A New 4<sup>th</sup> Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)

Bazuaye Frank Etin-Osa

doi:doi:10.11648/j.pamj.20180706.11

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A New 4^th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)

Bazuaye Frank Etin-Osa

Published in Pure and Applied Mathematics Journal (Volume 7, Issue 6)

Received: 13 November 2018 Accepted: 4 December 2018 Published: 2 January 2019

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Abstract

Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4^th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.

Published in	Pure and Applied Mathematics Journal (Volume 7, Issue 6)
DOI	10.11648/j.pamj.20180706.11
Page(s)	78-87
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

Hybrid Methods, Stability, Mean

References

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[2]	D. Dingwen and P. Tingting (2015). A Fourth-order Singly Diagonally Implicit Runge-Kutta Method for Solving One-dimensional Burgers’ Equation. IAENG International Journal of Applied Mathematics, 45, 4-11.
[3]	A. S Wusu, S. A Okunuga, A. B Sofoluwe (2012). A Third-Order Harmonic explicit Runge-Kutta Method for Autonomous initial value problems. Global Journal of Pure and Applied Mathematics. 8, 441-451.
[4]	A. S Wusu, and M. A Akanbi (2013). A Three stage Multiderivative Explicit Runge-kutta Method. American Journal ofComputationalMathematics. 3, 121-126.
[5]	M. AAkanbi (2011). On 3- stage Geometric Explicit Runge-Kutta Method for singular Autonomous initial value problems in ordinary differential equations computing. 92, 243-263.
[6]	J. D Evans and N. B Yaacob (1995). A Fourth Order Runge –Kutta Methods based on Heronian Mean Formula. International Journal of Computer Mathematics 58, 103-115.
[7]	J. D Evans and N. B Yaacob (1995) A Fourth Order Runge –Kutta Methods based on Heronian Mean Formula. International Journal of Computer Mathematics 59, 1-2.
[8]	J. D Evans and N. B Yaacob (1995). A New Fourth Order Runge –Kutta Methods based on Heronian Mean Formula. Department of Computer studies, Loughborough University of technology, Loughborough.
[9]	N. B Yaacob and B. Sangui (1998). A New Fourth Order Embedded Method based on Heronian Mean Mathematica Jilid, 1998, 1-6.
[10]	A. M. Wazwaz (1990). A Modified Third order Runge-kutta Method. Applied Mathematics Letter, 3(1990), 123-125.
[11]	J. D Evans and B. Sangui(1991). AComparison of Numerical O. D. E solvers based on Arithmetic and Geometric means. International Journal of Computer Mthematics, 32-35.
[12]	S. O. Fatunla (1988). Numerical Methods for initial value problems in ordinary Differential equations Academic Press, San Diago.
[13]	G. U, Agbeboh, U. S. U Aashikpelokhai., I. Aigbedion. (2007). Implementation of a New 4^th order RungeKutta Formula for solving initial value problems International Journal of Physical Siences. 2(4), 89-98.
[14]	Y Rini. Imran M, Syamsudhuha. A Third RungeKutta Method based on a LAinear combination of Arithmetic mean, harmonic mean and geometric mean. Applied and Computational Mathematics. 2014, 3(5), 231-234.
[15]	Ashirobo Serapon (2015). On the Derivation and Implementation of a Four Stage Harmonic ExplicitRunge-Kutta Method Ashirobo serapon. Applied Mathematics 6(4):694-699.
[16]	A. Ghazala and I. Ahmand, (2016). Solution of fourth order three-point boundary value problem using ADM and RKM. Journal of the Association of Arab Universities for Basic and Applied Sciences. 20, 61-67.
[17]	S. K. Khattri (2012). Eulers Number and some Means. Tamsui Oxford Journal of Information and Mathematical Sciences28 (2012), 369-377.
[18]	G. U. Agbeboh (2013). On the Stability Analysis of a 4th order Runge- kutta method based onGeometric mean. Mathematical Theory and Modeling, 4, 76-91.
[19]	S. O. Fatunla (1986). Numerical Treatment of singular IVPs. Computational Math. Application. 12:1109-1115.
[20]	J. D Lawson (1966)An order Five R-K Processes with Extended region of Absolute stability. SIAM J. Numer. Anal 4:372-380.

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Bazuaye Frank Etin-Osa. (2019). A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure and Applied Mathematics Journal, 7(6), 78-87. https://doi.org/10.11648/j.pamj.20180706.11

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ACS Style

Bazuaye Frank Etin-Osa. A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure Appl. Math. J. 2019, 7(6), 78-87. doi: 10.11648/j.pamj.20180706.11

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AMA Style

Bazuaye Frank Etin-Osa. A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure Appl Math J. 2019;7(6):78-87. doi: 10.11648/j.pamj.20180706.11

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@article{10.11648/j.pamj.20180706.11,
  author = {Bazuaye Frank Etin-Osa},
  title = {A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {6},
  pages = {78-87},
  doi = {10.11648/j.pamj.20180706.11},
  url = {https://doi.org/10.11648/j.pamj.20180706.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180706.11},
  abstract = {Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.},
 year = {2019}
}

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TY  - JOUR
T1  - A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)
AU  - Bazuaye Frank Etin-Osa
Y1  - 2019/01/02
PY  - 2019
N1  - https://doi.org/10.11648/j.pamj.20180706.11
DO  - 10.11648/j.pamj.20180706.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 78
EP  - 87
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20180706.11
AB  - Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.
VL  - 7
IS  - 6
ER  -

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Author Information

Bazuaye Frank Etin-Osa

Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Nigeria

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APA Style

Bazuaye Frank Etin-Osa. (2019). A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure and Applied Mathematics Journal, 7(6), 78-87. https://doi.org/10.11648/j.pamj.20180706.11

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ACS Style

Bazuaye Frank Etin-Osa. A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure Appl. Math. J. 2019, 7(6), 78-87. doi: 10.11648/j.pamj.20180706.11

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AMA Style

Bazuaye Frank Etin-Osa. A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs). Pure Appl Math J. 2019;7(6):78-87. doi: 10.11648/j.pamj.20180706.11

Copy | Download

@article{10.11648/j.pamj.20180706.11,
  author = {Bazuaye Frank Etin-Osa},
  title = {A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)},
  journal = {Pure and Applied Mathematics Journal},
  volume = {7},
  number = {6},
  pages = {78-87},
  doi = {10.11648/j.pamj.20180706.11},
  url = {https://doi.org/10.11648/j.pamj.20180706.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20180706.11},
  abstract = {Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - A New 4th Order Hybrid Runge-Kutta Methods for Solving Initial Value Problems (IVPs)
AU  - Bazuaye Frank Etin-Osa
Y1  - 2019/01/02
PY  - 2019
N1  - https://doi.org/10.11648/j.pamj.20180706.11
DO  - 10.11648/j.pamj.20180706.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 78
EP  - 87
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20180706.11
AB  - Recently, there has been a great deal of interest in the formulation of Runge-Kutta methods based on averages other than the conventional Arithmetic Mean for the numerical solution of Ordinary differential equations. In this paper, a new 4th Order Hybrid Runge-Kutta method based on linear combination of Arithmetic mean, Geometric mean and the Harmonic mean to solve first order initial value problems (IVPs) in ordinary differential equations (ODEs) is presented. Also the stability region for the method is shown. Moreover, the new method is compared with Runge-Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results indicate that the performance of the new method show superiority in terms of accuracy to some of other well known methods in literature and the stability investigation is in agreement with the known fourth order Runge-Kutta methods but with excellent stability region.
VL  - 7
IS  - 6
ER  -

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