Sains Malaysiana 41(5)(2012): 623-632

 

Solving Directly General Third Order Ordinary Differential Equations Using

Two-Point Four Step Block Method

(Penyelesaian Terus Persamaan Pembezaan Biasa Am Peringkat Tiga Menggunakan

Kaedah Blok Dua-Titik Empat Langkah)

 

Zanariah Abdul Majid* & Nurul Asyikin Azmi

Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor D.E.

Malaysia

 

Mohamed Suleiman & Zarina Bibi Ibrahaim

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

 

Received: 27 January 2010 / Accepted: 18 November 2011

 

ABSTRACT

Two-point four step direct implicit block method is presented by applying the simple form of Adams- Moulton method for solving directly the general third order ordinary differential equations (ODEs) using variable step size. This method is implemented to get the solutions of initial value problems (IVPs) at two points simultaneously in a block using four backward steps. The numerical results showed that the performance of the developed method is better in terms of maximum error at all tested tolerances and lesser total number of steps as the tolerances getting smaller compared to the existence direct method.

 

Keywords: Block method; higher order ordinary differential equations; two point

 

ABSTRAK

Kaedah blok tersirat secara terus bagi dua-titik empat langkah yang berasaskan aplikasi kaedah Adams-Moulton yang ringkas untuk menyelesaikan secara terus sistem persamaan pembezaan biasa (PPB) am peringkat ketiga menggunakan saiz langkah yang berubah. Kaedah ini dilaksanakan bagi mendapatkan penyelesaian masalah nilai awal (MNA) pada dua titik secara serentak di dalam blok dengan menggunakan empat langkah sebelumnya. Hasil berangka menunjukkan bahawa kaedah blok yang dibangunkan adalah lebih baik daripada segi ralat maksimum pada semua toleran yang di uji dan kurang jumlah bilangan langkah apabila toleran semakin kecil jika dibandingkan dengan kaedah secara terus sedia ada.

 

Kata kunci: Kaedah blok; dua titik; persamaan pembezaan biasa peringkat tinggi

REFERENCES

Awoyemi, D.O. 2003. A P-Stable linear multistep method for solving general third order ordinary differential equations. Int. J. Comput. Mathematics 80(8): 985–991.

Awoyemi, D.O & Idowu, O.M. 2005. A class of hybrid collocation methods for third-order ordinary differential equations. Int. J. Comput. Mathematics 82(10):1287–1293.

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

Majid, Z.A., Azmi, N.A. & Suleiman, M.B. 2009. Solving second order ordinary differential equations using two point four step direct implicit block method. European Journal of Scientific Research 31(1): 29 -36.

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

Majid, Z.A., Suleiman, M.B & Azmi, N.A. 2010. Variable step size block method for solving directly third order ordinary differential equations. Far East Journal of Mathematical Sciences 41(1): 63 – 73.

Olabode, B.T. & Yusuph, Y 2009 A new block method for special third order ordinary differential equations. Journal of Mathematics and Statistics 5(3): 167-170.

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

Suleiman, M.B. 1989. Solving higher order odes directly by the direct integration method, Applied Mathematics And Computation 33: 197-219.

Yap, L.K, Ismail, F., Suleiman, M.B & Amin, S.M. 2008. Block methods based on newton interpolations for solving special second order ordinary differential equations directly. Journal of Mathematics and Statistics 4(3): 174 -180.

 

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

 

 

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