Series Solution for Painlevé Equation II

Authors

  • Fazle MABOOD School of Mathematical Sciences, Universiti Sains Malaysia, Penang
  • Waqar Ahmad KHAN Department of Engineering Sciences, National University and Technology, Karachi
  • Ahmad Izani Md ISMAIL School of Mathematical Sciences, Universiti Sains Malaysia, Penang
  • Ishak HASHIM School of Mathematical Sciences, Universiti Kebangsaan Malaysia

Keywords:

Optimal homotopy asymptotic method, Painlevé equation, Nonlinear ODE

Abstract

The Painlev'e equations are second order ordinary differential equations which can be grouped into six families, namely Painlev'e equation I, II,…, VI. This paper presents the series solution of second Painlevé equation via optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solution. Comparison of the obtained solution via OHAM is provided with those obtained by Homotopy Perturbation Method (HPM), Adomian Decomposition Method (ADM), Sinc-collocation and Runge-Kutta 4 methods. It is revealed that there is an excellent agreement between OHAM and other published data which confirm the effectiveness of the OHAM.

doi:10.14456/WJST.2015.43

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References

M Dehghan and F Shakeri. The numerical solution of the second Painlevé equation. Numer. Meth. Part. Diff. Equat. 2008; 25, 1238-59.

R Ellahi, S Abbasbandy, T Hayat and A Zeeshan. On comparison for series and numerical solutions for second Painlevé equation II. Numer. Meth. Part. Diff. Equat. 2009; 26, 1070-8.

A Saadatmandi. Numerical study of second Painlevé equation. Comm. Numer. Anal. 2012; 2012, Article ID cna-00157.

SS Behzadi. Convergence of iterative methods for solving Painlevé equation. Appl. Math. Sci. 2010; 4, 1489-507.

H Esmail and A Peyrovi. The use of variational iteration method and homotopy perturbation method for Painlevé equation I. Appl. Math. Sci. 2009; 3, 1861-71.

H Esmail and A Peyrovi. Homotopy perturbation method for second Painlevé equation and comparison with analytic continuation method and Chebyshev series method. Int. Math. Forum. 2010; 5, 629-37.

AP Bassom, PA Clarkson and AC Hicks. Numerical studies of the fourth Painlevé equation. J. Appl. Math. 1993; 50, 167-93.

CW Clenshaw and HJ Norton. The solutions of non-linear ordinary differential equations in Chebyshev Series. Comput. J. 1963; 6. 88-92.

DG Dastidar and SK Majumdar. The solution of Painlevé equations in Chebyshev series. F.C. Auluk, F.N. 1972; 4, 155-60.

M Mazzocco and MY Mo. The Hamiltonian structure of the second Painlevé hierarchy. Nonlinearity 2007; 20, 2845-82.

V Marinca, N Herisanu and I Nemes. Optimal homotopy asymptotic method with application to thin film flow. Cent. Eur. J. Phys. 2008; 6, 648-53.

F Mabood, AIM Ismail and I Hashim. The application of optimal homotopy asymptotic method for the approximate solution of Riccati equation. Sains Malays. 2013; 42, 863-7.

MS Hashmi, N Khan and S Iqbal. Optimal Homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind. Appl. Math. Comput. 2012; 218, 10982-9.

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Published

2014-01-29

How to Cite

MABOOD, F., KHAN, W. A., ISMAIL, A. I. M., & HASHIM, I. (2014). Series Solution for Painlevé Equation II. Walailak Journal of Science and Technology (WJST), 12(10), 941–947. Retrieved from https://wjst.wu.ac.th/index.php/wjst/article/view/832