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

Ononogbo Chibuike Benjamin; Airemen Ikhuoria Edward; Ezurike Ugochi Julia

doi:doi:10.11648/j.ae.20220601.13

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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

Ononogbo Chibuike Benjamin, Airemen Ikhuoria Edward, Ezurike Ugochi Julia

Published in Applied Engineering (Volume 6, Issue 1)

Received: 3 May 2022 Accepted: 20 May 2022 Published: 16 June 2022

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Abstract

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.

Published in	Applied Engineering (Volume 6, Issue 1)
DOI	10.11648/j.ae.20220601.13
Page(s)	13-23
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

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

References

[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.

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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

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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

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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

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@article{10.11648/j.ae.20220601.13,
  author = {Ononogbo Chibuike Benjamin and Airemen Ikhuoria Edward and Ezurike Ugochi Julia},
  title = {Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations},
  journal = {Applied Engineering},
  volume = {6},
  number = {1},
  pages = {13-23},
  doi = {10.11648/j.ae.20220601.13},
  url = {https://doi.org/10.11648/j.ae.20220601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ae.20220601.13},
  abstract = {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.},
 year = {2022}
}

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TY  - JOUR
T1  - Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations
AU  - Ononogbo Chibuike Benjamin
AU  - Airemen Ikhuoria Edward
AU  - Ezurike Ugochi Julia
Y1  - 2022/06/16
PY  - 2022
N1  - https://doi.org/10.11648/j.ae.20220601.13
DO  - 10.11648/j.ae.20220601.13
T2  - Applied Engineering
JF  - Applied Engineering
JO  - Applied Engineering
SP  - 13
EP  - 23
PB  - Science Publishing Group
SN  - 2994-7456
UR  - https://doi.org/10.11648/j.ae.20220601.13
AB  - 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.
VL  - 6
IS  - 1
ER  -

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Author Information

Ononogbo Chibuike Benjamin

Department of Mathematics, Ambrose Alli University, Ekpoma, Nigeria
Airemen Ikhuoria Edward

Department of Mathematics, Ambrose Alli University, Ekpoma, Nigeria
Ezurike Ugochi Julia

Department of Basic Science, Federal Polytechnic, Auchi, Nigeria

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Plain Text BibTeX RIS

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

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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

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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

Copy | Download

@article{10.11648/j.ae.20220601.13,
  author = {Ononogbo Chibuike Benjamin and Airemen Ikhuoria Edward and Ezurike Ugochi Julia},
  title = {Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations},
  journal = {Applied Engineering},
  volume = {6},
  number = {1},
  pages = {13-23},
  doi = {10.11648/j.ae.20220601.13},
  url = {https://doi.org/10.11648/j.ae.20220601.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ae.20220601.13},
  abstract = {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.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Numerical Algorithm for One and Two-Step Hybrid Block Methods for the Solution of First Order Initial Value Problems in Ordinary Differential Equations
AU  - Ononogbo Chibuike Benjamin
AU  - Airemen Ikhuoria Edward
AU  - Ezurike Ugochi Julia
Y1  - 2022/06/16
PY  - 2022
N1  - https://doi.org/10.11648/j.ae.20220601.13
DO  - 10.11648/j.ae.20220601.13
T2  - Applied Engineering
JF  - Applied Engineering
JO  - Applied Engineering
SP  - 13
EP  - 23
PB  - Science Publishing Group
SN  - 2994-7456
UR  - https://doi.org/10.11648/j.ae.20220601.13
AB  - 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.
VL  - 6
IS  - 1
ER  -

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