### International Journal ofPhysical Sciences

• Abbreviation: Int. J. Phys. Sci.
• Language: English
• ISSN: 1992-1950
• DOI: 10.5897/IJPS
• Start Year: 2006
• Published Articles: 2572

## Quartic spline method for solving second-order boundary value problems

##### Eisa A. Al-Said1, Muhammad A. Noor1,2*, Anwar H. Almualim4, B. Kokkinis3 and John Coletsos3
1Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia. 2Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. 3Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece. 4Women Students Medical Studies (Sciences Sections), Mathematics Department, King Saud University, Riyadh 11495, Saudi Arabia.
Email: [email protected]

•  Accepted: 21 March 2011
•  Published: 02 September 2011

Abstract

In this paper, uniform quartic spline polynomial functions are used to develop some consistency relations which are then used to derive a numerical method for approximating the solution and its first, second, third and fourth derivatives of second order boundary value problems. The present method is capable of producing fourth order accurate approximations for the solution, as well as its first and second derivatives, and the second order accurate approximations for its third and fourth derivatives. The main consistency relation for the study’s quartic spline method is the same relation derived by Usmani et al. (1987) using quartic spline function. Their spline method has a stability problem which affects the accuracy of the computed approximations, whereas for the study’s new method, such stability problem does not appear. The present method produces more accurate approximations for the first and third derivatives of the solution than those produced by the other quartic spline method. A numerical example is included to demonstrate the efficiency and implementation of the proposed method.

Key words: Boundary value problems, quartic spline function, finite difference approximations, convergence analysis.

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