Sains Malaysiana 45(6)(2016): 1007–1012

 

Quintic Spline Method for Solving Linear and Nonlinear Boundary Value Problems

(Kaedah Splin Kuintik untuk Menyelesaikan Masalah Nilai Sempadan Linear dan Tak Linear)

 

OSAMA ALA'YED*, TEH YUAN YING & AZIZAN SAABAN

 

School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah Darul Aman, Malaysia

 

Diserahkan: 15 Januari 2015/Diterima: 27 Ogos 2015

 

ABSTRACT

In this article, a fourth order quintic spline method has been developed to obtain numerical solutions for second order boundary value problems with Dirichlet boundary conditions. The developments of the quintic spline method and convergence analysis were presented. Three test problems have been considered for comparison purposes. The numerical results showed that the quintic spline method is more accurate compared to existing cubic spline method when solving nonlinear second order boundary value problems but vice versa when solving linear second order boundary value problems.

 

Keywords: Boundary value problem; quintic spline method; shooting method; spline interpolation

 

ABSTRAK

Dalam artikel ini, satu kaedah splin kuintik peringkat keempat telah dibangunkan untuk memperoleh penyelesaian berangka bagi masalah nilai sempadan peringkat kedua dengan syarat sempadan Dirichlet. Pembangunan kaedah splin kuintik dan analisis penumpuan telah dibentangkan. Tiga masalah ujian telah diambil kira untuk tujuan perbandingan. Keputusan berangka menunjukkan bahawa kaedah splin kuintik adalah lebih tepat berbanding dengan kaedah splin kubik sedia ada apabila menyelesaikan masalah nilai sempadan tak linear peringkat kedua, tetapi sebaliknya apabila menyelesaikan masalah nilai sempadan linear peringkat kedua.

 

Kata-kata kunci: Interpolasi splin; kaedah splin kuintik; kaedah tembakan; masalah nilai sempadan

RUJUKAN

Ala'yed, O., Teh, Y.Y. & Saaban, A. 2013. Algorithm for the solution of bratu-type equation based on quintic spline method. Paper presented at the 1st Innovation and Analytics Conference and Exhibition (IACE) 2013. 29 December. UUM Sintok, Malaysia.

Al-Said, E.A., Noor, M.A., Almualim, A.H., Kokkinis, B. & Coletsos, J. 2011. Quartic spline method for solving second-order boundary value problems. International Journal of Physical Sciences 6(17): 4208-4212.

Burden, R.L. & Faires, J.D. 2001. Numerical Analysis. 7th ed. Boston: Prindle Weber and Schmidt.

Caglar, H., Caglar, N. & Elfaituri, K. 2006. B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems. Applied Mathematics and Computation 175(1): 72-79.

Chawla, M.M. & Subramanian, R. 1988. A new fourth-order cubic spline method for second-order nonlinear two-point boundary-value problems. Journal of Computational and Applied Mathematics 23(1): 1-10.

Fauzi, N.I.M. & Sulaiman, J. 2012. Half-sweep modified successive over relaxtion method for solving second order two-point boundary value problems using cubic spline. International Journal of Contemporary Mathematical Sciences 7(32): 1579-1589.

Hamid, N.N.A., Majid, A.A. & Ismail, A.I.M. 2012. Quartic B-spline interpolation method for linear two-point boundary value problem. World Applied Sciences Journal 17: 39-43.

Hamid, N.N.A., Majid, A.A. & Ismail, A.I.M. 2011. Extended cubic B-spline method for linear two-point boundary value problems. Sains Malaysiana 40(11): 1285-1290.

Keller, H.B. 1968. Numerical Methods for Two-Point Boundary- Value Problems. Massachusetts: Blaisdell Publishing Company.

Khuri, S.A. 2004. A new approach to Bratu’s problem. Appl. Math.Comput. 147: 131-136.

Ramadan, M.A., Lashien, I.F. & Zahra, W.K. 2007. Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems. Applied Mathematics and Computation 184(2): 476-484.

Rashidinia, J., Mohammadi, R. & Jalilian, R. 2008. Cubic spline method for two-point boundary value problems. International Journal of Engineering Science 19: 39-43.

 

 

*Pengarang untuk surat-menyurat; email: alayedo@yahoo.com

 

 

sebelumnya