On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

Spencer, Billie F.; Bergman, Lawrence A.

doi:10.1007/bf00120671

Cited by 247 publications

(101 citation statements)

References 26 publications

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“…With the periodic force taken into account, certain difficulties arise due to the time inhomogeneity of the corresponding stochastic process. Many results regarding FEM application on FP equation analysis can be found in [32] or [33]. For the most recent results concerning FEM application to SR problem, see [31], and additional details together with demonstrating examples, see [34].…”

Section: Methods Of Stochastic Resonance Investigationmentioning

confidence: 99%

Stochastic Resonance and Related Topics

Náprstek¹,

Fischer²

2017

Resonance

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The stochastic resonance (SR) is the phenomenon which can emerge in nonlinear dynamic systems. In general, it is related with a bistable nonlinear system of Duffing type under additive excitation combining deterministic periodic force and Gaussian white noise. It manifests as a stable quasiperiodic interwell hoppingbetweenbothstablestateswithasmall random perturbation. Classical definition and basic features of SR are regarded. The most important methods of investigation outlined are: analytical, semi-analytical, and numerical procedures of governing physical systems or relevant Fokker-Planck equation. Stochastic simulation is mentioned and experimental way of results verification is recommended. Some areas in Engineering Dynamics related with SR are presented together with a particular demonstration observed in the aeroelastic stability. Interaction of stationary and quasiperiodic parts of the response is discussed. Some nonconventional definitions are outlined concerning alternative operators and driving processes are highlighted. The chapter shows a large potential of specific basic, applied and industrial research in SR. This strategy enables to formulate new ideas for both development of nonconventional measures for vibration damping and employment of SR in branches, where it represents an operating mode of the system itself. Weaknesses and empty areas where the research effort of SR should be oriented are indicated.

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Section: Methods Of Stochastic Resonance Investigationmentioning

confidence: 99%

Stochastic Resonance and Related Topics

Náprstek¹,

Fischer²

2017

Resonance

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“…Its numerical solution is known to be computationally demanding especially in high dimensionalities and a number of solution procedures have been applied in the past. Traditionally it has been solved using the finite difference or finite element method [7,8,9], while a few researchers have also applied the path integration method [10,11] as well as a multiscale finite element approach [12]. Only lately researchers have used a radial basis function (RBF) approach [13,14] as well as a hybrid spectral-finite difference method [15].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Meshless Solution for the Fokker-Planck-Kolmogorov Equations of Probabilistic Elastoplasticity

Karapiperis¹,

Sett²,

Jeremić³

2015

Proceedings of the 1st International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. This paper describes the application of a numerical method for the solution of the nonlinear Fokker-Planck-Kolmogorov (FPK) equations of Probabilistic Elastoplasticity. We employ both a collocation and a Galerkin approach that utilize radial basis functions (RBFs) of various types, while time integration is achieved with the Crank-Nicolson scheme. The efficiency of the solution is demonstrated through simple linear elastic and elastic-perfectly plastic examples of various dimensionalities. In addition, we briefly address the accuracy and stability of the method with respect to the choice of RBF and the associated shape parameter. Finally, we provide a comparison with conventional solutions of the FPK equation to indicate the superiority of the approach. 273

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“…In further attempts, the finite element (FE) method has also been used [8]. This method allows to solve the unsteady equation or immediately the stationary equation through an eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, the finite element method encounters difficulties in modelling the far-field boundary conditions. To overcome this problem, Spencer [8] proposes to mesh a sufficiently large domain. In any case, the positivity of the pdf is not properly ensured: if elements are too large or the meshed domain too small some spurious waves can propagate through the state space and spoil the quality of the solution.…”

Section: Introductionmentioning

confidence: 99%

Transient Fokker-Planck Equation With SPH Method - Application in Seismic Design

Canor¹,

Denoël²

2014

Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. In reliability theory, the knowledge of the time evolution of the probability density function (pdf) of the response of a diffusive random system may be required. The time evolution of this pdf is described by the transient Fokker-Planck-Kolmogorov (FPK) Numerical implementation shows notable advantages of SPH method for solving FPK equation in an unbounded state-space compared with mesh-based method : (i) the conservation of total probability is explicitly written, (ii) no artefact is required in low density zones, (iii) the positivity of the pdf is ensured, (iv) the formalism offers straightforward extension to higher dimension systems, (v) the method is adapted for a large kind of initial conditions, even slightly dispersed distributions.In

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On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

Cited by 247 publications

References 26 publications

Stochastic Resonance and Related Topics

Stochastic Resonance and Related Topics

An Efficient Meshless Solution for the Fokker-Planck-Kolmogorov Equations of Probabilistic Elastoplasticity

Transient Fokker-Planck Equation With SPH Method - Application in Seismic Design

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