scholarly journals GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE

2005 ◽  
Vol 05 (02) ◽  
pp. L267-L274 ◽  
Author(s):  
ALEXANDER DUBKOV ◽  
BERNARDO SPAGNOLO
Keyword(s):  
White Noise ◽  
Wiener Process ◽  
Anomalous Diffusion ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Non Gaussian ◽  
Generalized Wiener Process ◽  
Fokker Planck

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

Approximate Solution of the Fokker–Planck Equation for a Multidegree of Freedom Frictionally Damped Bladed Disk Under Random Excitation

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek
Keyword(s):  
Approximate Solution ◽  
Weighting Function ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Random Excitation ◽  
Stationary Response ◽  
Bladed Disk ◽  
Fokker Planck Equation ◽  
Finite Differences Method ◽  
Fokker Planck

Bladed disks are subjected to different types of excitations, which cannot, in any, case be described in a deterministic manner. Fuzzy factors, such as slightly varying airflow or density fluctuation, can lead to an uncertain excitation in terms of amplitude and frequency, which has to be described by random variables. The computation of frictionally damped blades under random excitation becomes highly complex due to the presence of nonlinearities. Only a few publications are dedicated to this particular problem. Most of these deal with systems of only one or two degrees-of-freedom (DOFs) and use computational expensive methods, like finite element method or finite differences method (FDM), to solve the determining differential equation. The stochastic stationary response of a mechanical system is characterized by the joint probability density function (JPDF), which is driven by the Fokker–Planck equation (FPE). Exact stationary solutions of the FPE only exist for a few classes of mechanical systems. This paper presents the application of a semi-analytical Galerkin-type method to a frictionally damped bladed disk under influence of Gaussian white noise (GWN) excitation in order to calculate its stationary response. One of the main difficulties is the selection of a proper initial approximate solution, which is applicable as a weighting function. Comparing the presented results with those from the FDM, Monte–Carlo simulation (MCS) as well as analytical solutions proves the applicability of the methodology.

Anomalous diffusion: Fractional Fokker–Planck equation and its solutions

2003 ◽  
Vol 44 (5) ◽  
pp. 2179-2185 ◽  
Author(s):  
E. K. Lenzi ◽  
R. S. Mendes ◽  
Kwok Sau Fa ◽  
L. C. Malacarne ◽  
L. R. da Silva
Keyword(s):  
Anomalous Diffusion ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

1988 ◽  
Vol 52 (3-4) ◽  
pp. 979-1003 ◽  
Author(s):  
Sandro Faetti ◽  
Leone Fronzoni ◽  
Paolo Grigolini ◽  
Vincenzo Palleschi ◽  
Girolamo Tropiano
Keyword(s):  
Gaussian Noise ◽  
Projection Operator ◽  
Planck Equation ◽  
Operator Approach ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

scholarly journals Lévy anomalous diffusion and fractional Fokker–Planck equation

2000 ◽  
Vol 282 (1-2) ◽  
pp. 13-34 ◽  
Author(s):  
V.V. Yanovsky ◽  
A.V. Chechkin ◽  
D. Schertzer ◽  
A.V. Tur
Keyword(s):  
Anomalous Diffusion ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Description of the macroscopic dynamics of populations of phase elements with white non-Gaussian noise based on the circular cumulant approach

2021 ◽  
pp. 05-12
Author(s):  
A. V. Dolmatova ◽  
◽  
I. V. Tiulkina ◽  
D. S. Goldobin ◽  
◽  
...  
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Gaussian Noise ◽  
Gaussian White Noise ◽  
Macroscopic Description ◽  
Reduced Equations ◽  
Non Gaussian ◽  
Dynamics Of Populations ◽  
Stable Noise ◽  
Good Agreement

We use the method of circular cumulants, which allows us to construct a low-mode macroscopic description of the dynamics of populations of phase elements subject to non-Gaussian white noise. In this work, we have obtained two-cumulant reduced equations for alpha-stable noise. The application of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of numerical calculations for the ensemble of N = 1500 elements, the numericalsimulation of the chain of equations for the Kuramoto–Daido order parameters (Fourier modes of the probability density) with 200 terms (in the thermodynamic limit of an infinitely large ensemble) and the theoretical solution on the basis of the two-cumulant approximation are in good agreement with each other.

Cooperative target localization using multiple UAVs with out-of-sequence measurements

2017 ◽  
Vol 89 (1) ◽  
pp. 112-119 ◽  
Author(s):  
Xiaogang Wang ◽  
Wutao Qin ◽  
Yuliang Bai ◽  
Naigang Cui
Keyword(s):  
Time Delay ◽  
Probability Density ◽  
Gaussian Noise ◽  
Nonlinear Filtering ◽  
Planck Equation ◽  
Target Localization ◽  
Content Type ◽  
Fokker Planck Equation ◽  
Delay Measurement ◽  
Fokker Planck

Purpose The time delay would occurs when the measurements of multiple unmanned aerial vehicles (UAVs) are transmitted to the date processing center during cooperative target localization. This problem is often named as the out-of-sequence measurement (OOSM) problem. This paper aims to present a nonlinear filtering based on solving the Fokker–Planck equation to address the issue of OOSM. Design/methodology/approach According to the arrival time of measurement, the proposed nonlinear filtering can be divided into two parts. The non-delay measurement would be fused in the first part, in which the Fokker–Planck equation is utilized to propagate the conditional probability density function in the forward form. The time delay measurement is fused in the second part, in which the Fokker–Planck is used in the backward form approximately. The Bayes formula is applied in both parts during the measurement update. Findings Under the Bayesian filtering framework, this nonlinear filtering is not only suitable for the Gaussian noise assumption but also for the non-Gaussian noise assumption. The nonlinear filtering is applied to the cooperative target localization problem. Simulation results show that the proposed filtering algorithm is superior to the previous Y algorithm. Practical implications In this paper, the research shows that a better performance can be obtained by fusing multiple UAV measurements and treating time delay in measurement with the proposed algorithm. Originality/value In this paper, the OOSM problem is settled based on solving the Fokker–Planck equation. Generally, the Fokker–Planck equation can be used to predict the probability density forward in time. However, to associate the current state with the state related to OOSM, it would be used to propagate the probability density backward either.

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