In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated. Stability of the procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the method, an Algorithm is presented to solve this stiff problem. Results confirm the validity and the ability of this approach.
| Published in |
Applied and Computational Mathematics (Volume 4, Issue 1-1)
This article belongs to the Special Issue New Orientations in Applied and Computational Mathematics |
| DOI | 10.11648/j.acm.s.2015040101.12 |
| Page(s) | 6-11 |
| 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 |
Implicit Method, Taylor Series, Legendre Orthogonal Polynomial, Van Der Pol Equation, Lyapunov Function
| [1] | Butcher J.C. Numerical Methods for Ordinary Differential Equations. John Wiley,2003. |
| [2] | Frank R, Schneid J, Uberhuber C.W: Order results for implicit Runge-Kutta methods applied to stiff systems. SIAM J. Numer. Anal., 22, 515-534 (1985). |
| [3] | Hairer E, Lubich C, Roche M: Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations. BIT 28, 678-700 (1988). |
| [4] | Hairer E, Lubich C, Roche M: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer verlag (1989). |
| [5] | Hairer. E, Wanner. G & Nørsett S.P. " Solving Ordinary Differential Equations I, nonstiff problems ", Springer Series in Computational Mathematics 14, DOI 10.1007/978-3-642-05221-73, © Springer-Verlag Berlin Heidelberg 2010. |
| [6] | Jain M.J. Numerical Solution of Differential Equations. John Wiley & Sons (Asia) Pte Ltd(1979). |
| [7] | Kalman R. E. & Bertram J. F: "Control System Analysis and Design via the Second Method of Lyapunov", J. Basic Engrgvol.88 1960 pp.371; 394. |
| [8] | Lefschetz.s. Differential equation: Geometric theory, 2nd edition. Interscience, New York, 1963. |
| [9] | Lakshmikantham v, Leela s: Differential and integral inequalities: theory and applications, volume I, Acalemic Press.(1969) |
| [10] | Rama Mahana Rao.M. A note on an integral inequality, J. Indian Math, Soc. 27, 67-69, 1963. |
| [11] | Rama Mohana Rao.M. Ordinary differential equations : theory and applications, London : E. Arnold, 1981, ISBN : 9780713134520. |
APA Style
Jafar Biazar, Meysam Navidyan. (2014). Implicit Runge-Kutta Method for Van Der Pol Problem. Applied and Computational Mathematics, 4(1-1), 6-11. https://doi.org/10.11648/j.acm.s.2015040101.12
ACS Style
Jafar Biazar; Meysam Navidyan. Implicit Runge-Kutta Method for Van Der Pol Problem. Appl. Comput. Math. 2014, 4(1-1), 6-11. doi: 10.11648/j.acm.s.2015040101.12
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Download@article{10.11648/j.acm.s.2015040101.12,
author = {Jafar Biazar and Meysam Navidyan},
title = {Implicit Runge-Kutta Method for Van Der Pol Problem},
journal = {Applied and Computational Mathematics},
volume = {4},
number = {1-1},
pages = {6-11},
doi = {10.11648/j.acm.s.2015040101.12},
url = {https://doi.org/10.11648/j.acm.s.2015040101.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2015040101.12},
abstract = {In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated. Stability of the procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the method, an Algorithm is presented to solve this stiff problem. Results confirm the validity and the ability of this approach.},
year = {2014}
}
TY - JOUR T1 - Implicit Runge-Kutta Method for Van Der Pol Problem AU - Jafar Biazar AU - Meysam Navidyan Y1 - 2014/07/13 PY - 2014 N1 - https://doi.org/10.11648/j.acm.s.2015040101.12 DO - 10.11648/j.acm.s.2015040101.12 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 6 EP - 11 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.s.2015040101.12 AB - In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated. Stability of the procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the method, an Algorithm is presented to solve this stiff problem. Results confirm the validity and the ability of this approach. VL - 4 IS - 1-1 ER -
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