IMPLICIT FOUR-STEP APPROACH WITH APPLICATION TO NON-LINEAR THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS
Abstract
A unique and efficient implicit four-step approach with application to nonlinear third order ordinary differential equations is considered in this article. In the derivation of this method Collocation and Interpolation techniques were engaged and power series approximate solution was used as the interpolating polynomial. The third derivative of the power series was collocated at the entire grid points, while the interpolation was done at the first three points. Appropriate study of the basic properties of the method was done. The results generated when the new block method was applied on nonlinear third order ordinary differential equations are better in terms of accuracy than the existing methods.
References
J.O. Kuboye and Z. Omar (2015). Derivation of a six-step block method for direct solution of second order ordinary differential equations. Math. Comput. Appl, 20(4): 151–159.
R. Abdelrahim and Z. Omar (2016). Solving third order ordinary differential equations using hybrid block method of order five. International Journal of Applied Engineering Research, 10(24), 44307– 44310.
J. D. Lambert (1973). Computational methods in ordinary differential equations.
Sagir A. M. (2012). An accurate computation of block hybrid method for solving stiff ordinary differential equations. Journal of Mathematics, 4:18–21.
P. Henrici (1962). Discrete variable methods in ordinary differential equations.
S. J. Kayode, O. S. Ige, F. O. Obarhua and E. O. Omole (2018): An Order Six Stormer-cowell-type Method for Solving Directly Higher Order Ordinary Differential Equations. Asian Research Journal of Mathematics 11(3): 1-12.
Adoghe L. O, Ogunware B. G and Omole E. O. (2016): A family of symmetric implicit higher order methods for the solution of third order initial value problems in ordinary differential equations: Journal of Theoretical Mathematics & Applications, 6(3): 67-84.
A. O. Adeniran and A. E. Omotoye (2016), One Step Hybrid Block Method for the Numerical Solution of General Third Order Ordinary Differential Equations, International Journal of Mathematical Sciences, 2(5), pp. 1 – 12
Abdelrahim R., Omar Z., Ala’yed0.,and Batiha B., (2019), Hybrid third derivative block method for the solution of general second order initial value problems with generalized one step point, European Journal of Pure and Applied Mathematics, Vol. 12, No. 3, 1199-1214.
L. A. Ukpebor (2019): A 4-point block method for solving second order initial value problems in ordinary differential equations. American Journal of Computational and Applied Mathematics 2019, 9(3): 51-56. DOI: 10.5923/j.ajcam.20190903.01
Ogunware B. G, Adoghe L. O, Awoyemi D. O, Olanegan O. O., and Omole E. O(2018): Numerical Treatment of General Third Order Ordinary Differential Equations Using Taylor Series as Predictor, Physical Science International Journal, 17(3):1-8. DOI:10.9734/PSIJ/2018/22219.
N. M. Yao, A. Akinfenwa and S. N Jator (2011), A linear Multistep Hybrid Method with Continuous Coefficient for solving stiff Differential equation, Int. J. Comp. Mathematics, 5(2), 47-53.
D. Wend (1969). Existence and uniqueness of solutions of ordinary differential equations. Proceedings of the American Mathematical Society, page 2733.
Ogunware B. G, Omole E. O., and Olanegan O. O (2015): Hybrid and Non-Hybrid Implicit Schemes for Solving Third Order ODEs Using Block Method as Predictors. Journal of Mathematical Theory and Modelling (iiste) 5(3), 10 -25.
Omole E. O. and Ukpebor L. A. (2020), A Step by Step Guide on Derivation and Analysis of a new Numerical method for Solving Fourth-order Ordinary Differential Equations, Journal of Mathematics letter, 6(2): 13- 31, Doi: 10.11648/j.ml.20200602.12
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