Sains Malaysiana 46(5)(2017): 817–824

http://dx.doi.org/10.17576/jsm-2017-4605-16

 

Numerical Algorithm of Block Method for General Second Order ODEs using Variable

Step Size

(Algoritma Berangka Kaedah Blok bagi ODE Umum Peringkat Kedua Menggunakan Pemboleh Ubah Saiz Langkah)

 

NAZREEN WAELEH1* & ZANARIAH ABDUL MAJID2

 

1Faculty of Electronic & Computer Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Hang Tuah Jaya, 76100 Durian Tunggal, Melaka Bandaraya Bersejarah, Malaysia

 

2Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

 Diserahkan: 26 Januari 2016/Diterima: 1 November 2016

 ABSTRACT

This paper outlines an alternative algorithm for solving general second order ordinary differential equations (ODEs). Normally, the numerical method was designed for solving higher order ODEs by converting it into an n-dimensional first order equations with implementation of constant step length. Nevertheless, this involved a lot of computational complexity which led to consumption a lot of time. Consequently, a direct block multistep method with utilization of variable step size strategy is proposed. This method was developed for computing the solution at four points simultaneously and the derivation based on numerical integration as well as using interpolation approach. The convergence of the proposed method is justified under suitable conditions of stability and consistency. Five numerical examples are considered and some comparisons are made with the existing methods for demonstrating the validity and reliability of the proposed algorithm.

 

Keywords: Block method; general second order ordinary differential equations; variable step size

 

ABSTRAK

Kertas ini menggariskan satu algoritma alternatif untuk menyelesaikan persamaan pembezaan biasa (ODE) umum peringkat kedua. Kebiasaannya, kaedah berangka untuk menyelesaikan ODE peringkat tinggi direka dengan menukarkan ia ke dalam n-dimensi persamaan peringkat pertama dengan perlaksanaan panjang langkah kekal. Walau bagaimanapun, ini melibatkan kerumitan pengiraan yang membawa kepada penggunaan masa yang banyak. Oleh yang demikian, satu kaedah langsung pelbagai langkah blok dengan penggunaan strategi saiz langkah berubah dicadangkan. Kaedah ini dibangunkan bagi menghitung penyelesaian pada empat titik secara serentak dan terbitannya berdasarkan integrasi berangka serta menggunakan pendekatan interpolasi. Penumpuan kaedah yang dicadangkan dijustifikasi mengikut syarat kestabilan dan tekal yang sesuai. Terdapat lima contoh berangka dipertimbangkan dan beberapa perbandingan telah dibuat dengan kaedah yang sedia ada untuk menunjukkan kesahan dan kebolehpercayaan algoritma yang dicadangkan.

 

Kata kunci: Kaedah blok; persamaan pembezaan biasa umum peringkat kedua; saiz langkah berubah

 

RUJUKAN

Abdelrahim, R. & Omar, Z. 2016. Direct solution of second-order ordinary differential equation using a single-step hybrid block method of order five. Mathematical and Computational Applications 21(2): 12.

Anakira, N.R., Alomari, A.K. & Hashim, I. 2013. Numerical scheme for solving singular two-point boundary value problems. Journal of Applied Mathematics 2013: 1-8.

Cash, J.R. 1983. Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. Part I. The nonstiff case. Mathematics of Computation 40(161): 175-191.

Cash, J.R. & Girdlestone, S. 2006. Variable step Runge-Kutta- Nystršm methods for the numerical solution of reversible systems. Journal of Numerical Analysis, Industrial and Applied Mathematics 1(1): 59-80.

Fatunla, S.O. 1991. Block methods for second order ODEs. International Journal of Computer Mathematics 41(1-2): 55-63.

Hairer, E., Norsett, S.P. & Wanner, G. 1987. Solving Ordinary Differential Equations I : Nonstiff Problems. Berlin: Springer- Verlag.

Jator, S.N. 2012. A continuous two-step method of order 8 with a block extension for yÕÕ= f (x, y, yÕ). Applied Mathematics and Computation 219(3): 781-791.

Jikantoro, Y.D., Ismail, F. & Senu, N. 2015. Zero-dissipative trigonometrically fitted hybrid method for numerical solution of oscillatory problems. Sains Malaysiana 44(3): 473-482.

Kayode, S.J. 2008. An efficient zero-stable numerical method for fourth-order differential equations. International Journal of Mathematics and Mathematical Sciences 2008: 1-10.

Lambert, J.D. 1973. Computational Methods in Ordinary Differential Equations. New York: John Wiley & Sons. Inc.

Majid, Z.A. & Suleiman, M. 2006. Direct integration implicit variable steps method for solving higher order systems of ordinary differential equations directly. Sains Malaysiana 35(2): 63-68.

Majid, Z.A., Azmi, N.A., Suleiman, M. & Ibrahaim, Z.B. 2012. Solving directly general third order ordinary differential equations using two-point four step block method. Sains Malaysiana 41(5): 623-632.

Milne, W.E. 1953. Numerical Solution of Differential Equations. New York: John Wiley & Sons. Inc.

Omar, Z. & Suleiman, M. 2005. Solving higher order ordinary differential equations using parallel 2-point explicit block method. Matematika 21(1): 15-23.

Pandey, P.K. 2014. Rational finite difference approximation of high order accuracy for nonlinear two point boundary value problems. Sains Malaysiana 43(7): 1105-1108.

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

Waeleh, N., Majid, Z.A., Ismail, F. & Suleiman, M. 2011a. Numerical solution of higher order ordinary differential equations by direct block code. Journal of Mathematics and Statistics 8(1): 77-81.

Waeleh, N., Majid, Z.A. & Ismail, F. 2011b. A new algorithm for solving higher order IVPs of ODEs. Applied Mathematical Sciences 5(56): 2795-2805.

Yap, L.K. & Ismail, F. 2016. Ninth order block hybrid collocation method for second order ordinary differential equations. AIP Conference Proceedings 1705, 020006. doi: http://dx.doi. org/10.1063/1.4940254.

*Pengarang untuk surat-menyurat; email: nazreen@utem.edu.my

 

 

 

 

 

sebelumnya