A TREATMENT OF A MULTISTEP COLLOCATION METHOD FOR THE DIRECT SOLUTION OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Authors

Keywords:

Modified Backward Differentiation Formula, Block method, Second order Ordinary Differential Equations, Initial Value Problems, Collocation method

Abstract

A research has been conducted at the Department of Mathematics, Faculty of Science, Federal University, Gashua, Yobe State to study the treatment of multistep collocation method for the direct solution of second order ordinary differential equations using a class of modified Backward Differentiation Formular (BDF)-type with one super-future point.  The research has proposed the construction of a new method of solving second-order initial valued problem of ordinary differential equation. A step-number, , a number of discrete members are obtained and used in block through multi-step collocation approach. The stability properties of the newly constructed methods are investigated using written computer codes and its convergence are established. The numerical efficiency of the method has been tested on some treated second-order initial valued problems, in order to ascertain its suitability. The solutions of the problems are compared with the corresponding exact solutions and the associated absolute errors are presented. Tables and graph have been adopted in the presentation of results, and they have shown the efficiency of the new method for solving solution of second order ordinary differential equations.

Dimensions

Awoyemi D.O., Adebile E. A., Adesanya A.O., and Anake T.A. (2011). Modified block method for the direct solution of second order ordinary differential equations. International Journal of Applied Mathematics and Computation, 3(3), 181-188. https://www.researchgate.net/publication/268205669.

Badmus A.M. and Yahaya Y.A. (2019). An Accurate Uniform Order 6 Block Method for Direct Solution of General Second Order Ordinary Differential Equations. The Pacific Journal of Science and Technology, 10(2), 248-254. https://www.researchgate.net/publication/237637672

Butcher J.C. (2000). Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, 125, 1 29. https://doi.org/10.1016/B978-0-444-50617-7.50018-5.

Cash J.R. (2000). Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs. Journal of Computational and Applied Mathematics, 125, 117-130. https://doi.org/10.1016/S0377(00)00463-5

Chollom, J. P.; Ndam, J. N. & Kumleng, G. M. (2007). On some properties of the block linear multi-step methods. Science World Journal, 2(3), 11-17. https://doi.org/10.4314/swj.v2i3.51747

Kaze Atsi and Kumleng G. M. (2020). A family of modified backward differentiation formula (BDF) type block methods for the solution of stiff ordinary differential equations. International Journal of Statistics and Applied Mathematics, 5(2), 09-16. https://doi.org/10.3390/mca21020012

Kaze Atsi and Kumleng G. M. (2021). Block Extended Trapezoidal Rule of the Second Kind (Etr2) For the Direct Soltion of Second Order Initial Value Problems of Ordinary Differential Equations. IOSR Journal of Mathematics (IOSR-JM), 17(2), 10-14. https://doi.org/10.9790/5728-1702041014

Khadijah M. Abualnaja. (2015). A Block Procedure with Linear Multi-Step Methods Using Legendre Polynomials for Solving ODEs. Scientific Research Publishing Inc, 6, 717-723. https://doi.org/10.4236/am.2015.64067

Kuboye J. O., Omar Z., Abolarin O. E. and Abdelrahim R. (2018). Generalized Hybrid Block Method for Solving Second Order Ordinary Differential Equations Directly. Research and Reports on Mathematics, 2(2), 1-7. https://www.academia.edu/107854998

Kumleng G.M., Chollom J.P. and Longwap S. (2013). A Modified Block Adam Moulton (MOBAM) Method for the Solution of Stiff Initial Value Problems of Ordinary Differential Equations. Research Journal of Mathematics and Statistics, 5(4), 32-42. http://dx.doi.org/10.19026/rjms.5.5803

Obarhua F.O. (2019). An Efficient Two-Step Symmetric Hybrid Block Method for Solving Second-Order Initial Value Problems of Ordinary Differential Equations. International Journal of Research and Scientific Innovation (IJRSI), 6(11), 200-208. https://www.researchgate.net/publication/337928798

Raft Abdelrahim and Zurni Omar. (2016). Direct Solution of Second-Order Ordinary Differential Equation Using a Single-Step Hybrid Block Method of Order Five. Mathematical and Computational Applications, 21, 12. https://doi.org/10.3390/mca21020012

Ukpebor L. A. (2019). A 4-Point Block Method for Solving Second Order Initial Value Problems in Ordinary Differential Equations. American Journal of Computational and Applied Mathematics, 9(3), 51-56. https://doi.org/10.5923/j.ajcam.20190903.01

Published

31-12-2024

How to Cite

A TREATMENT OF A MULTISTEP COLLOCATION METHOD FOR THE DIRECT SOLUTION OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS. (2024). FUDMA JOURNAL OF SCIENCES, 8(6), 234-238. https://doi.org/10.33003/fjs-2024-0806-2857

Issue

Vol. 8 No. 6 (2024): FUDMA Journal of Sciences - Vol. 8 No. 6 (Special Issue)

Section

Research Articles
Copyright & Licensing

FUDMA Journal of Sciences

How to Cite

A TREATMENT OF A MULTISTEP COLLOCATION METHOD FOR THE DIRECT SOLUTION OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS. (2024). FUDMA JOURNAL OF SCIENCES, 8(6), 234-238. https://doi.org/10.33003/fjs-2024-0806-2857

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