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.

Effects of the Fractional Laplacian Order on the Nonlocal Elastic Rod Response

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Giuseppina Autuori ◽  
Federico Cluni ◽  
Vittorio Gusella ◽  
Patrizia Pucci
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Coefficient Of Variation ◽  
Fractional Laplacian ◽  
Numerical Procedure ◽  
Elastic Rod ◽  
The Mean ◽  
Distributed Forces ◽  
Nonlinear Fashion

In this paper, we yield with a nonlocal elastic rod problem, widely studied in the last decades. The main purpose of the paper is to investigate the effects of the statistic variability of the fractional operator order s on the displacements u of the rod. The rod is supposed to be subjected to external distributed forces, and the displacement field u is obtained by means of numerical procedure. The attention is particularly focused on the parameter s, which influences the response in a nonlinear fashion. The effects of the uncertainty of s on the response at different locations of the rod are investigated by the Monte Carlo simulations. The results obtained highlight the importance of s in the probabilistic feature of the response. In particular, it is found that for a small coefficient of variation of s, the probability density function of the response has a unique well-identifiable mode. On the other hand, for a high coefficient of variation of s, the probability density function of the response decreases monotonically. Finally, the coefficient of variation and, to a small extent, the mean of the response tend to increase as the coefficient of variation of s increases.

A New Microcontact Model Developed for Variable Fractal Dimension, Topothesy, Density of Asperity, and Probability Density Function of Asperity Heights

2006 ◽  
Vol 74 (4) ◽  
pp. 603-613 ◽  
Author(s):  
Jeng Luen Liou ◽  
Jen Fin Lin
Keyword(s):  
Fractal Dimension ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Gaussian Distribution ◽  
Fractal Theory ◽  
Radius Of Curvature ◽  
The Mean ◽  
Contact Surfaces ◽  
Non Gaussian

In the present study, the fractal theory is applied to modify the conventional model (the Greenwood and Williamson model) established in the statistical form for the microcontacts of two contact surfaces. The mean radius of curvature (R) and the density of asperities (η) are no longer taken as constants, but taken as variables as functions of the related parameters including the fractal dimension (D), the topothesy (G), and the mean separation of two contact surfaces. The fractal dimension and the topothesy varied by differing the mean separation of two contact surfaces are completely obtained from the theoretical model. Then the mean radius of curvature and the density of asperities are also varied by differing the mean separation. A numerical scheme is thus developed to determine the convergent values of the fractal dimension and topothesy corresponding to a given mean separation. The topographies of a surface obtained from the theoretical prediction of different separations show the probability density function of asperity heights to be no longer the Gaussian distribution. Both the fractal dimension and the topothesy are elevated by increasing the mean separation. The density of asperities is reduced by decreasing the mean separation. The contact load and the total contact area results predicted by variable D, G*, and η as well as non-Gaussian distribution are always higher than those forecast with constant D, G*, η, and Gaussian distribution.

Bifurcation Analysis of an Energy Harvesting System with Fractional Order Damping Driven by Colored Noise

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Yong-Ge Yang ◽  
Ya-Hui Sun ◽  
Wei Xu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Fractional Order ◽  
Colored Noise ◽  
Energy Harvester ◽  
Advanced Materials ◽  
Vibration Energy ◽  
Vibration Energy Harvester ◽  
The Mean

Vibration energy harvester, which can convert mechanical energy to electrical energy so as to achieve self-powered micro-electromechanical systems (MEMS), has received extensive attention. In order to improve the efficiency of vibration energy harvesters, many approaches, including the use of advanced materials and stochastic loading, have been adopted. As the viscoelastic property of advanced materials can be well described by fractional calculus, it is necessary to further discuss the dynamical behavior of the fractional-order vibration energy harvester. In this paper, the stochastic P-bifurcation of a fractional-order vibration energy harvester subjected to colored noise is investigated. Variable transformation is utilized to obtain the approximate equivalent system. Probability density function for the amplitude of the system response is derived via the stochastic averaging method. Numerical results are presented to verify the proposed method. Critical conditions for stochastic P-bifurcation are provided according to the change of the peak number for the probability density function. Then bifurcation diagrams in the parameter planes are analyzed. The influences of parameters in the system on the mean harvested power are discussed. It is found that the mean harvested power increases with the enhancement of the noise intensity, while it decreases with the increase of the fractional order and the correlation time.

scholarly journals Mixing and combustion at low heat release in large eddy simulations of a reacting shear layer

2021 ◽  
Author(s):  
J. X. Huang ◽  
W. A. McMullan
Keyword(s):  
Probability Density Function ◽  
Heat Release ◽  
Shear Layer ◽  
Probability Density ◽  
Density Function ◽  
Temperature Rise ◽  
Mixing Layer ◽  
Large Eddy ◽  
Inflow Condition ◽  
The Mean

AbstractIn this paper, the mixing and combustion at low-heat release in a turbulent mixing layer are studied numerically using large eddy simulation. The primary aim of this paper is to successfully replicate the flow physics observed in experiments of low-heat release reacting mixing layers, where a duty cycle of hot structures and cool braid regions was observed. The nature of the imposed inflow condition shows a dramatic influence on the mechanisms governing entrainment, and mixing, in the shear layer. An inflow condition perturbed by Gaussian white noise produces a shear layer which entrains fluid through a nibbling mechanism, which has a marching scalar probability density function where the most probable scalar value varies across the layer, and where the mean-temperature rise is substantially over-predicted. A more sophisticated inflow condition produced by a recycling and rescaling method results in a shear layer which entrains fluid through an engulfment mechanism, which has a non-marching scalar probability density function where a preferred scalar concentration is present across the thickness of the layer, and where the mean-temperature rise is predicted to a good degree of accuracy. The latter simulation type replicates all of the flow physics observed in the experiment. Extensive testing of subgrid-scale models, and simple combustion models, shows that the WALE model coupled with the Steady Laminar Flamelet model produces reliable predictions of mixing layer diffusion flames undergoing with fast chemistry.

