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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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