On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation

Agbeboh Goddy Ujagbe; Ehiemua Michael Ebhodaghe; Loko Perelah

doi:doi:10.11648/j.ajmcm.20200504.12

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On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation

Agbeboh Goddy Ujagbe, Ehiemua Michael Ebhodaghe, Loko Perelah

Published in American Journal of Mathematical and Computer Modelling (Volume 5, Issue 4)

Received: 27 September 2020 Accepted: 15 October 2020 Published: 23 October 2020

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Abstract

In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.

Published in	American Journal of Mathematical and Computer Modelling (Volume 5, Issue 4)
DOI	10.11648/j.ajmcm.20200504.12
Page(s)	102-108
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), 2020. Published by Science Publishing Group

Keywords

Rational Integrator, Encroachment Point, Binomial Theorem, Absolute Stability, Jordan Curve, A-stable, L-Stability

References

[1]	Aashikpelokhai U. S. U. (2000). A variable order numerical integrator based on rational interpolants. Journal of Nig. Mathematical Society. Vol. 19, 27-28.
[2]	Aashikpelokhai U. S. U and Momodu I. B. A (2006): “A high order rational integrator for stiff systems” Journal of Nigeria Annuals of Natural Sciences pg. 122–143, vol. 4 (1).
[3]	Elakhe A. O., Onianwa C. U. and Elakhe S. O. (2016). A new order five numerical rational Rational Integrator. Aiziza Journal of Science and Technology, Vol. 1, pp49-63.
[4]	Agbeboh, G. U. (2006). Comparison of some one-step integrator for solving singular initial value problems. Ph. D. Thesis, AAU.
[5]	Abhulimen, C. E. and Otunta, F. O. (2007): A Family of Two –Step Exponentially Fitted Multiderivable Methods for the Numerical Integration of Stiff IVPs in ODE. International Journal of Numerical Mathematics, Vol. 2, pp1-21.
[6]	Agbeboh, G. U. and Aashikpelokhai U.S.U. (2002). An analysis of order, thirteen rational integrator. Journal of Sci. Engr. Tech Vol. 9 (2), 4128-4145.
[7]	Aashikpelokhai U. S. U. and Elakhe, A. O. (2010). A moderate order numerical integrator for still differential system, Journal of Nigeria Association of Mathematical Physics. Vol. 17. Pp. 425-432.
[8]	Islam, M. A. (2015). A Comparative Study on Numerical Solutions of Initial Value Problems (IVP) for Ordinary Differential Equations (ODE) with Euler and Runge-Kutta Methods. American Journal of Computational Mathematics, Vol. 5, 393-404.
[9]	Agbeboh, G. U. Esekhaigbe, C. A. (2016): Transformation and Implementation of a Highly Efficient Fully Implicit Fourth-Order Runge-Kutta Method. International Journal of Innovative Research, Vol 5 (1), 171-183.
[10]	Elakhe, A. O. and Aashikpelokhai U.S.U (2013): Singulo Oscillatory – Stiff rational integrators. International Journal of Physical Sciences, vol. 8 (34), 1703–1715.
[11]	Fatunla S. O. and Aashikpelokhai, U.S.U (1994). A Fifth Order L-Stable Numerical Integrator, Scientific Comp pp 68-86.
[12]	Elakhe, A. O., Aashikpelokhai U.S.U. and Ebhomien P. A. (2011): A Dynamical Singulo-Stiff Rational Integrator. IRCAB Journal of Natural Applied Sciences. Vol. 1 pp73-79.
[13]	Awari, Y. S. (2017): Numerical Strategies for the System of First Order IVPs Using Block Hybrid Extended Trapezoidal Multistep Method of Second Kind for Stiff ODEs. Science Journal of Applied Mathematics and Statistics, Vol. 5 (5): 181-187.
[14]	Momodu, I. B. A. (2006): Solution of Singulo-stiff Ordinary Differential Equation. PhD Thesis, AAU, Ekpoma.
[15]	Fatunla, S. O. (1978): An Implicit Two-Part Numerical Integration Formula for Linear and Non-Linear Stiff System of ODE”, Mathematics of Computation 32, 1-11.

