Sains Malaysiana 40(6)(2011): 659–664

Predictor-Corrector Block Iteration Method for Solving

Ordinary Differential Equations

(Kaedah Lelaran Blok Peramal-Pembetul Bagi Menyelesaikan Persamaan Terbitan Biasa)

Zanariah Abdul Majid* & Mohamed Suleiman

Mathematics Department, Faculty of Science, Universiti Putra Malaysia 43400 Serdang, Selangor D.E., Malaysia

Received: 5 January 2010 / Accepted:  29 November 2010

ABSTRACT

Predictor-corrector two point block methods are developed for solving first order ordinary differential equations (ODEs) using variable step size. The method will estimate the solutions of initial value problems (IVPs) at two points simultaneously. The existence multistep method involves the computations of the divided differences and integration coefficients when using the variable step size or variable step size and order. The block method developed will be presented as in the form of Adams Bashforth - Moulton type and the coefficients will be stored in the code. The efficiency of the predictor-corrector block method is compared to the standard variable step and order non block multistep method in terms of total number of steps, maximum error, total function calls and execution times.

Keywords: Block method; ordinary differential equations; predictor corrector block

ABSTRAK

Kaedah dua titik blok peramal-pembetul telah dibangunkan bagi penyelesaian persamaan terbitan biasa peringkat pertama menerusi panjang langkah berubah. Kaedah ini akan memberi nilai penghampiran bagi masalah nilai awal pada dua titik secara serentak. Kaedah multilangkah yang sedia ada melibatkan pengiraan beza pembahagi dan pekali kamiran apabila menggunakan saiz langkah berubah atau saiz langkah berubah dan berperingkat. Kaedah blok yang dibangunkan adalah dalam bentuk Adams Bashforth – Moulton dan pekali akan disimpan di dalam kod. Keberkesanan kaedah blok peramal-pembetul akan di bandingkan dengan kaedah multilangkah bukan blok bagi panjang langkah dan peringkat berubah dari segi jumlah langkah, ralat maksimum, jumlah kiraan fungsi dan masa pelaksanaan.

Kata kunci: Blok peramal pembetul; kaedah blok; persamaan terbitan biasa

REFERENCES

Chu, M.T. & Hamilton, H. 1987. Parallel Solution of ODE’s by Multiblock Methods, SIAM J. Sci Stat. Comput. 8:342-354.

Lambert, J.D. 1993. Numerical Methods For Ordinary Differential Systems. The Initial Value Problem. New York: John Wiley & Sons, Inc.

Majid, Z.A. & Suleiman, M. 2006. 1-Point Implicit Code of Adams Moulton Type For Solving First Order Ordinary Differential Equations, Chiang Mei Journal of Sciences 33(2): 153-159.

Majid, Z.A, Suleiman, M. & Omar, Z. 2006. 3-Point Implicit Block Method for Solving Ordinary Differential Equations. Bulletin of the Malaysian Mathematical Sciences Society 29(1/2): 1-9.

Majid, Z.A., Suleiman, M., Ismail, F. & Othman, M. 2003. Two Point Implicit Block Method In Half Gauss Seidel For Solving Ordinary Differential Equations. Jurnal Matematika19(2): 91-100.

Omar, Z., 1999. Developing Parallel Block Methods For Solving Higher Order ODEs Directly, Ph.D. Thesis, University Putra Malaysia, Malaysia. (Unpublished)

Rosser, J.B. 1976. Runge Kutta For All Season, SIAM Rev. 9: 417-452.

Shampine, L.F. & Gordon, M.K. 1975. Computer Solution of Ordinary Differential Equations: The Initial Value Problem.San Francisco: W. H. Freeman and Company

Suleiman, M.B. 1979. Generalised Multistep Adams and Backward Differentiation Methods for the Solution of Stiff and Non-Stiff Ordinary Differential Equations. Ph.D. Thesis. University of Manchester, UK. (Unpublished)

Worland, P.B. 1976. Parallel Methods for the Numerical Solutions of Ordinary Differential Equations. IEEE Transactions on Computers 25: 1045-1048.

*Corresponding author; email: zanariah@math.upm.edu.my