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The authors investigate the asymptotic stability in quadratic mean of the trivial solution of the Gikhman–Ito stochastic diffusion functional differential equations in terms of the eigenvalues of the matrix constructed from the coefficients of these equations. [ABSTRACT FROM AUTHOR]

This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations.

This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations. [ABSTRACT FROM AUTHOR]

Stochastic differential equations are proposed on the basis of generalized Fokker-Planck-Kolmogorov equation. From the statements of boundary and initial value problems for probabilistic density, dispersion and differential entropy the conditions for stability of solutions and changes in scenarios of development of random phenomena of heat and nuclear particle transfer are obtained. [Projekat Ministartsva nauke Republike Srbije, br. 174005 i br. 174024]

Introduction. The moral and physical deterioration of medium voltage power cables with phase and zone paper-impregnated insulation requires implementation of quality systems and reliable nondestructive electric diagnostic. Informative indicator of the insulation is the time decay curve of the charging current. It reflects the processes of accumulation of space charges (absorption). The measurements are carried out the 15th since the second direct voltage supply, then – on the 30th and the second on the 60th second. The ratio of the parameters measured in these times gives the dimensionless criteria – absorption coefficients. Three measurements are made at different times, provide a more complete picture of the state of insulation than the measurement of the value of steady leakage (conduction-through), adopted in conventional prevention trials. Purpose. Research and testing methods of diagnostics of power cables with paper-impregnated insulation by absorption and phase characteristics of the belt insulation based on the total measurements. Methodology. A procedure for determining the individual characteristics of phase and zone paper-impregnated insulation based realized on the use of the equivalent circuit of a three-core cable in the metal shell and solved of an over determined system of linear algebraic equations by least squares. Results. The proposed method allows determining the absorption characteristics of the individual phase and zone insulation medium voltage power cables in the overall metal shell at a direct voltage. Individual characteristics reflect the characteristics of cables and allow a greater degree to assess the degree of aging of each of the components of paper-impregnated insulation. Originality. Regardless of the cable connection diagrams probing electric field grabs as the phase, and zone insulation. The cumulative nature of the measurement leads to the fact that the differences in the properties of insulation components are smoothed: the aggregate results of the measurements do not differ for the different schemes of the same type. The individual characteristics of isolation, defined on the basis of the proposed method are more differences than total, indicating that non-symmetrical modes of operation of the cable. Practical value. The values of individual characteristics power cables 6 kV are 3 times more total, because of what their direct measurement may be a problem. The total resistance of several insulation spaces connected in parallel, behind the individual. Smaller values of insulation resistance are measured more easily, especially on short samples cables.

In the 1970-1980s, V. K. Leont'ev and E. N. Gordeev proposed and studied an approach to the analysis of the stability of solutions. In a number of subsequent articles, this approach was developed and the stability of solutions was analyzed on its basis. The approach itself is of a rather general nature but was originally connected with the discrete optimization problems. Later similar results, although in different terms, were published for various classes of problems. We demonstrate the proximity of approaches both on the level of statements of problems and the interpretation of results. [ABSTRACT FROM AUTHOR]

A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley-Harper equation, which is frequently referred to as the generalized Kuramoto-Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré-Dulac normal forms. A normal form was constructed using a modified Krylov-Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived. [ABSTRACT FROM AUTHOR]

The numerical techniques are presented to solve the static and dynamic contact problems of deformable bodies having large rotations of the contact surfaces. The contact conditions on the possible contact surfaces are enforced by using the contact error vector, and an iterative scheme similar to augmented Lagrange multiplier method is employed to reduce the contact error vector monotonically. For dynamic contact problems using implicit time integration, a contact error vector is also defined by combining the displacement, velocity, and acceleration on the contact surface. The suggested iterative technique is implemented to ABAQUS by using the UEL subroutine UEL. In this work, after the computing procedures to solve the frictional contact problems are explained, the numerical examples are presented to compare the present solutions with those obtained by ABAQUS. [ABSTRACT FROM AUTHOR]