Pure and Applied Mathematics Journal

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

Received: 23 July 2020    Accepted: 17 August 2020    Published: 08 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.

DOI 10.11648/j.pamj.20200905.11
Published in Pure and Applied Mathematics Journal (Volume 9, Issue 5, October 2020)
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), 2024. Published by Science Publishing Group

Keywords

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

References
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[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.
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[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.
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Author Information
  • Department of Mathematics, Lagos State University, Lagos, Nigeria

  • Department of Mathematics, Federal College of Education (Technical), Lagos, Nigeria

  • Department of Mathematics, Lagos State University, Lagos, Nigeria

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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://download.sciencepg.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
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    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
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    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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