Sains Malaysiana 38(5)(2009): 717–721

 

Direct Solution of Second-order BVPs by Homotopy-perturbation Method

(Penyelesaian Secara Langsung MNS Berperingkat-Dua Melalui Kaedah Homotopi-usikan)

 

O. Abdulaziz1, M.S.H. Chowdhury2,

I. Hashim1* & S. Momani3

 

1Centre for Modelling & Data Analysis

School of Mathematical Sciences, Universiti Kebangsaan Malaysia

43600 UKM Bangi Selangor D.E., Malaysia

 

2Faculty of Engineering

International Islamic University Malaysia

Jalan Gombak, 53100 Kuala Lumpur, Malaysia

 

3Department of Mathematics

Mutah University, P.O. Box 7, Al-Karak, Jordan

 

Diserahkan: 20 Jun 2008 / Diterima: 20 November 2008

 

 

ABSTRACT

In this paper, systems of second-order boundary value problems (BVPs) are considered. The applicability of the homotopy-perturbation method (HPM) was extended to obtain exact solutions of the BVPs directly.

 

Keywords: Boundary value problems; homotopy-perturbation method

 

ABSTRAK

 

Dalam makalah ini, sistem masalah nilai sempadan (MNS) berperingkat dua dipertimbangkan. Kegunaan kaedah homotopi-usikan (KHU) diperluaskan bagi memperoleh penyelesaian tepat MNS tersebut secara langsung.

Kata kunci: Kaedah homotopi-usikan; masalah nilai sempadan

 

RUJUKAN

 

Abbasbandy, S. 2007a. A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos Solitons Fractals 31: 257-260.

Abbasbandy, S. 2007b. Application of He’s homotopy perturbation method to functional integral equations. Chaos Solitons Fractals 31: 1243-1247.

Abdulaziz, O., Hashim, I. & Momani, S. 2008. Application of homotopy-perturbation method to fractional IVPs. J. Comput. Appl. Math. 216: 574-584.

Abdulaziz, O., Hashim, I. & Ismail, E.S. 2009. Approximate analytical solution to fractional modified KdV equations. Mathl. Comput. Model. 49: 136-145.

Chen, S.H., Hu, J., Chen, L. & Wang, C.P. 2005. Existence results for η-point boundary value problem of second order ordinary differential equations. J. Comput. Appl. Math. 180: 425-432.

Cheng, X.Y. & Zhong, C.K. 2005. Existence of positive solutions for a second order ordinary differential system. J. Math. Anal. Appl. 312: 14-23.

Chowdhury, M.S.H. & Hashim, I. 2007a. Solutions of a class of singular second-order IVPs by homotopy-perturbation method. Phys. Lett. A 365: 439-447.

Chowdhury, M.S.H. & Hashim, I. 2007b. Solutions of time-dependent Emden-Fowler type equations by homotopy-perturbation method, Phys. Lett. A 368: 305-313.

Chowdhury, M.S.H. & Hashim, I., Abdulaziz, O. 2007. Application of homotopy-perturbation method to nonlinear population dynamics models. Phys. Lett. A 368: 251-258.

Dehghan, M. & Saadatmandi, A. 2007. The numerical solution of a nonlinear system of second-order boundary value problems using the sinc-collocation method, Mathl. Comput. Model. 46: 1434-1441.

Geng, F.Z. & Cui, M.G. 2007. Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327: 1167-1181.

He, J.H. 1999. Homotopy perturbation technique, Computer Meth. Appl. Mech. Eng. 178: 257-262.

He, J.H. 2000. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Intern. J. Nonlin. Mech. 35: 37-43.

He, J.H. 2006. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 350: 87-88.

Lomtatidze, A. & Malaguti, L. 2003. On a two-point boundary value problem for the second order ordinary differential equations with singularities, Nonlinear Anal. 52: 1553-1567.

Lu, J.F. 2007. Variational iteration method for solving a nonlinear system of second-order boundary value problems. Comput. Math. Appl. 54: 1133-1138.

Mawhin, J. & Tisdell, C. 2003. A note on the uniqueness of solutions to nonlinear, discrete, vector boundary value problems. Nonlinear Anal. Appl. 1: 789-798.

Odibat, Z.M. 2007. A new modifcation of the homotopy perturbation method for linear and nonlinear operators. Appl. Math. Comput. 189: 746-753.

Saadatmandi, A., Dehghan, M. & Eftekhari, A. 2009. Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems. Nonlin. Analy.: Real World Appl. 10: 1912-1922.

Thompson, H.B. & Tisdell, C. 2000. Systems of difference equations associated with boundary value problems for second order systems of ordinary differential equations. J. Math. Anal. Appl. 248: 333-347.

Thompson, H.B. & Tisdell, C. 2002. Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations. Appl. Math. Lett. 15(6): 761-766.

Yusufoglu, E. 2007. Homotopy perturbation method for solving a nonlinear system of second order boundary value problem, Int. J. Nonlin. Sci. Numer. Simul. 8: 353-358.

 

 

*Pengarang untuk surat-menyurat; email: ishak_h@ukm.my

 

 

 

 

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