EXPONENTIAL–POLYNOMIAL CLOSURE METHOD FOR SOLVING TRUNCATED KOLMOGOROV–FELLER EQUATION

2012 ◽  
Vol 09 (01) ◽  
pp. 1240018 ◽  
Author(s):  
H. T. ZHU ◽  
G. K. ER ◽  
V. P. IU ◽  
K. P. KOU
Keyword(s):  
Numerical Analysis ◽  
Probability Density Function ◽  
Nonlinear Systems ◽  
Duffing Oscillator ◽  
Exponential Polynomial ◽  
Impulse Excitation ◽  
Non Gaussian ◽  
Polynomial Closure ◽  
Closure Method ◽  
Probability Density Function Pdf

The probability density function (PDF) solution of the response is formulated for nonlinear systems under discrete Poisson impulse excitation. The PDF solution is governed by the Kolmogorov–Feller (KF) equation, which is approximately solved by the exponential–polynomial closure (EPC) method. A Duffing oscillator is further investigated in the case of either Gaussian or non-Gaussian distributed amplitude of Poisson impulse to show the effectiveness of the EPC method in these cases. The numerical analysis shows that the EPC method with the polynomial order being 6 presents a good result compared with the simulated result, even in the tails of the PDF of the oscillator response.

Probabilistic Solutions of the Stochastic Oscillators With Even Nonlinearity in Displacement

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Guo-Kang Er ◽  
Siu-Siu Guo ◽  
Vai Pan Iu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Results ◽  
Exponential Polynomial ◽  
The Mean ◽  
Polynomial Closure ◽  
Stochastic Oscillators ◽  
Closure Method

The probabilistic solutions of the nonlinear stochastic oscillators with even nonlinearity in displacement are investigated with the exponential-polynomial closure method. Numerical results show that the results obtained from the exponential-polynomial closure method agree well with the simulated solution in the presented case, even if the mean of displacement is nonzero and the probability density function of the displacement is nonsymmetric about its mean.

Transient probabilistic solutions of stochastic oscillator with even nonlinearities by exponential polynomial closure method

2021 ◽  
pp. 107754632098777
Author(s):  
Kun Wang ◽  
Zhihui Zhu ◽  
Lei Xu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Kolmogorov Equation ◽  
Exponential Polynomial ◽  
Stochastic Oscillator ◽  
Polynomial Closure ◽  
Closure Method ◽  
Transient Probability Density ◽  
Transient Probability

The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.

Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 179-185 ◽  
Author(s):  
R. J. Chang ◽  
G. E. Young
Keyword(s):  
Nonlinear Systems ◽  
Goodness Of Fit ◽  
Duffing Oscillator ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Statistical Linearization ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Non Gaussian ◽  
Parametric And External Excitations

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

scholarly journals Analysis of FM demodulator output noise with applications to FM telemetry

2006 ◽  
Vol 2006 ◽  
pp. 1-13 ◽  
Author(s):  
Rajendra Kumar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Noise Power ◽  
Signal To Noise ◽  
Output Noise ◽  
Simplified Analysis ◽  
Modulation Schemes ◽  
Non Gaussian ◽  
Probability Density Function Pdf

We present an analysis for evaluating the probability density function (pdf) of the noise at the output of the frequency demodulator. It is shown that the noise is non-Gaussian and that for low to medium signal-to-noise power ratios, its pdf differs very significantly from the Gaussian pdf commonly assumed in simplified analysis. These results are very important for analyzing the performance of the PCM/FM type of modulation schemes used in telemetry systems as illustrated in the paper.

Modified Path Integral Solution of Fokker–Planck Equation: Response and Bifurcation of Nonlinear Systems

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Path Integral ◽  
Duffing Oscillator ◽  
Planck Equation ◽  
Harmonic Excitation ◽  
Integration Scheme ◽  
Pi Method ◽  
Non Gaussian ◽  
Fokker Planck

Response of nonlinear systems subjected to harmonic, parametric, and random excitations is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or colored noise excitation (delta correlated) is governed by both the forward Fokker–Planck (FP) and the backward Kolmogorov equations. This paper presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric, and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss–Legendre integration scheme. The effects of white noise intensity, amplitude, and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviors of a hardening Duffing oscillator are also investigated.

Mathematical modeling of conversion of non-Gaussian random processes, signals and noise in linear and nonlinear systems

2018 ◽  
pp. 66-78
Author(s):  
V. M. Artyushenko ◽  
V. I. Volovach
Keyword(s):  
Mathematical Modeling ◽  
Nonlinear Systems ◽  
Random Processes ◽  
Mathematical Transformation ◽  
Positive And Negative Feedback ◽  
Non Linear Systems ◽  
Linear And Nonlinear ◽  
Non Gaussian ◽  
Linear And Nonlinear Systems ◽  
Inertial Systems

The questions connected with mathematical modeling of transformation of non-Gaussian random processes, signals and noise in linear and nonlinear systems are considered and analyzed. The mathematical transformation of random processes in linear inertial systems consisting of both series and parallel connected links, as well as positive and negative feedback is analyzed. The mathematical transformation of random processes with polygamous density of probability distribution during their passage through such systems is considered. Nonlinear inertial and non-linear systems are analyzed.

A Study of the NOx Emission in a Regenerative Industrial Furnace

2001 ◽  
Author(s):  
Qing Jiang ◽  
Chao Zhang
Keyword(s):  
Mixture Fraction ◽  
Combustion Process ◽  
Nox Emission ◽  
Moment Closure ◽  
Combustion Model ◽  
Industrial Furnace ◽  
Low Emission ◽  
Reduction Of Nox ◽  
Closure Method ◽  
Probability Density Function Pdf

Abstract A study of the nitrogen oxides (NOx) emission and combustion process in a gas-fired regenerative, high temperature, low emission industrial furnace has been carried out numerically. The effect of two additives, methanol (CH3OH) and hydrogen peroxide (H2O2), to fuel on the NOx emission has been studied. A moment closure method with the assumed β probability density function (PDF) for mixture fraction is used in the present work to model the turbulent non-premixed combustion process in the furnace. The combustion model is based on the assumption of instantaneous full chemical equilibrium. The results showed that CH3OH is effective in the reduction of NOx in a regenerative industrial furnace. However, H2O2 has no significant effect on the NOx emission.

scholarly journals Dynamic Reliability Analysis Method of Degraded Mechanical Components Based on Process Probability Density Function of Stress

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Peng Gao ◽  
Liyang Xie
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Solar Array ◽  
Dynamic Reliability ◽  
Reliability Models ◽  
Practical Engineering ◽  
Mechanical Components ◽  
Reliability Computation ◽  
Probability Density Function Pdf

It is necessary to develop dynamic reliability models when considering strength degradation of mechanical components. Instant probability density function (IPDF) of stress and process probability density function (PPDF) of stress, which are obtained via different statistical methods, are defined, respectively. In practical engineering, the probability density function (PDF) for the usage of mechanical components is mostly PPDF, such as the PDF acquired via the rain flow counting method. For the convenience of application, IPDF is always approximated by PPDF when using the existing dynamic reliability models. However, it may cause errors in the reliability calculation due to the approximation of IPDF by PPDF. Therefore, dynamic reliability models directly based on PPDF of stress are developed in this paper. Furthermore, the proposed models can be used for reliability assessment in the case of small amount of stress process samples by employing the fuzzy set theory. In addition, the mechanical components in solar array of satellites are chosen as representative examples to illustrate the proposed models. The results show that errors are caused because of the approximation of IPDF by PPDF and the proposed models are accurate in the reliability computation.

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