Sains Malaysiana 43(7)(2014): 1105–1108

Rational Finite Difference Approximation of High Order Accuracy for

Nonlinear Two Point Boundary Value Problems

(Penghampiran Beza Terhingga Rasional Ketepatan Peringkat Tinggi  untuk Masalah Nilai

Dua Titik Sempadan Tak Linear)

P.K. PANDEY*

Department of Mathematics, College of Applied Sciences, PB No. 1905, PC 211, Salalah

Sultanate of Oman

Department of Mathematics, Dyal Singh College (Univ. of Delhi), Lodhi Road,

New Delhi - 110003, India

Diserahkan: 16 Mei 2013/Diterima: 16 September 2013

ABSTRACT

In this paper, we present a new method for solving nonlinear general two point boundary value problems. A method based on finite differences and rational function approximation and we call this method as rational approximation method. A rational approximation method is applied to construct the numerical solution for two point boundary value problems. The novel method is tested on three model problems. Thus the numerical results obtained for these model problems show the performance and efficiency of the developed method.

Keywords: Boundary value problems; fourth-order method; rational approximation

ABSTRAK

Dalam kertas ini, kami memberikan kaedah baru bagi menyelesaikan masalah nilai dua titik sempadan tak linear umum. Kaedah berdasarkan beza terhingga dan penghampiran fungsi rasional ini dikenali sebagai kaedah penghampiran rasional. Kaedah penghampiran rasional yang digunakan untuk membina penyelesaian berangka bagi masalah dua titik sempadan. Kaedah baru ini diuji pada tiga model masalah. Oleh itu keputusan berangka yang diperoleh bagi masalah model ini menunjukkan prestasi dan keberkesanan kaedah yang dibangunkan.

Kata kunci: Kaedah peringkat empat; masalah nilai sempadan; penghampiran rasional

RUJUKAN

Baxley, J.V. 1981. Nonlinear Two Point Boundary Value Problems. In Ordinary and Partial Differential Equations, edited by Everrit, W.N. & Sleeman, B.D. New York: Springer- Verlag. pp. 46-54.

Chawla, M.M. 1978. High-accuracy tridiagonal finite difference approximation for nonlinear two point boundary value problems. J. Inst. Maths. Applics22: 203- 209.

Collatz, L. 1966. The Numerical Treatment of Differential Equations. New York: Springer- Verlag.

Jain, M.K. 1984. Numerical Solution of Differential Equations (2/e). New Delhi: Wiley Eastern Limited.

Jain, M.K., Iyenger, S.R.K. & Jain, R.K. 1987. Numerical Methods for Scientific and Engineering Computation (2/e). New Delhi: Wiley Eastern Ltd.

Keller, H.B. 1968. Numerical Methods for Two Point Boundary Value Problems. Waltham, Mass.: Blasdell Publ. Co.

Lambert, J.D. 1991. Numerical Methods for Ordinary Differential Systems. England: John Wiley.

Mickens, R.E. 1994. Nonstandard Finite Difference Models of Differential Equations. Singapore: World Scientific.

Okosun, K.O. & Ademiluyi, R.A. 2007. A three step rational methods for integration of differential equations with singularities. Research Journal of Applied Sciences 2(1): 84-88.

Pandey, P.K. 2013. A non-classical finite difference method for solving two point boundary value problems. Pacific Journal of Science and Technology 14(2): 147-152.

Ramos, H. 2007. A nonstandard explicit integration scheme for initial value problems. Applied Mathematics and Computation 189(1): 710-718.

Stoer, J. & Bulirsch, R. 1991. Introduction to Numerical Analysis (2/e). Berlin Heidelberg: Springer-Verlag.

Tirmizi, I.A. & Twizel, E.H. 2002. High order finite difference methods for nonlinear second order two point boundary value problems. Appl. Math. Lett. 15: 897-902.

Ying, T.Y. & Yaacob, N. 2013. One-step exponential-rational methods for the numerical solution of first order initial value problems. Sains Malaysiana42(6): 845-853.

*Pengarang untuk surat-menyurat; email: pramod_10p@hotmail.com