , and the auxiliary function are so properly chosen, then the series (2.4) converges at , then one has which must be one of the solutions of the original nonlinear equations, as proved by liao. Define the vector Differentiating (2.2), times with respect to the embedding parameter and then setting and finally dividing them by , we have the so-called kth-order deformation equation subject to the initial conditions where It should be emphasized that for is governed by the linear equation (2.8) under the linear boundary conditions that come from original problem, which can be easily solved by symbolic computation software such as Mathematica. When , and (2.2) becomes which is used mostly in the homotopy perturbation method [1620].

For the following numerical examples, we use ; .

3. Two-Component Evolutionary System of Order 3: Type I

In this section, we consider system (1.1) subject to For application of the homotopy analysis method, we choose the initial approximations as Employing HAM with the mentioned parameters in Section 2, we have the following zero-order deformation equations: Subsequently solving the Nth order deformation equations, we obtain We use an 10-term approximation and set Comments and illustrations upon the obtained results can be observed in Figures 1 and 2.

4. Two-Component Evolutionary System of Order 3: Type II

In this section, we consider system (1.2) subject to For application of the homotopy analysis method, we choose the initial approximations as Employing HAM with mentioned parameters in Section 2, we have the following zero-order deformation equations: Subsequently solving the Nth order deformation equations, we obtain and so on. We use an 10-term approximation and set Comments and illustrations upon the obtained results can be observed in Figures 3 and 4.

5. Two-Component Evolutionary System of Order 2

In this section, we consider system (1.3) subject to We proceed in the same manner and choose the initial approximations as The zero-order deformation equations are Subsequently, we obtain and so on. We use an 11-term approximation and set The obtained h-curve and the HAM solutions of and are given in Figures 5 and 6.

6. Discussion and Concluding Remarks

In this paper, we obtained soliton solutions for homogeneous KdV systems of second and third order by means of homotopy analysis method. The convergence region for the obtained, approximation, is determined by the parameter h as shown in Figures 1, 3, and 5, respectively, for systems ((1.1), (1.2), and (1.3)).

Homotopy analysis method provides us a convenient freely chosen with parameter h in contrast to the Homotopy perturbation method, where h is assumed to be . In this work, h has been chosen to be , respectively, for systems ((1.1), (1.2), and (1.3)). It is worth noting that the choice of the parameter in this paper is considered based on the obtained stability region of the -curve for each system; for example, in system (1.1) the stability region of falls between and and we considered the midpoint of this interval.

Finally, the absolute errors for the obtained approximate solution of system (1.1) are given in Tables 1 and 2.







References

  1. S. J. Liao, Proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis], Shanghai Jiao Tong University, 1992.
  2. S. Abbasbandy, E. Babolian, and M. Ashtiani, “Numerical solution of the generalized Zakharov equation by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4114–4121, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. E. Babolian and J. Saeidian, “Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1984–1992, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. S. J. Liao, Beyond Perturbation: An Introduction to Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  5. A. M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, UK, 1992.
  6. S. J. Liao, “Notes on the homotopy analysis method: some definitions and theorems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 983–997, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. S. Abbasbandy and A. Shirzadi, “A new application of the homotopy analysis method: solving the Sturm-Liouville problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 112–126, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. S. Abbasbandy, J. L. López, and R. López-Ruiz, “The homotopy analysis method and the Liénard equation,” International Journal of Computer Mathematics, vol. 88, no. 1, pp. 121–134, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. R. C. Aziz, I. Hashim, and S. Abbasbandy, “Effects of thermocapillarity and thermal radiation on flowand heat transfer in a thin liquid film on an unsteady stretching sheet,” Mathematical Problems in Engineering, vol. 2012, Article ID 127320, 14 pages, 2012. View at: Publisher Site | Google Scholar
  10. A. Bekir, “Applications of the extended tanh method for coupled nonlinear evolution equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 9, pp. 1748–1757, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. M. Alquran, “Solitons and periodic solutions to nonlinear partial differential equations by the Sine-Cosine method,” Applied Mathematics and Information Sciences, vol. 6, no. 1, pp. 85–88, 2012. View at: Google Scholar
  12. M. Alquran, K. Al-Khaled, and H. Ananbeh, “New soliton solutions for systems of nonlinear evolution equations by the rational Sine-Cosine method,” Studies in Mathematical Sciences, vol. 3, no. 1, pp. 1–9, 2011. View at: Google Scholar
  13. S. Shukri and K. Al-Khaled, “The extended tanh method for solving systems of nonlinear wave equations,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1997–2006, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. M. Alquran, R. Al-Omary, and Q. Katatbeh, “New explicit solutions for homogeneous Kdv equations of third order by trigonometric and hyperbolic function methods,” Applications and Applied Mathematics, vol. 7, no. 1, pp. 211–225, 2012. View at: Google Scholar
  15. A. Jabbari and H. Kheiri, “Homotopy analysis and homotopy Padé methods for (2+1)-dimensional Boiti-Leon-Pempinelli system,” International Journal of Nonlinear Science, vol. 12, no. 3, pp. 291–297, 2011. View at: Google Scholar
  16. A. M. A. El-Sayed, A. Elsaid, and D. Hammad, “A reliable treatment of homotopy perturbation method for solving the nonlinear Klein-Gordon equation of arbitrary (fractional) orders,” Journal of Applied Mathematics, vol. 2012, Article ID 581481, 13 pages, 2012. View at: Publisher Site | Google Scholar
  17. M. Alquran and M. Mohammad, “Approximate solutions to system of nonlinear partial differential equations using homotopy perturbation method,” International Journal of Nonlinear Science, vol. 12, no. 4, pp. 485–497, 2011. View at: Google Scholar
  18. J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. View at: Publisher Site | Google Scholar
  20. J. H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2012 Mohammed Ali 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views781
Downloads596
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.