A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

Adoghe Lawrence Osa; Omole Ezekiel Olaoluwa

doi:doi:10.11648/j.acm.20190803.11

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A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations

Adoghe Lawrence Osa, Omole Ezekiel Olaoluwa

Published in Applied and Computational Mathematics (Volume 8, Issue 3)

Received: 30 January 2019 Accepted: 17 March 2019 Published: 12 August 2019

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Abstract

In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were 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 method.

Published in	Applied and Computational Mathematics (Volume 8, Issue 3)
DOI	10.11648/j.acm.20190803.11
Page(s)	50-57
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), 2019. Published by Science Publishing Group

Keywords

Collocation, Interpolation, Approximate Solution, Continuous Block Method, Discrete Block Method, Convergence

References

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[3]	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.
[4]	Adoghe L. O and Omole E. O (2018): Comprehensive Analysis of 3-Quarter-Step Collocation Method for Direct Integration of Second Order Ordinary Differential Equations Using Taylor Series Function. ABACUS (Mathematics Science Series) Vol. 44, N0 2, pp. 311-321.
[5]	Awoyemi D. O (2001): A new Sixth –Order Algorithm for General Second Order Ordinary Differential Equations. International Journal of Computer mathematics, Vol 77, pp. 117-124.
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[8]	Fatokun, J. O (2007): Continuous Approach for deriving Self –starting multistep methods for initial value problems in ordinary differential Equations. Journal of Engineering and Applied Sciences Vol. 2 (3) pp. 504-508.
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[10]	Henrici, P. (1962): Discrete Variable method in ordinary differential equations, John Wiley and Sons, New York.
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[14]	Mohammed U. and R. B Adeniyi (2014): A Three Step Implicit Hybrid Linear Multistep Method for the Solution of Third Order Ordinary Differential Equations. Gen. Math. Notes, Vol. 25, No. 1, pp. 62-74.
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[17]	Tumba, P., Sabo, J. & Hamadina, M., (2018): Uniformly Order Eight Implicit Second Derivative Method for Solving Second- Order Stiff Ordinary Differential Equations ODEs. Academic Journal of Applied Mathematical Sciences, 4: 43-48.
[18]	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.
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Adoghe Lawrence Osa, Omole Ezekiel Olaoluwa. (2019). A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Applied and Computational Mathematics, 8(3), 50-57. https://doi.org/10.11648/j.acm.20190803.11

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ACS Style

Adoghe Lawrence Osa; Omole Ezekiel Olaoluwa. A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Appl. Comput. Math. 2019, 8(3), 50-57. doi: 10.11648/j.acm.20190803.11

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AMA Style

Adoghe Lawrence Osa, Omole Ezekiel Olaoluwa. A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Appl Comput Math. 2019;8(3):50-57. doi: 10.11648/j.acm.20190803.11

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@article{10.11648/j.acm.20190803.11,
  author = {Adoghe Lawrence Osa and Omole Ezekiel Olaoluwa},
  title = {A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {3},
  pages = {50-57},
  doi = {10.11648/j.acm.20190803.11},
  url = {https://doi.org/10.11648/j.acm.20190803.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190803.11},
  abstract = {In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were 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 method.},
 year = {2019}
}

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TY  - JOUR
T1  - A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations
AU  - Adoghe Lawrence Osa
AU  - Omole Ezekiel Olaoluwa
Y1  - 2019/08/12
PY  - 2019
N1  - https://doi.org/10.11648/j.acm.20190803.11
DO  - 10.11648/j.acm.20190803.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 57
PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.acm.20190803.11
AB  - In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were 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 method.
VL  - 8
IS  - 3
ER  -

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Author Information

Adoghe Lawrence Osa

Department of Mathematics, Ambrose Alli University, Ekpoma, Edo State, Nigeria
Omole Ezekiel Olaoluwa

Department of Mathematics, Federal University of Technology, Akure, Nigeria

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APA Style

Adoghe Lawrence Osa, Omole Ezekiel Olaoluwa. (2019). A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Applied and Computational Mathematics, 8(3), 50-57. https://doi.org/10.11648/j.acm.20190803.11

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ACS Style

Adoghe Lawrence Osa; Omole Ezekiel Olaoluwa. A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Appl. Comput. Math. 2019, 8(3), 50-57. doi: 10.11648/j.acm.20190803.11

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AMA Style

Adoghe Lawrence Osa, Omole Ezekiel Olaoluwa. A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations. Appl Comput Math. 2019;8(3):50-57. doi: 10.11648/j.acm.20190803.11

Copy | Download

@article{10.11648/j.acm.20190803.11,
  author = {Adoghe Lawrence Osa and Omole Ezekiel Olaoluwa},
  title = {A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations},
  journal = {Applied and Computational Mathematics},
  volume = {8},
  number = {3},
  pages = {50-57},
  doi = {10.11648/j.acm.20190803.11},
  url = {https://doi.org/10.11648/j.acm.20190803.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20190803.11},
  abstract = {In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were 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 method.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - A Fifth-fourth Continuous Block Implicit Hybrid Method for the Solution of Third Order Initial Value Problems in Ordinary Differential Equations
AU  - Adoghe Lawrence Osa
AU  - Omole Ezekiel Olaoluwa
Y1  - 2019/08/12
PY  - 2019
N1  - https://doi.org/10.11648/j.acm.20190803.11
DO  - 10.11648/j.acm.20190803.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 50
EP  - 57
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20190803.11
AB  - In this paper, block method was developed using method of collocation and interpolation of power series as approximate solution to give a system of non linear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic properties of the discrete block method were 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 method.
VL  - 8
IS  - 3
ER  -

Copy | Download