Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems. (English) Zbl 1154.65361
Summary: The response of a structural system to white noise excitation (delta-correlated) constitutes a Markov vector process whose transitional probability density function (TPDF) is governed by both the forward Fokker-Planck and backward Kolmogorov equations. The numerical solution of these equations by finite element and finite difference methods for dynamical systems of engineering interest has been hindered by the problem of dimensionality. In this paper, numerical solution of the stationary and transient form of the Fokker-Planck (FP) equation corresponding to two state nonlinear systems is obtained by the standard sequential finite element method (FEM) using \(C^0\) shape functions and the Crank-Nicholson time integration scheme. The method is applied to Van-der-Pol and Duffing oscillators providing good agreement between results obtained by it and exact results. An extension of the finite difference discretization scheme developed by B. F. Spencer jun., L. A. Bergman and S. F. Wojtkiewicz [High fidelity numerical solutions of the Fokker-Planck equation, Proc. ICOSSAR’97, 7th Int. Conf. on Structural Safety and Reliability, Kyoto, Japan (1994)] is also presented. This paper presents an extension of the finite difference method for the solution of FP equation up to four dimensions. The difficulties associated in extending these methods to higher dimensional systems are discussed.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 35K55 | Nonlinear parabolic equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
Keywords:
Fokker-Planck equation; finite element method; finite difference method; random vibration; nonlinear stochastic dynamics; Kolmogorov equations; Crank-Nicholson time integration; Van-der-Pol and Duffing oscillatorsReferences:
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