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The numericao solution of the TVD Runge-Kutta and the WENO scheme to the FPK equations to nonlinear dynamic system with random excitation and Gauss white noise excitation are discussing.
The response and reliability of system is important, which can be analyzed by using the transition probability density of system. However, at present, only special nonliear dynamical systems can get the exact solution.
Many scholars dedicate to study of the numerical solution of FPK equation and obtain many methods. For example, the main finite element method and finite difference method, the path integral method, equivalent linearization method, Gaussian closure method, perturbation method, the gram Charlier expansion method, equivalent nonlinear system method, stochastic averaging method. However, these methods have their shortcoming: the Calculation quantity of the finite element method is usually very large, and the tail probability density is not accurate; equivalent linearization method and Gauss's method is not applicable to strongly nonlinear systems or systems with random parametric excitation, for the steady-state probability density of the system response is often non Gauss type; perturbation method is only applicable to weakly nonlinear systems; the Gram-Charlier expansion method may lead to the case that the probability density is negative; equivalent nonlinear system method requires that the properties of the two nonlinear systems are very close; the stochastic averaging method is only applicable to the case of weak damping and weak excitation.
The author of this paper proposes a new method, which combines the TVD Runge-Kutta scheme with the WENO scheme. Discretizates the FPK equation to time using the TVD Runge-Kutta method and to space using the WENO method. And the new method has high order accuracy and non oscillatory. Numerical example shows that the new method can obtains more accurately the probability density function.
Author:WenJie Wang, Lecturer, school of science of Chang’An university, Xi’An, China. Research direction: the numerical solution of FPK equation for nonlinear stochastic dynamical systems. Feng Jianhu, Professor, school of science of Chang’An university, Xi’An, China. Research direction: the entropy compatible solutions of the hyperbolic Conservation laws equations and the high resolution scheme. Xu Wei, Professor, school of science of Northwestern Polytechnical University, Xi’An, China. Research direction: Nonlinear stochastic dynamical systems.