Vol 11, No 2 (2017)

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.

We study a system of nonlinear equations describing unsteady flows of a viscoelastic fluid of the Oldroyd type in a bounded three-dimensional domain under the mixed boundary conditions: the Navier slip condition is set on one part of the boundary, while on the other part, the no-slip condition is set. We prove the theorem concerning the existence, uniqueness, and energy estimates for weak solutions.

A method is proposed for obtaining lower bounds for the length of the shortest cover and complexity of the minimal cover based on the notion of independent family of sets. For the problem of minimization of Boolean functions, we provide the functions and construct coverings by faces of the set of unit vertices for which the suggested lower bounds can be achieved in the case of five or more variables. The lower bounds, based on independent sets, are unreachable and cannot be used as sufficient conditions for minimality of such functions.

We develop amatrix-graph approach to estimating themixing properties of bijective shift registers over a set of binary vectors. Such shift registers generalize, on the one hand, the class of ciphers based on the Feistel network and, on the other hand, the class of transformations of additive generators (the additive generators are the base for the Fish, Pike, andMush algorithms). It is worth noting that the original schemes of additive generators are found insecure due to their weak mixing properties. The article contains the results of investigations for the mixing properties of modified additive generators. For the mixing directed graph of a modified additive generator, we define the sets of arcs and cycles, obtain primitivity conditions, and give a bound for the exponent. We show that, the determination of parameters for the modified additive generator allows us to achieve a full mixing in a number of iterations that is substantially less than the number of vertices in the mixing digraph.

Using the equations of state for fractured-porous media that describe the sorptioninduced deformation of coal, we develop a geomechanical model for radial gas influx to a borehole drilled in a coal bed with the concurrent evolution of stress field in the borehole environment. A numerical-and-analytical method is proposed for solving the corresponding system of equations for poroelastic media. A relation is found between the volume of slack withdrawn in the borehole (when opening up the gas-bearing seams), the sorption-and-storage capacities of coal, the permeability k, and the horizontal component σ_h of the natural stress field. We demonstrate the solvability of the inverse boundary-coefficient problem of determining k and σ_h on the basis of pressure in the closed borehole. We substantiate an express-method for estimating the permeability by the measurements of pressure in the borehole operating in the “pressure drop” mode.

Under consideration is the equilibrium of a composite plate containing a through vertical crack of variable length at the interface between thematrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion, by the Kirchhoff–Love model. Some formula is obtained for the derivative of the energy functional with respect to the crack length.

The two-layer flows of a liquid and a gas in a horizontal channel are investigated under condition of given gas flow rate. Evaporation on the thermocapillary interface is taken into account. An exact solution is constructed of the Navier–Stokes equations in the Boussinesq approximation, taking into account the Dufour effect in the gas-vapor layer. Within the framework of linear theory, the stability of the obtained solutions and the characteristics of the arising perturbations are studied. The influence is considered of the thickness of the liquid layer and the magnitude of the longitudinal temperature gradient on the structure of the basic flow and perturbations.

Under study is numerical solution of the problems of two-phase filtration. The formulation of the problem is given in terms of velocity, pressure and saturation. To approximate the velocity and pressure, the mixed finite elementmethod is used. The flux schemes are applied for discretization of the convection term in the saturation equation. We present the results of numerical solution of a model problem for heterogeneous media.