Investigation of error in evaluation of spectral statistical characteristics of numerical solution of stochastic dynamic system

In this paper, an algorithm for numerical simulations is developed for calculating a discrete dynamic system with a stochastic perturbation and an analysis of the quality of numerical solutions is carried out. For this, an algorithm for the numerical solution of a second-order differential equation with a stochastic right-hand side was developed and this algorithm was implemented as a program. The next step was to carry out a set of computational studies by varying the parameters of numerical integration with the subsequent assessment of their impact on the error and accuracy of simulations. To estimate the spectral density, the Welch periodogram method was used. To check the quality of simulations and assess the accuracy of solutions, it is proposed to compare the results of numerical integration and subsequent digital processing with analytical solutions that are known for the linear problem, given by the equation. As a result of the work, a comparative analysis of the dispersion of displacements relative to the lengths of signals from a different number of blocks was carried out, into which the signal is divided for the Welch method; the confidence interval of the error at different signal lengths and the confidence interval of the error with a different number of blocks at a certain signal length. Comparison of the variance with a different number of blocks showed that with a signal length of 30 s and from 90 s, there is a slight scatter of the variance values within an error of ± 5%.

Dynamics of a Rumor Propagation Model with Stochastic Perturbation On Homogeneous Social Networks

The rapid development of information society highlights the important role of rumors in social communication, and its propagation has a significant impact on human production and life. The investigation of the influence of uncertainty on rumor propagation is an important issue in the current communication study. Due to incomprehension about others and the stochastic properties of the users' behavior, the transmission rate between individuals on social network platforms is usually not a constant value. In this paper, we propose a new rumor propagation model on homogeneous social networks from the deterministic structure to the stochastic structure. Firstly, a unique global positive solution of rumor propagation model is obtained. Then, we verify that the extinction and persistence of stochastic rumor propagation model are restricted by some conditions. IfR *0< 1 and the noise intensity s i (i = 1,2,3) satisfies some certain conditions, rumors will extinct with a probability one. If R *0 > 1, rumor-spreading individuals will continue to exist in the system, which means the rumor will prevail for a long time. Finally, through some numerical simulations, the validity and rationality of the theoretical analysis are effectively verified.

Fractal–fractional and stochastic analysis of norovirus transmission epidemic model with vaccination effects

In this paper, we investigate an norovirus (NoV) epidemic model with stochastic perturbation and the new definition of a nonlocal fractal–fractional derivative in the Atangana–Baleanu–Caputo (ABC) sense. First we present some basic properties including equilibria and the basic reproduction number of the model. Further, we analyze that the proposed stochastic system has a unique global positive solution. Next, the sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. Furthermore, the fractal–fractional dynamics of the proposed model under Atangana–Baleanu–Caputo (ABC) derivative of fractional order “$${p}$$ p ” and fractal dimension “$${q}$$ q ” have also been addressed. Besides, coupling the non-linear functional analysis with fixed point theory, the qualitative analysis of the proposed model has been performed. The numerical simulations are carried out to demonstrate the analytical results. It is believed that this study will comprehensively strengthen the theoretical basis for comprehending the dynamics of the worldwide contagious diseases.

Stationary solutions in thermodynamics of stochastically forced fluids

We study the full Navier–Stokes–Fourier system governing the motion of a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbation. The system is supplemented with non-homogeneous Neumann boundary conditions for the temperature and hence energetically open. We show that, in contrast with the energetically closed system, there exists a stationary solution. Our approach is based on new global-in-time estimates which rely on the non-homogeneous boundary conditions combined with estimates for the pressure.

Statistical analyses of the energy demand and thermal comfort for multiple uncertain input parameters, performed using transformed variable and perturbation method

In the calculations of buildings’ thermal comfort, the input parameters are usually considered as strictly determined values. Numerous of them may be characterized by certain probability density functions. In the energy related problems, the uncertainty analyses are usually performed using the Monte Carlo method. However, this method requires multiple calculations and, therefore, may be very time-consuming. In the proposed work, two approaches are applied for the probabilistic studies: the stochastic perturbation method and the transformed random variables method. The stochastic analysis is based on the response functions and their derivatives with respect to all random input parameters. The relation between the thermal comfort and the input (random) variables have been calculated using the Energy Plus software. Afterwards, the response functions were estimated using the polynomial regression. The expected value and central moments of the response functions were calculated by means of the perturbation method and the transformed random variable theorem. The latter method allowed to obtain, using the same response functions, the implicit form of probability distributions function of the output parameter.