New Seven-Step Numerical Method for Direct Solution of Fourth Order Ordinary Differential Equations
DOI:
https://doi.org/10.5614/j.math.fund.sci.2016.48.2.1Keywords:
block method, collocation, direct solution, interpolation, ordinary differential equations, seven-stepAbstract
A new numerical method for solving fourth order ordinary differential equations directly is proposed in this paper. Interpolation and collocation were employed in developing this method using seven steps. The use of the approximated power series as an interpolation equation was adopted in deriving the method. The basic properties of the new method such as zero-stability, consistency, convergence and order are established. The numerical results indicate that the new method gives better accuracy than the existing methods when it is applied to fourth ordinary differential equations.
References
Awari, Y.S. & Abada, A.A., A Class of Seven Point Zero Stable Continuous Block Method for Solution of Second Order Ordinary Differential Equation, International Journal and Mathematics and Statistics Invention (IJMSI), 2(1), pp. 47-54, 2014.
Anake, T.A., Awoyemi, D.O. & Adesanya, A.O., One-Step Implicit Hybrid Block Method for the Direct Solution of General Second Order Ordinary Differential Equations, IEANG International Journal of Applied Mathematics, 42(4), IJAM_42_4_04, 2012.
Bun, R.A. & Vasil'ev, E.D., A Numerical Method for Solving Differential Equation of Any Orders, Comp. Math. Phys., 32(3), pp. 317-330, 1992.
Brugnano, L. & Trigiante, D., Solving Differential Problems by Multistep IVPs and BVP Methods, Gordon and Breach Science Publishers, 1998.
Lambert, J.D., Computational Methods in Ordinary Differential Equations, John Wiley & Sons Inc. 1973.
Senu, N., Suleiman, M., Ismail, F. & Othman, M., A Singly Diagonally Implicit Runge-kutta-Nystrom Method for Solving Oscillatory Problems, IEANG International Journal of Applied Mathematics, 41(2), IJAM _41_2_11, 2011.
Omar, Z., Parallel Block Methods for Solving Higher Order Ordinary Differential Equation Directly. Doctor Thesis in the Faculty of Science and Environmental Studies Universiti Putra Malaysia, Unpublished, 1999.
Omar, Z. & Kuboye, J.O., Computation of an Accurate Implicit Block Method for Solving Third Order Ordinary Differential Equations Directly, Global Journal of Pure and Applied Mathematics, 11(1), pp. 177-186, 2015.
Omar, Z. & Kuboye, J.O., Developing Block Method of Order Seven for Solving Third Order Ordinary Differential Equations Directly using Multistep Collocation Approach. International Journal of Applied Mathematics and Statisticsa,, 53(3), pp. 165-173, 2015.
Adesanya, A.O., Momoh, A.A., Alkali, M.A. & Tahir, A., A Five Steps Block Method for the Solution of fourth Order Ordinary Differential Equations, International Journal of Engineering Research and Applications (IJERA), 2(4), pp. 991-998, 2012.
Butcher, J.C, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, New York, 2003.
Kayode, S.J., An Efficient Zero Stable Method for Fourth Order Ordinary Differential Equations, Int. J. Math. Sci, pp. 1-10, 2008. dx.doi.org/10.1155/2008/364021
Mohammed, U., A Six Step Block Method for Solution of Fourth Order Ordinary Differential Equations, The Pacific Journal of Science and Technology, 11(1), pp. 259-265, 2010.
Olabode, B.T., A Six Step Scheme for the Solution of Fourth Order Ordinary Differential Equations, The Pacific Journal of Science and Technology, 10(1), pp. 143-148, 2009.
