A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations

Adegoke Stephen Olaniyan; Omolara Fatimah Bakre; Moses Adebowale Akanbi

doi:doi:10.11648/j.pamj.20200905.11

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A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations

Adegoke Stephen Olaniyan, Omolara Fatimah Bakre, Moses Adebowale Akanbi

Published in Pure and Applied Mathematics Journal (Volume 9, Issue 5)

Received: 23 July 2020 Accepted: 17 August 2020 Published: 8 September 2020

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Abstract

In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.

Published in	Pure and Applied Mathematics Journal (Volume 9, Issue 5)
DOI	10.11648/j.pamj.20200905.11
Page(s)	84-90
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

Implicit Runge-Kutta, Heronian Mean, Absolute Stability, Convergence, Ordinary Differential Equation

References

[1]	Ababneh O. Y. and Ahmad R. “Construction of Third-Order Diagonal Implicit Runge-Kutta method for Stiff Problems”. Chinese Physics Letters, Vol. 26, No. 8, Article ID 080503, 2009.
[2]	Ademiluyi R. A., Babatola P. O. and Kayode S. J. “A new class of implicit Rational Runge-Kutta method for integration of stiff ODEs”. Journal of Nigerian Mathematical Society, 21, (2002), 27-41.
[3]	Ademiluyi R. A., Babatola P. O. and Kayode S. J.“Semi-implicit Rational Runge-Kutta formulas of approximation of stiff initial value problems in ODEs”. Journal of Nigerian Mathematical Science and Education, (2001), 1 1 25.
[4]	Agam, S. A. and Yahaya, Y. A. (2014). A highly efficient implicit Runge-Kutta method for first order ordinary differential equations. African Journal of Mathematics and Computer Science Research, 7 (5), 55-60.
[5]	Akanbi M. A. (2011). “On 3-Stage Geometric Explicit Runge-Kutta Methods for Singular Initial value Problems in Ordinary Differential equations”. Computing, Springer Journals, Vol. 92, No. 3, 243263.
[6]	Akanbi, M, A. and Okunuga, S. A. (2008). “On the Convergence and Stability of 2stage Multiderivative Explicit Runge-Kutta Methods”. Journal of Nigerian Mathematical Society, Vol. 27.123143.
[7]	Butcher, J. C. (2000). Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, 125 (1-2), 1-29.
[8]	Chen D. J. L., “The Efficiency of Singly-Implicit Runge-Kutta Methods for Stiff Diffential Equations”. Numerical Algorithm, Vol. 65, No. 3, pp. 533-554, 2014.
[9]	Dahlquist G. (1956).”Convergence and Stability in the Numerical Integration of ODEs”. Math. Scand. 4: 33 3 53.
[10]	Evans D. J. and Yaacob N. B. (1995). “A Fourth Order Runge-Kutta Method Based on the Heronian Mean Formula”. International Journal of Computer Mathematics, 58, 103 3 115.
[11]	Evans D. J. and Yaacob N. B. (1995).”A New Fourth Order Runge-Kutta Method Based on the Contraharmonic Mean”. International Journal of Computer Mathematics, 57, 249 9 256.
[12]	Jameson A. “Evaluation of Fully Implicit Runge-Kutta Schemes for Unsteady Flow Calculations”. Journal of Scientific Computing, Vol. 73, No. 2-3, pp. 819-852, 2017.
[13]	Jawias N. I. C., Ismail F., Suleiman M. and Jaafar A.“Fourth Order Four-Stage Diagonally Implicit Runge-Kutta method for linear Ordinary Differential Equations”. Malaysian Journal of Mathematical Sciences, vol. 4, pp. 95 5 105, 2010.
[14]	Lambert J. D. (1991).“Numerical Methods for Ordinary Differential Systems”. John Wiley and sons, England.
[15]	Liao W. and Yan Y., “Singly Diagonally Implicit Runge-Kutta Method for Time-Dependent Reaction-Diffusion equation”. Numerical Methods for Partial Differential equations, Vol. 27, No. 6, pp. 1423-1441, 2011.
[16]	Liu M. Y., Zhang L. and Zhang C. F. “Study on Banded Implicit Runge-Kutta for Solving Stiff Differential Equations”. Hindawi Mathematical Problems in Engineering, Volume 2019, Article ID 4850872, 8 Pages.
[17]	Ponalagusamy P. J. A. and Chandra M. (2011), “Development of New Fifth-Order Fifth Stage Runge-Kutta Method on Heronian Mean”. International Journal of Engineering Science for Advance Computing, Vol. 2, pp. 162 2 197.
[18]	Wusu A. S., Akanbi M. A. and Bakre O. F. (2015). “On the Derivation and Implementation of a Four-Stage HarmonicExplicit Runge-Kutta Method”. Applied Mathematics, 2015, 6, 694 4 699.
[19]	Wusu A. S., Okunuga S. A. and Sofoluwe A. B. (2012). “A Third-Order Harmonic Explicit Runge-Kutta Method for Autonomous Initial Value Problems”. Global Journal of Pure and Applied Mathematics, 8, 441 1 451.
[20]	Yaacob N. B. and Sanugi B. (1995). “A New Fourth Order Embedded Method Based on the Harmonic Mean”. Mathematika, 14, 1 16.

