New Generalized Algorithm for Developing k-Step Higher Derivative Block Methods for Solving Higher Order Ordinary Differential Equations

Oluwaseun Adeyeye, Zurni Omar

Abstract


This article presents a new generalized algorithm for developing k-step (m+1)th derivative block methods for solving mth order ordinary differential equations. This new algorithm utilizes the concept from the conventional Taylor series approach of developing linear multistep methods. Certain examples are shown to show the simplicity involved in the usage of this new generalized algorithm.


Keywords


block methods; generalized algorithm; higher derivative; higher order; k-step; Taylor series

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References


Badmus, A.M., Yahaya, Y.A. & Pam, Y.C., Adams Type Hybrid Block Methods Associated with Chebyshev Polynomial for the Solution of Ordinary Differential Equations, British Journal of Mathematics & Computer Science, 6(6), pp. 464-474, 2015.

Ibrahim, Z.B., Othman, K.I. & Suleiman, M., Implicit r-point Block Backward Differentiation Formula for Solving First-order Stiff ODEs, Applied Mathematics and Computation, 186(1), pp. 558-565, 2007.

Jator, S.N. & Li, J., A Self-Starting Linear Multistep Method for A Direct Solution of the General Second-order Initial Value Problem, International Journal of Computer Mathematics, 86(5), pp. 827-836, 2009.

Kuboye, J.O. & Omar, Z., Numerical Solution of Third Order Ordinary Differential Equations Using A Seven-step Block Method, International Journal of Mathematical Analysis, 9(15), pp. 743-745, 2015.

Lambert, J.D., Computational Methods in Ordinary Differential Equations, Wiley, London, 1973.

Majid, Z.A., See, P.P. & Suleiman, M., Solving Directly Two Point Non Linear Boundary Value Problems using Direct Adams Moulton Method, Journal of Mathematics and Statistics, 7(2), pp. 124-128, 2011.

Okuonghae, R.I. & Ikhile, M.N.O., On the Construction of High Order A (α)-stable Hybrid Linear Multistep Methods for Stiff IVPs in ODEs, Numerical Analysis and Applications, 5(3), pp. 231-241, 2012.

Omar, Z. & Adeyeye, O., K-step Block Method Algorithm for Arbitrary Order Ordinary Differential Equations, International Journal of Mathematical Analysis, 10(11), pp. 545-552, 2016.

Omar, Z. & Alkasassbeh, M.F., Generalized One-Step Third Derivative Implicit Hybrid Block Method for the Direct Solution of Second Order Ordinary Differential Equation, Applied Mathematical Sciences, 10(9), pp. 417-430, 2016.

Sunday, J., Odekunle, M.R., Adesanya, A.O. & James, A.A., Extended Block Integrator for First-order Stiff and Oscillatory Differential Equations, American Journal of Computational and Applied Mathematics, 3, pp. 283-290, 2013.

Butcher, J.C., Numerical Methods for Ordinary Differential Equations, Wiley, West Sussex, 2008.

Fatunla, S.O., Numerical Methods for Initial Value Problems in Ordinary Differential Equation, Academic Press, New York, 1988.

Kayode, S.J. & Obarhua, F.O., Continuous y-function Hybrid Methods for Direct Solution of Differential Equations, International Journal of Differential Equations and Applications, 12(1), pp. 37-48, 2013.

Jator, S.N. & Li, J., Boundary Value Methods via A Multistep Method with Variable Coefficients for Second Order Initial and Boundary Value Problems, International Journal of Pure and Applied Mathematics, 50(3), pp. 403-420, 2009.

Sahi, R.K., Jator, S.N. & Khan, N.A., Continuous Fourth Derivative Method for Third Order Boundary Value Problems, International Journal of Pure and Applied Mathematics, 85(2), pp. 907-923, 2013.




DOI: http://dx.doi.org/10.5614%2Fj.math.fund.sci.2018.50.1.4

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