Fungsi Densitas Peluang Temporal (T-PDF) dari Dinamika Euro terhadap Dolar Amerika Serikat

2019 ◽  
Vol 3 (1) ◽  
pp. 17
Author(s):  
Triyana Muliawati ◽  
. Kartono ◽  
Edi Cahyono
Keyword(s):  
Exchange Rate ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Global Economy ◽  
Global Financial Crisis ◽  
Polynomial Interpolation ◽  
United States Dollar ◽  
The Mean ◽  
The Exchange Rate

In this paper we discuss the dynamics of exchange rate of Euro (EUR) relative to United States dollar (USD). The data is observed from daily data in the period 1 January 2005 to 31 December 2012. In this period of global financial crisis which is affected the global economy. Therefore, an understanding of the exchange rate of EUR relative to USD is required. The Data is presented in the form of a diagram of a candlestick (candle). Statistical analysis on a mean of exchange rate of EUR relative to USD is applied to each monthly candle representation. The mean may vary per candle, which means that the mean as a function of time. A function of mean relative to time so commonly referred to as a moving average, or a trend. The continous of a trend is approached with linear interpolation and  polynomial interpolation is based on the mean of the candle every month. The mean and standard deviation of exchange rate of EUR relative to USD generates a probability density function (pdf). A pdf is based on the assumption that the dynamics is normally distributed. The mean is dependent on time is called temporal probability density function (t-pdf). The trend of the dynamics of exchange rate of  EUR relative to USD is implicitly represented in the t-pdf. By knowing t-pdf will help investors know the dynamics of the exchange rate of the EUR against the USD. Keywords: dynamics, exchange rate, candlestick, t-pdf

scholarly journals On the theory of advective effects on biological dynamics in the sea. III. The role of turbulence in biological–physical interactions

2007 ◽  
Vol 464 (2091) ◽  
pp. 555-572 ◽  
Author(s):  
Louis Goodman ◽  
Allan R Robinson
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Euphotic Zone ◽  
Biological Interaction ◽  
State Variables ◽  
Turbulent Layer ◽  
Wide Range ◽  
The Mean ◽  
Interaction Term

A nonlinear model for biological and physical dynamical interactions in a laminar upwelling flow field in parts I and II of this study is extended to turbulent flow. In the previous studies, a prescription for obtaining quadrature solutions to the fundamental biodynamical equations was developed. In this study, we use a probability density function approach on these solutions to obtain statistics of the biodynamical state variables and their self-interaction for the case of turbulent advection. To illustrate the theory, a simple nutrient ( N ), phytoplankton ( P ) problem is considered, that of upwelling into a surface turbulent layer. Biological interaction is modelled as bilinear, representing the uptake of N by P in a uniform light euphotic zone. A random walk model is used to obtain the appropriate probability density function for the advective turbulent field. The mean quantities, , , as well as the biological interaction term are calculated. The term has two contributions, , and the turbulence-induced interaction term, . It is shown that the often neglected turbulence-induced coupling term is of the order and opposite in sign. This results in, over a wide range of Peclet numbers, the mean interaction term being significantly smaller than either of its constituent terms, and .

scholarly journals The Probability Distribution of Sea Surface Wind Speeds. Part I: Theory and SeaWinds Observations

2006 ◽  
Vol 19 (4) ◽  
pp. 497-520 ◽  
Author(s):  
Adam Hugh Monahan
Keyword(s):  
Standard Deviation ◽  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Surface Wind ◽  
Sea Surface ◽  
Wind Speeds ◽  
Sea Surface Wind ◽  
The Mean

Abstract The probability distribution of sea surface wind speeds, w, is considered. Daily SeaWinds scatterometer observations are used for the characterization of the moments of sea surface winds on a global scale. These observations confirm the results of earlier studies, which found that the two-parameter Weibull distribution provides a good (but not perfect) approximation to the probability density function of w. In particular, the observed and Weibull probability distributions share the feature that the skewness of w is a concave upward function of the ratio of the mean of w to its standard deviation. The skewness of w is positive where the ratio is relatively small (such as over the extratropical Northern Hemisphere), the skewness is close to zero where the ratio is intermediate (such as the Southern Ocean), and the skewness is negative where the ratio is relatively large (such as the equatorward flank of the subtropical highs). An analytic expression for the probability density function of w, derived from a simple stochastic model of the atmospheric boundary layer, is shown to be in good qualitative agreement with the observed relationships between the moments of w. Empirical expressions for the probability distribution of w in terms of the mean and standard deviation of the vector wind are derived using Gram–Charlier expansions of the joint distribution of the sea surface wind vector components. The significance of these distributions for improvements to calculations of averaged air–sea fluxes in diagnostic and modeling studies is discussed.

On Estimation of a Functional of a Probability Density Function

1993 ◽  
Vol 43 (1-2) ◽  
pp. 13-24
Author(s):  
L. O. Odongo ◽  
M. Samanta
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Mean Square Error ◽  
Regularity Conditions ◽  
Mean Square ◽  
Simulation Studies ◽  
Kernel Estimate ◽  
Primary 62G07 ◽  
The Mean

The problem of estimating the integral of the square of a probability density function is considered, It is shown that under some regularity conditions the kernel estimate of this functional is asymptotically normally distributed. An expression for the smoothing parameter that minimizes the mean square error of the estimate is derived. Results of simulation studies are included. AMS (1980) Subject Classification: Primary 62G07 Secondary 60FOS.

Sign in / Sign up

Export Citation Format

Share Document