### Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean

Received: 16 May 2019     Accepted: 13 June 2019     Published: 6 August 2019
Abstract

Over the years, the Quadrature Algorithm as a method of solving initial value problems in ordinary differential equations is known to be of low accuracy compared to other well known methods. However, It has been shown that the method perform well when applied to moderately stiff problems. In this present study, the nonlinear method based on the Heronian Mean (HeM), of the function value for the solution of initial value problems is developed. Stability investigation is in agreement with the known Trapezoidal method.

 Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 2) DOI 10.11648/j.ijamtp.20190502.12 Page(s) 45-51 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

Harmonic Mean, Stability, Stiff Problems, Geometric Mean, Accuracy

References
 [1] 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 [2] A. S Wusu, and M. A Akanbi (2013). A Three stage Multiderivative Explicit Rungekutta Method. American Journal of Computational Mathematics. 3, 121-126 [3] M. A Akanbi. (2011). On 3- stage Geometric Explicit RungeKutta Method for singular Autonomous initial value problems in ordinary differential equations Computing.92, 243-263 [4] 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 [5] N. B Yaacob and B. Sangui (1998). A New Fourth Order Embedded Method based on Heronian Mean Mathematica Jilid, 1998, 1-6. [6] Wazwaz A. M. (1990). A Modified Third order Runge-kutta Method. Applied Mathematics Letter, 3(1990), 123-125 [7] F. E Bazuaye (2018). A new 4th order Hybrid Runge-Kutta Methods for solving initial value problems. Pure and Applied Mathematics Journal 7(6): 78-87 [8] A. R Gourlay (1970). A note on trapezoidal methods for the solution of initial value problems. Math. Comput.. 24(1), 629-633. [9] Rini Y. Imran M, Syamsudhuha. (2014). A Third RungeKutta Method based on a Linear combination of Arithmetic mean, harmonic mean and geometric mean. Applied and Computational Mathematics. 2014, 3(5), 231-234 [10] Ashiribo Senapon Wusu1, Moses Adebowale Akanbi1, Bakre Omolara Fatimah (2015). On the derivation and implementation of four stage Harmonic Explicit Runge –kutta Method. Applied Mathematics. 6, 694-699 [11] D. J Evans (1988). A Stable Nonlinear Mid-Point Formula for Solving O. D. E. s. Appl. Math. L&t. Vol. 1, No. 2, pp. 165-169. [12] M. CHANDRU, R. PONALAGUSAMY, P. J. A. ALPHONSE (2017). A New fifth order weighted Runge –kutta methods based on the Heronian mean for initial value problems in ordinary differential equations. J. Appl. Math. & Informatics Vol. 35 (2017), 191 – 204. [13] Evans J. D and Sangui B. B (1988). A nonlinear Trapezoidal Formula for the solution of initial value problems. Comput. Math. Applic. 15 (1), 77-79 [14] Evans J. D and Sangui B. B (1991). A Comparison of Numerical O. D. E solvers based on Arithmetic and Geometric means. International Journal of Computer Mthematics, 32-35 [15] S. O. Fatunla (1986). Numerical Treatment of singular IVPs. Computational Math. Application. 12:1109-1115. [16] J. D Lawson (1966) An order Five R-K Processes with Extended region of Absolute stability. SIAM J. Numer. Anal 4:372-380.
• APA Style

Bazuaye Frank Etin-Osa. (2019). Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean. International Journal of Applied Mathematics and Theoretical Physics, 5(2), 45-51. https://doi.org/10.11648/j.ijamtp.20190502.12

ACS Style

Bazuaye Frank Etin-Osa. Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean. Int. J. Appl. Math. Theor. Phys. 2019, 5(2), 45-51. doi: 10.11648/j.ijamtp.20190502.12

AMA Style

Bazuaye Frank Etin-Osa. Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean. Int J Appl Math Theor Phys. 2019;5(2):45-51. doi: 10.11648/j.ijamtp.20190502.12

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title = {Solution of an Initial Value Problemin Ordinary Differential Equations Using the Quadrature Algorithm Based on the Heronian Mean},
journal = {International Journal of Applied Mathematics and Theoretical Physics},
volume = {5},
number = {2},
pages = {45-51},
doi = {10.11648/j.ijamtp.20190502.12},
url = {https://doi.org/10.11648/j.ijamtp.20190502.12},
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abstract = {Over the years, the Quadrature Algorithm as a method of solving initial value problems in ordinary differential equations is known to be of low accuracy compared to other well known methods. However, It has been shown that the method perform well when applied to moderately stiff problems. In this present study, the nonlinear method based on the Heronian Mean (HeM), of the function value for the solution of initial value problems is developed. Stability investigation is in agreement with the known Trapezoidal method.},
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AB  - Over the years, the Quadrature Algorithm as a method of solving initial value problems in ordinary differential equations is known to be of low accuracy compared to other well known methods. However, It has been shown that the method perform well when applied to moderately stiff problems. In this present study, the nonlinear method based on the Heronian Mean (HeM), of the function value for the solution of initial value problems is developed. Stability investigation is in agreement with the known Trapezoidal method.
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Author Information
• Department of Mathematics and Statistics, University of Port Harcourt, Port Harcourt, Rivers State, Nigeria

• Sections