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
Volume 5, Issue 3, June 2016, Pages: 160-164
Received: Jun. 19, 2016;
Accepted: Jun. 27, 2016;
Published: Jul. 23, 2016
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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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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.
Nonlinear System, FPK Equations, The Finite Difference Method, The TVD Runge-Kutta Scheme, The ENO Scheme, The WENO Scheme
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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, Applied and Computational Mathematics.
Vol. 5, No. 3,
2016, pp. 160-164.
Copyright © 2016 Authors retain the copyright of this article.
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Y. K. Lin and G. Q. Cai. Probabilistic Structural Dynamics: Advanced Theory and Applications, 1995. (New York: MeGraw-Hill).
Zhu Weiqiu. Nonlinear Stochastic Dynamics and Controls---Frame of Hamilton Theory, 2003. (Beijing: Science Press) (in Chinese).
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).
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).
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).
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).
Liu Ruxun, Shu Qiwang. Some new methods in computational fluid dynamics [M]. Beijing: Science Press, 2003: 42-106.
Zhang Senwen. The solution of nonlinear staionary FPK equation using wavelet method [J]. Journal of Jinan University, 2002, 1: 29-33.(in Chinese).
Li Likang, Yu Chonghua, Zhu Zhenghua. Numerical method for Partial Differential Equation [M]. Shanghai: Fudan University Press, 1999. (in Chiinese).
Ye Dayi, Li Qingyang. Numerical method [M]. Beijing: High Education Press. (in Chinese).