The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension

Wang Wenjie; Feng Jianhu; Xu Wei

doi:doi:10.11648/j.acm.20160503.20

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The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension

Wang Wenjie, Feng Jianhu, Xu Wei

Published in Applied and Computational Mathematics (Volume 5, Issue 3)

Received: 19 June 2016 Accepted: 27 June 2016 Published: 23 July 2016

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Abstract

Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.

Published in	Applied and Computational Mathematics (Volume 5, Issue 3)
DOI	10.11648/j.acm.20160503.20
Page(s)	160-164
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

Nonlinear System, FPK Equations, The Finite Difference Method, The TVD Runge-Kutta Scheme, The ENO Scheme, The WENO Scheme

References

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[2]	Zhu Weiqiu. Nonlinear Stochastic Dynamics and Controls---Frame of Hamilton Theory, 2003. (Beijing: Science Press) (in Chinese).
[3]	Zhao Chaoying, Tan Weihan and Guo Qizhi. The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light, 2003 Acta Phys. Sin. 52 2694 (in Chinese).
[4]	Wang Ping, Yang Xine and Song X iaohui. Exact solution for a harmonic oscillator with a time-dependent inverse square potential by path-integral, 2003 Acta Phys. Sin. 52 2957 (in Chinese).
[5]	Xu Wei, He Qiu, Rong Haiwu and Fang Tong. Global analysis of stochastic bifurcation in a Duffing-van der Pol system, 2003 Acta Phys. Sin. 52 1365 (in Chinese).
[6]	Sun Zhongkui, Xu Wei and Yang Xiaoli. A new analytic approximate technique for strongly nonlinear dynamic systems. Journal of Dynamics and Control, 2005, 2 (3): 29-35 (in Chinese).
[7]	Liu Ruxun, Shu Qiwang. Some new methods in computational fluid dynamics [M]. Beijing: Science Press, 2003: 42-106.
[8]	Zhang Senwen. The solution of nonlinear staionary FPK equation using wavelet method [J]. Journal of Jinan University, 2002, 1: 29-33.(in Chinese).
[9]	Li Likang, Yu Chonghua, Zhu Zhenghua. Numerical method for Partial Differential Equation [M]. Shanghai: Fudan University Press, 1999. (in Chiinese).
[10]	Ye Dayi, Li Qingyang. Numerical method [M]. Beijing: High Education Press. (in Chinese).

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Wang Wenjie, Feng Jianhu, Xu Wei. (2016). The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Applied and Computational Mathematics, 5(3), 160-164. https://doi.org/10.11648/j.acm.20160503.20

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ACS Style

Wang Wenjie; Feng Jianhu; Xu Wei. The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Appl. Comput. Math. 2016, 5(3), 160-164. doi: 10.11648/j.acm.20160503.20

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AMA Style

Wang Wenjie, Feng Jianhu, Xu Wei. The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Appl Comput Math. 2016;5(3):160-164. doi: 10.11648/j.acm.20160503.20

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@article{10.11648/j.acm.20160503.20,
  author = {Wang Wenjie and Feng Jianhu and Xu Wei},
  title = {The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {3},
  pages = {160-164},
  doi = {10.11648/j.acm.20160503.20},
  url = {https://doi.org/10.11648/j.acm.20160503.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.20},
  abstract = {Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.},
 year = {2016}
}

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TY  - JOUR
T1  - The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension
AU  - Wang Wenjie
AU  - Feng Jianhu
AU  - Xu Wei
Y1  - 2016/07/23
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160503.20
DO  - 10.11648/j.acm.20160503.20
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 160
EP  - 164
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160503.20
AB  - Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.
VL  - 5
IS  - 3
ER  -

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Author Information

Wang Wenjie

School of Science of Chang’an University, Xi’an, China
Feng Jianhu

School of Science of Chang’an University, Xi’an, China
Xu Wei

School of Science of Northwestern Polytechnic University, Xi’an, China

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Plain Text BibTeX RIS

APA Style

Wang Wenjie, Feng Jianhu, Xu Wei. (2016). The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Applied and Computational Mathematics, 5(3), 160-164. https://doi.org/10.11648/j.acm.20160503.20

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ACS Style

Wang Wenjie; Feng Jianhu; Xu Wei. The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Appl. Comput. Math. 2016, 5(3), 160-164. doi: 10.11648/j.acm.20160503.20

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AMA Style

Wang Wenjie, Feng Jianhu, Xu Wei. The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension. Appl Comput Math. 2016;5(3):160-164. doi: 10.11648/j.acm.20160503.20

Copy | Download

@article{10.11648/j.acm.20160503.20,
  author = {Wang Wenjie and Feng Jianhu and Xu Wei},
  title = {The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {3},
  pages = {160-164},
  doi = {10.11648/j.acm.20160503.20},
  url = {https://doi.org/10.11648/j.acm.20160503.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160503.20},
  abstract = {Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - The Numerical Solution of the TVD Runge-Kutta and WENO Scheme to the FPK Equations to Nonlinear System of One-Dimension
AU  - Wang Wenjie
AU  - Feng Jianhu
AU  - Xu Wei
Y1  - 2016/07/23
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160503.20
DO  - 10.11648/j.acm.20160503.20
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 160
EP  - 164
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160503.20
AB  - Firstly, it was studied to the Fokker-Planck-Kolmogorov (FPK) equations for nonlinear stochastic dynamic system. Secondly, it was discussed to the third-order TVD Runge-Kutta difference scheme totime for differitial equations and the fifth-order WENO scheme for differitial operators. And combined he third-order TVD Runge-Kutta difference scheme with the fifth-order WENO scheme, obtained the numerical solution for FPK equations using the TVD Runge-Kutta WENO scheme. Finally, the numerical solution was compared with the analytic solution for FPK equations. The numerical method is shown to give accurate results and overcomes the difficulties of other methods, such as: the big value of probability density function at tail etc.
VL  - 5
IS  - 3
ER  -

Copy | Download