and Computational Methods

doi: 10.18698/2309-3684-2019-4-5068

The article investigates two approaches to the calculation of electric circuits in the conditions of uncertainty in the initial data or the influence of random influences, noise of different nature. The first approach is based on the application of interval and bilateral methods and is to some extent an example of robust methods. The second approach uses the stochastic Fokker-Planck equation and is the most informative. Although the application of the Fokker-Planck equation is limited by some conditions (random effects have the character of white noise or are subject to the Gaussian distribution law). This method has a wide scope of applicability. In turn, there are various methods for solving this differential equation. The article considers the solution of the corresponding problem using the finite difference method.

Examples of modeling of specific electrical and electronic circuits are given.

Арутюнян Р.В., Арутюнян Т.Р. Моделирование и методы расчета электрических цепей с приближёнными характеристиками. Математическое моделирование и численные методы, 2019, № 4, с. 50–68.

doi: 10.18698/2309-3684-2021-2-6885

The article deals with the analysis of nonlinear dynamic and stationary systems based on Volterra integro–functional series and various classes of quadrature formulas. A mathematical model of the input–output type is used, which does not take into account the specific physical nature of the dynamic process, which is commonly called a black box. The methods of the article are applicable to the main variants of the Volterra integral–functional decomposition, including for the case of stationary dynamical systems, a vector input signal. An example of an optimization problem based on the considered integrative series is given. It is noted that when analyzing and optimizing nonlinear dynamical systems by the method of integro–functional series, the problem of calculating multidimensional integrals may arise. The article considers the application of the combined method based on the Volterra integrative series and grid methods for solving the corresponding one -— and multidimensional integral equations for the analysis of nonlinear dynamic and stationary systems. This article considers the case when a certain set of implementations of input and output signals is known, which can be in principle random processes. According to these data, the kernels are found in the decomposition based on the solution of the corresponding linear multidimensional Fredholm integral equation of the first kind. The corresponding problem belongs to the incorrectly posed ones and the regularization method according to A.N. Tikhonov is used to solve it. The article proposes to apply the quasi Monte–Carlo method, characterized by satisfactory convergence, in this problem in the case of large dimensions. The computational qualities in the considered problem of a semi-statistical method for solving integral equations of large dimension, the quasi Monte–Carlo method, the method of central rectangles (cells) and the quadrature formulas of Gauss–Legendre are studied. The approaches under consideration allow us to expand the range of problems to be solved in the theory of analysis and optimization of systems, since methods are proposed that are practically acceptable for large dimensions of integral equations in conditions of limited information about the system.

Абас Висам Махди Абас, Арутюнян Р.В. Моделирование нелинейных динамических и стационарных систем на основе интегро–функциональных рядов Вольтерры и различных классов квадратурных формул. Математическое моделирование и численные методы, 2021, № 2, с. 68–85.

doi: 10.18698/2309-3684-2017-4-1730

The purpose of the paper was to formulate and study the system of kinetic equations modeling the process of diffusion filtration based on a stochastic approach. Within the research we proved the theorem of existence and uniqueness of the solution with respect to the case of continuous density, obtained the solutions in uniformly convergent and asymptotic series and examined its behavior at infinity. Moreover, we considered the specific cases of density of the Delta-function type and uniform distribution. As a result, the finite-difference scheme for solving the corresponding Cauchy problem on finite time intervals is built and justified. The results of computer simulation are also given.

Arutyunyan R.V. Modeling of stochastic filtration processes in lattice systems. Mathematical Modeling and Computational Methods, 2017, №4 (16), pp. 17-30