Sains Malaysiana 42(6)(2013): 845–853

One-Step Exponential-rational Methods for the Numerical Solution of First Order Initial Value Problems

(Kaedah Eksponen-Nisbah Satu Langkah Bagi Penyelesaian Masalah Nilai Awal

Peringkat Pertama Secara Berangka)

Teh Yuan Ying*

School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia

06010 Sintok, Kedah Darul Aman, Malaysia

Nazeeruddin Yaacob

Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia

81310 Skudai, Johor Darul Ta’zim, Malaysia

Diserahkan: 27 Jun 2012/Diterima: 2 Oktober 2012

ABSTRACT

In this study, a new class of exponential-rational methods (ERMs) for the numerical solution of first order initial value problems has been developed. Developments of third order and fourth order ERMs, as well as their corresponding local truncation error have been presented. Each ERM was found to be consistent with the differential equation and L-stable. Numerical experiments showed that the third order and fourth order ERMs generates more accurate numerical results compared with the existing rational methods in solving first order initial value problems.

Keywords: Exponential function; exponential-rational method; problem whose solution possesses singularity; rational function; rational method

ABSTRAK

Dalam kajian ini, satu kelas kaedah eksponen-nisbah untuk penyelesaian masalah nilai awal peringkat pertama secara berangka telah dibangunkan. Pembangunan kaedah eksponen-nisbah yang berperingkat ketiga dan keempat telah dibentangkan bersama dengan ralat pangkasan setempat yang sepadan. Setiap kaedah eksponen-nisbah didapati adalah tekal dengan persamaan pembezaan dan mempunyai kestabilan jenis L. Eksperimen secara berangka telah menunjukkan bahawa kaedah eksponen-nisbah berperingkat ketiga dan keempat menjana keputusan berangka yang lebih tepat berbanding dengan kaedah-kaedah nisbah yang sedia ada dalam menyelesaikan masalah nilai awal peringkat pertama.

Kata kunci: Fungsi eksponen; fungsi nisbah; kaedah eksponen-nisbah; kaedah nisbah; masalah dengan penyelesaian yang mempunyai singular

RUJUKAN

Butcher, J.C. 2003. Numerical Methods for Ordinary Differential Equations. West Sussex: John Wiley & Sons, Ltd.

Fatunla, S.O. 1976. A new algorithm for numerical solution of ordinary differential equations. Computers and Mathematics with Applications 2(3/4): 247-253.

Fatunla, S.O. 1982. Non linear multistep methods for initial value problems. Computers and Mathematics with Applications 8(3): 231-239.

Fatunla, S.O. 1986. Numerical Treatment of Singular Initial Value Problems. Computers and Mathematics with Applications 12B(5/6): 1109 – 1105.

Ikhile, M.N.O. 2001. Coefficients for studying one-step rational schemes for IVPs in ODEs: . Computers and Mathematics with Applications 41: 769-781.

Ikhile, M.N.O. 2002. Coefficients for studying one-step rational schemes for IVPs in ODEs: II. Computers and Mathematics with Applications 44: 545-557.

Ikhile, M.N.O. 2004. Coefficients for studying one-step rational schemes for IVPs in ODEs: III. Computers and Mathematics with Applications 47: 1463-1475.

Lambert, J.D. 1973. Computational Methods in Ordinary Differential Equations. London: John Wiley & Sons. Ltd.

Lambert, J.D. 1974. Two unconventional classes of methods for stiff systems. In Stiff Differential Equations, edited by Willoughby, R. A. New York: Plenum Press.

Lambert, J.D. 1991. Numerical Methods for Ordinary Differential Systems. Chichester: John Wiley & Sons. Ltd.

Lambert, J.D. & Shaw, B. 1965. On the numerical solution of y´ = f (x,y) by a class of formulae based on rational y´ approximation. Mathematics of Computation 19: 456-462.

Luke, Y.L., Fair, W. & Wimp, J. 1975. Predictor-corrector formulas based on rational interpolants. Computers and Mathematics with Applications 1: 3-12.

Okosun, K.O. & Ademiluyi, R.A. 2007a. A two step second order inverse polynomial methods for integration of differential equations with singularities. Research Journal of Applied Sciences 2(1): 13-16.

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

Ramos, H. 2007. A non-standard explicit integration scheme for initial-value problems. Applied Mathematics and Computation 189: 710-718.

van Niekerk, F.D. 1987. Non-linear one-step methods for initial value problems. Computers and Mathematics with Applications 13(4): 367-371.

van Niekerk, F.D. 1988. Rational one-step methods for initial value problems. Computers and Mathematics with Applications 16(12): 1035-1039.

Wambecq, A. 1976. Nonlinear methods in solving ordinary differential equations. Journal of Computational and Applied Mathematics 2(1): 27-33.

Wu, X.Y. 1998. A sixth-order a-stable explicit one-step method for stiff systems. Computers and Mathematics with Applications 35(9): 59-64.

Yaakub, A.R. & Evans, D.J. 2003. New L-stable modified trapezoidal methods for the initial value problems. International Journal of Computer Mathematics 80(1): 95-104.

*Pengarang untuk surat-menyurat; email: yuanying@uum.edu.my