Using the method of a priori estimates, we studied the operator-difference equations that approximate the differential problems of heat convection for an incompressible liquid in the variables “stream function-vorticity,” and consider the issues of convergence of the iterative schemes for solving the equations. The boundary values for the vorticity are chosen in the form of Thom’s formulas. We give a boundedness estimate and a condition for uniqueness of the solution of the finite-difference problem. Using an auxiliary function of vorticity, we transform the finite-difference equations under consideration to the relations with homogeneous boundary conditions. Some implicit iterative algorithms are proposed for numerically solving the finite-difference equations; the estimates for the rate of convergence are obtained for these algorithms provided that the conditions equivalent to the condition of uniqueness are satisfied. The behavior of iterations is analyzed in the case of the Stokes linear problem. To illustrate the advantages of the considered iterative algorithms, a problem in a closed domain with side heating is considered. The calculations are carried out for the iterative algorithm of the type of variable directions.

Moving-boundary problems for incompressible flows often require repeated mesh generation when body-fitted finite element methods are used. This may become costly for flows with moving rigid obstacles, since the numerical solution from the previous time level also has to be transferred to the new finite element space. In this paper, we study a fictitious-domain finite element method for the two-dimensional incompressible Navier–Stokes equations with rigid obstacles. The aim is to analyze a penalized formulation on a fixed background domain and to compare it with a classical body-fitted finite element approach. The flow problem is extended to a larger fixed domain, and the effect of the solid domain is imposed by a nonlinear penalty term depending on the parameters $\epsilon$ and $\beta$. A fully discrete finite element scheme is constructed, and stability and convergence estimates are obtained for the fixed-obstacle auxiliary penalized problem. The moving obstacle case is examined numerically. Several numerical tests are presented. A manufactured auxiliary problem is used to check the spatial convergence of the method, the influence of the penalty parameter, and the behavior of the solution inside the fictitious domain. We also consider two moving obstacle examples: a rotating disk moving near a no-slip wall and two rotating disks moving along symmetric trajectories. The fictitious-domain solutions are compared with body-fitted finite element solutions, and the main flow patterns are reproduced without remeshing. The results indicate that the fictitious-domain finite element method is a practical alternative for moving obstacle flow problems. It is not universally faster than body-fitted methods, but it becomes advantageous when the geometry changes in time, since the fixed background mesh avoids repeated remeshing and the transfer of the previous solution between nonmatching meshes.

This paper presents an analytical solution of the differential equations of motion of a material point in the plane perpendicular to the plane of the gravitating disk. The differential equations of the problem under study and the applied Gilden's method are described in the works of A. Poincaré. Differential equations refer to nonlinear equations. The analysis of methods for solving nonlinear differential equations was carried out. The methodology of applying the Gilden method to the solution of the differential equations under consideration can be applied in studies of the problem of the motion of celestial bodies in the “disk-material point” system in perpendicular planes. To identify the various properties of the gravitating disk, an analytical review of the state of the problem of the motion of a material point in the field of a gravitating disk is carried out. Summing up the presented review on the problem under study, a conclusion is made. The substantive formulation of the problem is described, which is formulated as follows: the study of the influence of disk-shaped bodies on the motion of a material point and methods for their solution.

Currently, data traffic is growing rapidly, and ensuring optimal network performance and effective data flow management have become the most important tasks. In this context, the quality of network service plays a crucial role in achieving these goals. This article suggests an approach to solving the problem of efficient service in ISDN. Namely, optimization of resource distribution between channel switching and packet switching subnets in ISDN to calculate optimal quality of service characteristics. In the process of ISDN design analysis, an optimization problem is compiled, where the evaluation of the packet-switched subnet service is used as an objective function, and the evaluation of the circuit-switched subnet service is used as one of the constraints for this task. To calculate the main characteristics of a packet-switched subnet, the subnet is considered as a service system with a delay. During the study, the methods of optimal movement of the generalized channel boundary between the subnets of channel switching and packet switching were identified, depending on the data parameters and the state of the integrated network, which made it possible to develop an optimal mathematical model of optimal control of the generalized boundary. To calculate the bandwidth for channel switching and packet switching subnets, an algorithm has been compiled to implement the resulting model and a program in C++ has been compiled. The study of the generalized boundary and the dynamic redistribution of bandwidth between subnets represents a new approach to network optimization. The results are based on the use of the classical Erlang formula for systems with service failures and on load distribution plans, which makes it possible to effectively manage the maintenance process in the network

The Boussinesq equations are widely used to model incompressible thermally driven flows, but their numerical simulation becomes more challenging in domains containing fixed or moving rigid obstacles, particularly when the prescribed obstacle motion is uncertain. In this work, we develop a Brinkman-type fictitious-domain formulation for the incompressible Boussinesq system on a fixed computational domain. Brinkman penalization is used to impose the prescribed solid velocity, while a thermal penalty term enforces the temperature inside the immersed bodies. The resulting problem is discretized by a finite element method using skew-symmetric convective forms and grad-div stabilization. The uncertainty in the obstacle motion is introduced through a random oscillation amplitude and treated by a non-intrusive stochastic collocation method based on Gauss–Legendre quadrature. Stability and convergence of the fully discrete scheme are established. Numerical experiments for moving heated and hot–cold obstacles demonstrate the influence of the uncertain oscillation amplitude on the Nusselt number, kinetic energy, mean vorticity, and the spatial distributions of the mean and standard deviation of the solution fields. The results indicate that the proposed method provides a stable numerical framework for simulating incompressible thermally coupled flows with moving immersed obstacles under uncertainty in the prescribed obstacle motion.