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Adegoke Stephen Olaniyan, Omolara Fatimah Bakre, Moses Adebowale Akanbi. (2020). A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(5), 84-90. https://doi.org/10.11648/j.pamj.20200905.11

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ACS Style

Adegoke Stephen Olaniyan; Omolara Fatimah Bakre; Moses Adebowale Akanbi. A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(5), 84-90. doi: 10.11648/j.pamj.20200905.11

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AMA Style

Adegoke Stephen Olaniyan, Omolara Fatimah Bakre, Moses Adebowale Akanbi. A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure Appl Math J. 2020;9(5):84-90. doi: 10.11648/j.pamj.20200905.11

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@article{10.11648/j.pamj.20200905.11,
  author = {Adegoke Stephen Olaniyan and Omolara Fatimah Bakre and Moses Adebowale Akanbi},
  title = {A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {5},
  pages = {84-90},
  doi = {10.11648/j.pamj.20200905.11},
  url = {https://doi.org/10.11648/j.pamj.20200905.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200905.11},
  abstract = {In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.},
 year = {2020}
}

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TY  - JOUR
T1  - A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations
AU  - Adegoke Stephen Olaniyan
AU  - Omolara Fatimah Bakre
AU  - Moses Adebowale Akanbi
Y1  - 2020/09/08
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200905.11
DO  - 10.11648/j.pamj.20200905.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 84
EP  - 90
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200905.11
AB  - In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.
VL  - 9
IS  - 5
ER  -

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Author Information

Adegoke Stephen Olaniyan

Department of Mathematics, Lagos State University, Lagos, Nigeria
Omolara Fatimah Bakre

Department of Mathematics, Federal College of Education (Technical), Lagos, Nigeria
Moses Adebowale Akanbi

Department of Mathematics, Lagos State University, Lagos, Nigeria

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Plain Text BibTeX RIS

APA Style

Adegoke Stephen Olaniyan, Omolara Fatimah Bakre, Moses Adebowale Akanbi. (2020). A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure and Applied Mathematics Journal, 9(5), 84-90. https://doi.org/10.11648/j.pamj.20200905.11

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ACS Style

Adegoke Stephen Olaniyan; Omolara Fatimah Bakre; Moses Adebowale Akanbi. A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure Appl. Math. J. 2020, 9(5), 84-90. doi: 10.11648/j.pamj.20200905.11

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AMA Style

Adegoke Stephen Olaniyan, Omolara Fatimah Bakre, Moses Adebowale Akanbi. A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations. Pure Appl Math J. 2020;9(5):84-90. doi: 10.11648/j.pamj.20200905.11

Copy | Download

@article{10.11648/j.pamj.20200905.11,
  author = {Adegoke Stephen Olaniyan and Omolara Fatimah Bakre and Moses Adebowale Akanbi},
  title = {A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations},
  journal = {Pure and Applied Mathematics Journal},
  volume = {9},
  number = {5},
  pages = {84-90},
  doi = {10.11648/j.pamj.20200905.11},
  url = {https://doi.org/10.11648/j.pamj.20200905.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20200905.11},
  abstract = {In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - A 2-Stage Implicit Runge-Kutta Method Based on Heronian Mean for Solving Ordinary Differential Equations
AU  - Adegoke Stephen Olaniyan
AU  - Omolara Fatimah Bakre
AU  - Moses Adebowale Akanbi
Y1  - 2020/09/08
PY  - 2020
N1  - https://doi.org/10.11648/j.pamj.20200905.11
DO  - 10.11648/j.pamj.20200905.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 84
EP  - 90
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20200905.11
AB  - In recent times, the use of different types of mean in the derivation of explicit Runge-Kutta methods had been on increase. Researchers have explored explicit Runge-Kutta methods derivation by using different types of mean such as geometric mean, harmonic mean, contra-harmonic mean, heronian mean to name but a few; as against the conventional explicit Runge-Kutta methods which was viewed as arithmetic mean. However, despite efforts to improve the derivation of explicit Runge-Kutta methods with use of other types of mean, none has deemed it fit to extend this notion to implicit Runge-Kutta methods. In this article, we present the use of heronian mean as a basis for the construction of implicit Runge-Kutta method in a way of improving the conventional method which is arithmetic mean based. Numerical results was conducted on ordinary differential equations which was compared with the conventional two-stage fourth order implicit Runge-Kutta (IRK4) method and two-stage third order diagonally implicit Runge-Kutta (DIRK3) method. The results presented confirmed that the new scheme performs better than these numerical methods. A better Qualitative properties using Dalquist test equation were established.
VL  - 9
IS  - 5
ER  -

Copy | Download