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 |
Riccati Equation, Symmetry Group, Infinitesimal Generator, Runge-Kutta
| [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. |
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
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
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
Share This Article
Twitter
Linked In
Facebook
Copy
Download@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}
}
TY - JOUR T1 - Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation AU - Sami H. Altoum AU - Salih Y. Arbab Y1 - 2016/04/15 PY - 2016 N1 - https://doi.org/10.11648/j.acm.20160502.15 DO - 10.11648/j.acm.20160502.15 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 64 EP - 72 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20160502.15 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 -
Email: