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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
Received: 15 January 2015/Accepted: 27 August 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
REFERENCES
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. 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.
*Corresponding author; email: alayedo@yahoo.com
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