Vol 12, No 3 (2018)

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.

We consider a bilevel “defender-attacker” model built on the basis of the Stackelberg game. In this model, given is a set of the objects providing social services for a known set of customers and presenting potential targets for a possible attack. At the first step, the Leader (defender) makes a decision on the protection of some of the objects on the basis of his/her limited resources. Some Follower (attacker), who is also limited in resources, decides then to attack unprotected objects, knowing the decision of the Leader. It is assumed that the Follower can evaluate the importance of each object and makes a rational decision trying to maximize the total importance of the objects attacked. The Leader does not know the attack scenario (the Follower’s priorities for selecting targets for the attack). But, the Leader can consider several possible scenarios that cover the Follower’s plans. The Leader’s problem is then to select the set of objects for protection so that, given the set of possible attack scenarios and assuming the rational behavior of the Follower, to minimize the total costs of protecting the objects and eliminating the consequences of the attack associated with the reassignment of the facilities for customer service. The proposed model may be presented as a bilevelmixed-integer programming problem that includes an upper-level problem (the Leader problem) and a lower-level problem (the Follower problem). The main efforts in this article are aimed at reformulation of the problem as some one-level mathematical programming problems. These formulations are constructed using the properties of the optimal solution of the Follower’s problem, which makes it possible to formulate necessary and sufficient optimality conditions in the form of linear relations.

Under discussion are the requirements from the tools for analysis of long-term strategies for the development of energy systems and the possibility of using combinatorial modeling methods for studying discrete solutions in this development. The basis of the methods is the representation of the development of the modeled system in the form of a directed graph whose nodes correspond to possible states of the system at certain time, whereas the links characterize the admissibility of transitions from one state to another. The methods of combinatorial modeling, as a visual form of representation of dynamic discrete branching alternatives, allow us to simulate a long-term process of the development of the investigated system under various possible external and internal conditions and then determine a rational strategy for its development. We describe an algorithm for formation of the graph of feasible options for the development of the system. The approach is illustrated by an example. The described algorithms are implemented in the software package for a distributed computing environment.

We present a numerical algorithm for determining the inhomogeneities of permittivity from the strength modulus of a scattered electric field. The algorithm was tested on simulated noisy data and revealed its practical operability.

Under consideration is the nonlinear dynamical system describing the near-resonance motion in the atmosphere of a spacecraft with small aerodynamic, mass, and inertial asymmetries. The aim of the study is to obtain an expression for estimating the probability of capture into the principal resonance. It is shown that the expression for estimating the probability of capture into resonance makes it possible to determine the magnitude of asymmetries of the spacecraft both in the realization of capture into and in passing through the principal resonance. The veracity of the obtained estimate of the probability of capture into resonance is confirmed by the results of numerical simulation in the problem of descent of a spacecraft in the atmosphere of Mars.

The problem is considered about the vertical continuous impact and subsequent free deceleration of a circular cylinder semi-immersed in a liquid. The specificity of this problem is that, under certain conditions, some areas of low pressure near the body appear and the attached cavities are formed. The separation zones and the motion law of the cylinder are unknown in advance and have to be determined in solving the problem. The study of the problem is conducted by a direct asymptotic method effective for small spans of time. Some nonlinear problem with unilateral constraints is formulated that is solved together with the equation defining the law of motion of the cylinder. In the case when the space above the external free surface of a liquid is filled with a gas with low pressure (vacuum), an analytical solution of the problem is constructed. To determine the main hydrodynamic characteristics (the separation point and acceleration of the cylinder), we derive a system of transcendental equations with elementary functions. The solution of this system agrees well with the results obtained by the direct numerical method.

Under consideration is the problem of continuation of the wave field from the boundary of a half-plane inside it. We obtain stability estimate for the solution of the corresponding Cauchy problem.

We consider the retrospective inverse problem that consists in determining the initial solution of the one-dimensional heat conduction equation with a given condition at the final instant of time. The solution of the problem is given in the form of the Poisson integral and is numerically realized by means of a quadrature formula leading to a system of linear algebraic equations with dense matrix. The results of numerical experiments are presented and show the efficiency of the numerical method including the case of the final condition with random errors.