A New Implicit Block Method for Solving Second Order Ordinary Differential Equations Directly

Year 2015, Volume: 28 Issue: 4, 689 - 694, 01.09.2015

Zurni Omar John Kuboye

Abstract

This article considers the derivation of an implicit block method for the solution of initial value problems of ordinary differential equations directly. The method of interpolation and collocation is adopted in developing the method where approximated power series is used as an interpolation polynomial and its second derivative is collocated at the selected grid points where k=5. The method developed is zero stable, consistent and convergent. The generated numerical results show that the new method is better when compared with the existing methods of the same step-length in terms of error.

Keywords

Power series, Interpolation, Collocation, Block method, Second order initial value problems.

References

• Badmus, A. M and Yahaya, Y. A. An accurate uniform order six block method for direct solution of general second order ordinary differential equations. Pacific Journal of Science and Technology, 10(2), 248-254 (2009).
• Gear, C. W. The numerical integration of ordinary differential equations. Math. Comp., 21: 146 – 156 (1966).
• Gear, C. W. The stability of numerical methods for second order ordinary differential equations. SIAM J. Numer. Analy, 1987, 15(1), 118 – 197 (1987).
• Henrici, P. Discrete variable methods in ordinary differential equations, (1962).
• Lambert, J. D. Computational Methods in Ordinary Differential Equations. Introductory Mathematics for Scientists and Engineers. Wiley, (1973).
• Mohammed, U. A class of implicit five-step block method for general second order ordinary differential equations. Journal of Nigerian Mathematical Society (JNMS). 30:25 – 39 (2011).
• Mohammed, U. and Adeniyi R.B. Derivation of block hybrid backward difference formula (HBDF) through multistep collocation for solving second order differential equations. The Pacific Journal of Science and Technology, 15(3), 89 – 95 (2014).
• Omar, Z. Developing Parallel 3-Point Implicit Block Method for solving second Order Ordinary Differential Equations Directly. IJMS, 11(1), 91 – 103 (2004).