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A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean

Received: 11 July 2014     Accepted: 12 September 2014     Published: 30 September 2014
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Abstract

We present a new third order Runge Kutta method based on linear combination of arithmetic mean, geometric mean and harmonic mean to solve a first order initial value problem. We also derive the local truncation error and show the stability region for the method. Moreover, we compare the new method with Runge Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results show that the performance of the new method is the same as known third order Runge-Kutta methods.

Published in Applied and Computational Mathematics (Volume 3, Issue 5)
DOI 10.11648/j.acm.20140305.16
Page(s) 231-234
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), 2014. Published by Science Publishing Group

Keywords

Initial Value Problems, Runge Kutta Method, Arithmetic Mean, Harmonic Mean, Geometric Mean

References
[1] O.Y. Ababneh and R. Rozita, New Third Order Runge Kutta Method Based on Contraharmonic Mean for Stiff Problems, Applied Mathematical Sciences, 3(2009), 365-376.
[2] G. Dahlquist and A. Bjorck, Numerical Method, Prentice-Hall,Inc., New York, 1974.
[3] D.J. Evans, 1989. New Runge-Kutta Methods For Initial Value Problems, Applied Mathematics Letter, 2(1989), pp.25--28.
[4] S.K. Khattri, “Euler's Number and Some Means”, Tamsui Oxford Journal of Information and Mathematical Sciences, 28(2012), pp.369--377.
[5] K.A. Ross, Elementary Analysis, Springer, New York, 1980.
[6] L. P. Shampine, Numerical of Ordinary Diferential Equation, Chapman and Hall, New York, 1994.
[7] A.G. Ujagbe, “On the Stability Analysis of a Geometric Mean 4th Order Runge-Kutta Formula, Mathematical Theory and Modelling, 3(2013), pp.76-91.
[8] A.M. Wazwaz, “A Modified Third Order Runge-Kutta Method”, Applied Mathematics Letter, 3(1990), pp.123-125.
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  • APA Style

    Rini Yanti, M Imran, Syamsudhuha. (2014). A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean. Applied and Computational Mathematics, 3(5), 231-234. https://doi.org/10.11648/j.acm.20140305.16

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    ACS Style

    Rini Yanti; M Imran; Syamsudhuha. A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean. Appl. Comput. Math. 2014, 3(5), 231-234. doi: 10.11648/j.acm.20140305.16

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    AMA Style

    Rini Yanti, M Imran, Syamsudhuha. A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean. Appl Comput Math. 2014;3(5):231-234. doi: 10.11648/j.acm.20140305.16

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  • @article{10.11648/j.acm.20140305.16,
      author = {Rini Yanti and M Imran and Syamsudhuha},
      title = {A Third Runge Kutta Method Based on a Linear Combination of Arithmetic Mean, Harmonic Mean and Geometric Mean},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {5},
      pages = {231-234},
      doi = {10.11648/j.acm.20140305.16},
      url = {https://doi.org/10.11648/j.acm.20140305.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.16},
      abstract = {We present a new third order Runge Kutta method based on linear combination of arithmetic mean, geometric mean and harmonic mean to solve a first order initial value problem. We also derive the local truncation error and show the stability region for the method. Moreover, we compare the new method with Runge Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results show that the performance of the new method is the same as known third order Runge-Kutta methods.},
     year = {2014}
    }
    

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    AU  - Rini Yanti
    AU  - M Imran
    AU  - Syamsudhuha
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    T2  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20140305.16
    AB  - We present a new third order Runge Kutta method based on linear combination of arithmetic mean, geometric mean and harmonic mean to solve a first order initial value problem. We also derive the local truncation error and show the stability region for the method. Moreover, we compare the new method with Runge Kutta method based on arithmetic mean, geometric mean and harmonic mean. The numerical results show that the performance of the new method is the same as known third order Runge-Kutta methods.
    VL  - 3
    IS  - 5
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Author Information
  • Numerical Computing Group, Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia

  • Numerical Computing Group, Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia

  • Numerical Computing Group, Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia

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