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Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation

Received: 17 February 2016     Accepted: 25 March 2016     Published: 15 April 2016
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Abstract

This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically.

Published in Applied and Computational Mathematics (Volume 5, Issue 2)
DOI 10.11648/j.acm.20160502.15
Page(s) 64-72
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), 2016. Published by Science Publishing Group

Keywords

Riccati Equation, Symmetry Group, Infinitesimal Generator, Runge-Kutta

References
[1] W. T. Reid, Riccati Differential Equations (Mathematics in science and engineering), New York: Academic Press, 1972.
[2] F. Dubois, A. Saidi, Unconditionally Stable Scheme for Riccati Equation, ESAIM Proceeding. 8(2000), 39-52.
[3] A. A. Bahnasawi, M. A. El-Tawil and A. Abdel-Naby, Solving Riccati Equation using Adomians Decomposition Method, App. Math. Comput. 157(2007), 503-514.
[4] T. Allahviraloo, Sh. S. Bahzadi. Application of Iterative Methods for Solving General Riccati Equation, Int. J. Industrial Mathematics, Vol. 4, ( 2012) No. 4, IJIM-00299.
[5] Supriya Mukherjee, Banamali Roy. Solution of Riccati Equation with Variable Co-efficient by Differential Transform Method, Int. J. of Nonlinear Science Vol.14, (2012) No.2, pp. 251-256.
[6] Taiwo, O. A., Osilagun J. A. Approximate Solution of Generalized Riccati Differential Equation by Iterative Decomposition Algorithm, International Journal of Engineering and Innivative Technology(IJEIT) Vol. 1(2012) No. 2, pp. 53-56.
[7] J. Biazar, M. Eslami. Differential Transform Method for Quadratic Riccati Differential Equation, vol. 9 (2010) No.4, pp. 444-447.
[8] Cristinel Mortici. The Method of the Variation of Constants for Riccati Equations, General Mathematics, Vol. 16(2008) No.1, pp. 111-116.
[9] B. Gbadamosi, O. adebimpe, E. I. Akinola, I. A. I. Olopade. Solving Riccati Equation using Adomian Decomposition Method, International Journal of Pure and Applied Mathematics, Vol. 78(2012) No. 3, pp. 409-417.
[10] Olever. P. J. Application of Lie Groups to Differential Equations. New York Springer-Verlag, (1993).
[11] Al Fred Grany. Modern Differential Geometry of Curves and Surfaces, CRC Press, (1998).
[12] Aubin Thierry. Differential Geometry, American Mathematical Society, (2001).
[13] Nail. H. Ibragimov, Elementry Lie Group Analysis and Ordinary Differential Equations, John Wiley Sons New York, (1996).
[14] T. R. Ramesh Rao, "The use of the A domain Decomposition Method for Solving Generalized Riccati Differential Equations" Proceedings of the 6th IMT-GT Conference on Mathematics, Statistics and its Applications (ICMSA2010) Universiti Tunku Abdul Rahman, Kuala Lumpur, Malaysia pp. 935-941.
[15] B. Batiha, M. S. M. Noorani and I. Hashim, " Application of Variational Iteration Method to a General Riccati Equation" International Mathematical Forum, Vol.2, no. 56, pp. 2759–2770, 2007.
Cite This Article
  • APA Style

    Sami H. Altoum, Salih Y. Arbab. (2016). Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation. Applied and Computational Mathematics, 5(2), 64-72. https://doi.org/10.11648/j.acm.20160502.15

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    ACS Style

    Sami H. Altoum; Salih Y. Arbab. Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation. Appl. Comput. Math. 2016, 5(2), 64-72. doi: 10.11648/j.acm.20160502.15

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    AMA Style

    Sami H. Altoum, Salih Y. Arbab. Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation. Appl Comput Math. 2016;5(2):64-72. doi: 10.11648/j.acm.20160502.15

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  • @article{10.11648/j.acm.20160502.15,
      author = {Sami H. Altoum and Salih Y. Arbab},
      title = {Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {2},
      pages = {64-72},
      doi = {10.11648/j.acm.20160502.15},
      url = {https://doi.org/10.11648/j.acm.20160502.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160502.15},
      abstract = {This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically.},
     year = {2016}
    }
    

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    T1  - Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation
    AU  - Sami H. Altoum
    AU  - Salih Y. Arbab
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    N1  - https://doi.org/10.11648/j.acm.20160502.15
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    AB  - This paper introduced Lie group method as a analytical method and then compared to RK4 and Euler forward method as a numerical method. In this paper the general Riccati equation is solved by symmetry group. Numerical comparisons between exact solution, Lie symmetry group and RK4 on these equations are given. In particular, some examples will be considered and the global error computed numerically.
    VL  - 5
    IS  - 2
    ER  - 

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Author Information
  • Department of Mathematics, University College of Qunfudha, Umm Alqura University, Makkah, KSA

  • Engineering College, Albaha University, Albaha, KSA

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