Sains Malaysiana 41(4)(2012): 489-492

Numerical Solution of First Order Stiff Ordinary Differential Equations using

Fifth Order Block Backward Differentiation Formulas

(Penyelesaian Berangka bagi Persamaan Pembezaan Biasa Kaku Peringkat

Satu Menggunakan Blok Formula Beza ke Belakang Peringkat Lima)

Nor Ain Azeany Mohd Nasir¹*, Zarina Bibi Ibrahim¹*, Khairil Iskandar Othman² & Mohamed Suleiman¹

¹Department of Mathematics, Faculty of Science, Universiti Putra Malaysia

43400 UPM Serdang, Selangor D.E. Malaysia

²Department of Mathematics, Faculty of Computer and Mathematical Sciences

Universiti Teknologi MARA, 40450 Shah Alam, Selangor D.E. Malaysia

Received: 4 August 2010 / Accepted: 7 October 2011

ABSTRACT

This paper describes the development of a two-point implicit code in the form of fifth order Block Backward Differentiation Formulas (BBDF(5)) for solving first order stiff Ordinary Differential Equations (ODEs). This method computes the approximate solutions at two points simultaneously within an equidistant block were computed. Numerical results are presented to compare the efficiency of the developed BBDF(5) to the classical one-point Backward Differentiation Formulas (BDF). The results indicated that the BBDF(5) outperformed the BDF in terms of total number of steps, accuracy and computational time.

Keywords: Block method; ordinary differential equation

ABSTRAK

Kertas ini membincangkan pembentukan kod tersirat dua titik dalam bentuk Blok Formula Beza Ke Belakang peringkat lima (BBDF(5)) bagi menyelesaikan Persamaan Pembezaan Biasa (PPB) kaku peringkat pertama. Kaedah ini mengira penyelesaian penghampiran dua titik serentak dalam jarak blok yang sama. Keputusan berangka diberi untuk membandingkan kaedah BBDF(5) dengan kaedah Formula Beza Ke Belakang klasik (BDF). Keputusan kajian menunjukkan bahawa BBDF(5) mengatasi BDF dalam hal jumlah langkah, kesalahan maksima dan masa pengkomputeraan.

Kata kunci: Kaedah blok; persamaan pembezaan biasa

REFERENCES

Brannan, R.J. & William, E.B., 2007. Differential Equations: An Introduction to Modern Methods and Applications. New York: John Wiley & Sons.

Chu, M.T. & Hamilton, H. 1987. Parallel Solution of ODEs by Multi-block methods. Siam Journal on Scientific and Statistical Computing 8(1): 342-353.

Fatunla S.O., 1990. Block Methods for Second Order. ODEs, International Journal of Computer Mathematics 40: 55-63.

Ibrahim, Z.B., Othman, K.I. & Suleiman, M.B. 2007. Implicit r-point block backward differentiation formula for solving first-order stiff ODE, Applied Mathematics and Computation 186: 558- 565.

Lambert, J.D. 1991. Numerical Methods for Ordinary Differential Equations: The Initial Value Problems. New York: John Wiley & Sons.

Majid, Z.A. 2004. Parallel Block Methods for Solving Ordinary Differential Equations. PhD thesis, Universiti Putra Malaysia. (Unpublisher)

Rosser, J.B. 1967. Runge-Kutta for all seasons. Siam Review 9(3):417-452.

Voss, D. & Abbas, S. 1997. Block Predictor-Corrector Scheme for the Parallel Solution of ODEs. Computers & Mathematics with Applications 33(6): 63-72.

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