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Agbeboh Goddy Ujagbe, Ehiemua Michael Ebhodaghe, Loko Perelah. (2020). On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. American Journal of Mathematical and Computer Modelling, 5(4), 102-108. https://doi.org/10.11648/j.ajmcm.20200504.12

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ACS Style

Agbeboh Goddy Ujagbe; Ehiemua Michael Ebhodaghe; Loko Perelah. On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. Am. J. Math. Comput. Model. 2020, 5(4), 102-108. doi: 10.11648/j.ajmcm.20200504.12

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AMA Style

Agbeboh Goddy Ujagbe, Ehiemua Michael Ebhodaghe, Loko Perelah. On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. Am J Math Comput Model. 2020;5(4):102-108. doi: 10.11648/j.ajmcm.20200504.12

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@article{10.11648/j.ajmcm.20200504.12,
  author = {Agbeboh Goddy Ujagbe and Ehiemua Michael Ebhodaghe and Loko Perelah},
  title = {On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {5},
  number = {4},
  pages = {102-108},
  doi = {10.11648/j.ajmcm.20200504.12},
  url = {https://doi.org/10.11648/j.ajmcm.20200504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20200504.12},
  abstract = {In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.},
 year = {2020}
}

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TY  - JOUR
T1  - On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation
AU  - Agbeboh Goddy Ujagbe
AU  - Ehiemua Michael Ebhodaghe
AU  - Loko Perelah
Y1  - 2020/10/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ajmcm.20200504.12
DO  - 10.11648/j.ajmcm.20200504.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20200504.12
AB  - In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.
VL  - 5
IS  - 4
ER  -

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Author Information

Agbeboh Goddy Ujagbe

Department of Mathematics, Ambrose Alli Unuversity, Ekpoma, Nigeria
Ehiemua Michael Ebhodaghe

Department of Mathematics, Ambrose Alli Unuversity, Ekpoma, Nigeria
Loko Perelah

Department of Mathematics, Bayelsa State College of Education, Sagbama, Nigeria

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APA Style

Agbeboh Goddy Ujagbe, Ehiemua Michael Ebhodaghe, Loko Perelah. (2020). On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. American Journal of Mathematical and Computer Modelling, 5(4), 102-108. https://doi.org/10.11648/j.ajmcm.20200504.12

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ACS Style

Agbeboh Goddy Ujagbe; Ehiemua Michael Ebhodaghe; Loko Perelah. On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. Am. J. Math. Comput. Model. 2020, 5(4), 102-108. doi: 10.11648/j.ajmcm.20200504.12

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AMA Style

Agbeboh Goddy Ujagbe, Ehiemua Michael Ebhodaghe, Loko Perelah. On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation. Am J Math Comput Model. 2020;5(4):102-108. doi: 10.11648/j.ajmcm.20200504.12

Copy | Download

@article{10.11648/j.ajmcm.20200504.12,
  author = {Agbeboh Goddy Ujagbe and Ehiemua Michael Ebhodaghe and Loko Perelah},
  title = {On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {5},
  number = {4},
  pages = {102-108},
  doi = {10.11648/j.ajmcm.20200504.12},
  url = {https://doi.org/10.11648/j.ajmcm.20200504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20200504.12},
  abstract = {In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - On the Stability Analysis of Rational Integrator Method for the Solution of Initial Value Problems in Ordinary Differential Equation
AU  - Agbeboh Goddy Ujagbe
AU  - Ehiemua Michael Ebhodaghe
AU  - Loko Perelah
Y1  - 2020/10/23
PY  - 2020
N1  - https://doi.org/10.11648/j.ajmcm.20200504.12
DO  - 10.11648/j.ajmcm.20200504.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 102
EP  - 108
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20200504.12
AB  - In all numerical methods, it is necessary to ascertain the validity of any particular scheme. And this is possible to determine, by verifying the nature of the stability of that scheme. So the general stability function definition is given, from where an investigation is carried out on a class of rational integrator of order 15, to establish the region of absolute stability of the scheme, by constructing the Jordan curve. In the process of expanding the rational function, binomial theorem as well as the idea of combination process were introduced to ease the computation by using Maple-18 package. The simplification of the general rational integrator formula, is constructed from two processes namely through complex function, and then through polar analysis, The Jordan curve is constructed with the help of MATLAB package. Furthermore, it was discovered that the region of instability is on the positive side of the complex plane, while the region of absolute stability is outside the Jordan curve. Finally, it is further established that the encroachment point, τ, lie within the interval ± 140.6. And the encroachment point is visible from the corresponding values of ø and R at the extremes. The stability curve revealed that the integrator is not only A-stable, but also L- stable.
VL  - 5
IS  - 4
ER  -

Copy | Download