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Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations

In this paper, we developed a numerical Algorithm for one and two-step hybrid block methods for the numerical solution of first order initial value problems in ordinary differential equations using method of collocation and interpolation of Taylor’s series as approximate solution to give a system of non linear equations which was solved to give a hybrid block method. To further justify the usability and effectiveness of this new hybrid block method, the basic properties of the hybrid block scheme was investigated and found to be zero-stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing methods. The errors displayed after solving some selected initial value problems, revealed that, it is better to increase L (Derivative) rather than the step length k as shown in our numerical results. Also, It was difficult to satisfy the zero-stability for larger k. In addition, the new method converges faster with lesser time of computation, which address the setback associated with other methods in the literature. Finally, the new method has order of accuracy for one-step as order Ten while order Eighteen for two-step.

Block Method, Initial Value Problems, Hybrid, One-Step, Two-Step, Taylor Series

APA Style

Ononogbo Chibuike Benjamin, Airemen Ikhuoria Edward, Ezurike Ugochi Julia. (2022). Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Applied Engineering, 6(1), 13-23. https://doi.org/10.11648/j.ae.20220601.13

ACS Style

Ononogbo Chibuike Benjamin; Airemen Ikhuoria Edward; Ezurike Ugochi Julia. Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Appl. Eng. 2022, 6(1), 13-23. doi: 10.11648/j.ae.20220601.13

AMA Style

Ononogbo Chibuike Benjamin, Airemen Ikhuoria Edward, Ezurike Ugochi Julia. Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations. Appl Eng. 2022;6(1):13-23. doi: 10.11648/j.ae.20220601.13

Copyright © 2022 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Yakusak, N. S. & Adeniyi, R. B. (2015). A Four-Step Hybrid Block Method for First Order Initial Value Problems in Ordinary Differential Equations. AMSE JOURNALS –2015-Series: Advances A; Vol. 52; No 1; pp 17-30.
2. Timothy, A. A, David, O. A & Adetola,. A. A. (2012). A One Step Method for the Solution of General Second Order Ordinary Differential Equations. International Journal of Science and Technology, Volume 2 No. 4.
3. Dahlquist, G. (1959). Stability and Error Bounds in the Numerical Integration of ODEs. Transcript 130, Royal Institute of Technology, Stockholm.
4. Awoyemi, D. O. & Idowu, O. M. (2005). A class of hybrid collocation method for third order ordinary differential equation. Intern. J. Comp. Math. 82 (10), 1287-1293.
5. Gear, C. W. (1964): Hybrid methods for initial value problems in Ordinary Differential Equations. SIAM J. of Numer. Anal, 2, 69-86.
6. Lambert. J. D. (1973). Computational Methods in Ordinary Differential Equations. 1973 pp 114-149, John Wiley and Sons, London New York Sydney Toronto.
7. Shokri, A. & Shokri, A. A. (2013). The new class of implicit L-stable hybrid Obrechkoff method for thenumerical solution of first order initial value problems. Comput. Phys. Commun., 184: 529-531.
8. Zurni Omar & Oluwaseun adeyeye (2016). Numerical solution of first order initial value problems using a self-starting implicit two-step obrechkoff-type block method. Journal of mathematics and statistics, 12 (2): 27-134.
9. Fatunla, S. O. (1988). Numerical Methods for Initial Value Problems in Ordinary Differential Equation, Academic Press, New York.
10. Ononogbo,. C. B., Airemen,. I. E., Julia,. U. E. & Ishie,. I. H. (2021). An Implicit One-Step Third Derivative Hybrid Block Method For The Numerical Solution Of First Order Initial Value Problems In Ordinary Differential Equations. International Advanced Research Journal in Science, Engineering and Technology Vol. 8, Issue 8, August 2021 DOI: 10.17148/IARJSET.2021.8825.
11. Adoghe, L. O. Ukpebor, L. A., Ononogbo, C. B & Airemen, E. (2021). A One- Step Hybrid Obrechkoff- Type Block Method For First Order Initial Value Problems In Ordinary Differential Equations.
12. Ononogbo, C. B., AiremenIkhuoria, E., Ezurike, U. J., & Ugbolo. C. (2020). Implementation of a New Fourth Order Kutta’s Formula for Solving Initial Value Problems in Ordinary Differential Equations. International Journal of Scientific Research & Engineering Trends Volume 6, Issue 5, Sept-Oct-2020, ISSN (Online): 2395-566X.
13. Raymond, D., Skwamw, Y. & Sunday, J. (2019). An Implicit Two-Step One Off-Grid Point Third Derivative Hybrid Block Method for the Direct Solution of Second-Order Ordinary Differential Equations. Academic Journal of Applied Mathematical Sciences, Vol. 4, Issue. 2, pp: 8-14.
14. Emmanuel A. Areo, Nosimot O. Adeyanju & Sunday J. Kayode (2020). Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods. FUOYE Journal of Engineering and Technology (FUOYEJET), Vol. 5, Issue 2, 2579-0617.
15. Oluwaseun Adeyeye & Zurni Omar (2017). Hybrid Block Method for Direct Numerical Approximation of Second Order Initial Value Problems Using Taylor Series Expansions. American Journal of Applied Sciences, 14 (2): 309.315. DOI: 10.3844/ajassp.
16. Farhan, M. , Omar, Z., Mebarek-Oudina, F, Raza, J. , Shah, Z. , Choudhari, R. V & Makinde, O. D. (2020). Computational Mathematics and Modeling, Vol. 31, No. 1, DOI 10.1007/s10598-020-09480-0.
17. Ukpebor, L. A and Adoghe, L. O. (2019). Continuous Fourth Derivative Block Method For Solving Stiff Systems First Order Ordinary Differential Equations (Odes). Abacus (Mathematics Science Series) Vol. 44, No 1, Aug. 2019.