Vol 13, No 2 (2019)

We consider a mathematical model of market competition between two parties. The parties sequentially bring their products to the market while aiming to maximize profit. The model is based on the Stackelberg game and formulated as a bilevel integer mathematical program. The problem can be reduced to the competitive facility location problem (CompFLP) with a prescribed choice of suppliers which belongs to a family of bilevel models generalizing the classical facility location problem. For the CompFLP with a prescribed choice of suppliers, we suggest an algorithm of finding a pessimistic optimal solution. The algorithm is an iterative procedure that successively strengthens an estimating problem with additional constraints. The estimating problem provides an upper bound for the objective function of the CompFLP and is resulted from the bilevel model by excluding the lower-level objective function. To strengthen the estimating problem, we suggest a new family of constraints. Numerical experiments with randomly generated instances of the CompFLP with prescribed choice of suppliers demonstrate the effectiveness of the algorithm.

We present a first polynomial algorithm with guaranteed approximation ratio for the asymmetric maximization version of the asymmetric 3-Peripatetic Salesman Problem (3-APSP). This problem consists in finding the three edge-disjoint Hamiltonian circuits of maximal total weight in a complete weighted digraph. We prove that the algorithm has guaranteed approximation ratio 3/5 and cubic running-time.

The simple assembly line balancing problem (SALBP) is considered. We describe the special class of problems with an infinitely large stability radius of the optimal balance. For other tasks we received the lower and the upper reachable estimates of the stability radius of optimal balances in the case of an independent perturbation of the parameters of the problem.

Some mathematical model is proposed of a flow in a long channel with compliant walls. This model allows us to describe both stationary and nonstationary (self-oscillatory) regimes of motion. The model is based on a two-layer representation of the flow with mass exchange between the layers. Stationary solutions are constructed and their structure is under study. We perform the numerical simulation of various flow regimes in presence of a local change of the wall stiffness. In particular, the solutions are constructed that describe the formation of a monotonic pseudoshock and the development of nonstationary self-oscillations.

We consider the problem of finding the generalized functionally invariant solutions to Maxwell’s equations. The solutions found contain some functional arbitrariness that can be used for determining the parameters of Maxwell’s system (the dielectric and magnetic constants).

The inverse boundary value problem of aerohydrodynamics is considered in the new formulation: Find the shape of an airfoil streamlined by a potential flow of an inviscid incompressible fluid under assumption that the velocity potential is given as a function of the abscissa of the profile point, as well as the values of the velocity are available on the leading edge of the airfoil and for the unperturbed flow around the unknown profile. The formulas are derived by which the coordinates of the points of the unknown profile can be calculated and the distribution of the velocity value along the obtained profile can be found. It is shown that, depending on the given value of the velocity of the unperturbed forward flow, the problem is either uniquely solvable or has two solutions with different slope angles of the velocity vector of the forward flow with respect to the real axis.

Under study is the problem of spatial free deceleration of a rigid body in a resisting medium. It is assumed that a axisymmetric homogeneous body interacts with the medium only through the front part of its outer surface that has the shape of flat circular disk. Under the simplest assumptions about the impact forces from the medium, it is demonstrated that any oscillations of bounded amplitude are impossible. In this case, any precise analytical description of the force-momentum characteristics of the medium impact on the disk is missing. For this reason, we use the method of “embedding” the problem into a wider class of problems. This allows us to obtain some relatively complete qualitative description of the body motion. For the dynamical systems under consideration, it is possible to obtain particular solutions as well as the families of phase portraits in the three-dimensional space of quasi-velocities which consist of a countable set of the phase portraits that are trajectory-nonequivalent and have different nonlinear qualitative properties.

The mathematical statement is given for the problem of filtration of a viscous fluid in a deformable porous medium that possesses predominantly viscous properties. Some theorems are proved on local solvability and existence of a global-in-time solution in the Hölder classes for the problem.

Mathematical simulation is carried out of the dynamics of an acute inflammatory process in the central zone of necrotic myocardial damage. Some mathematical model of the dynamics of the monocyte-macrophages and cytokines is presented and the numerical algorithm is developed for solving an inverse coefficient problem for a stiff nonlinear system of ordinary differential equations (ODEs). The methodological studies showed that the solution obtained by the genetic BGA algorithm agrees well with the solutions obtained by the gradient and ravine methods. Adequacy of the simulation results is confirmed by their qualitative and quantitative agreement with the laboratory data on the dynamics of inflammatory process in the case of infarction in the left ventricle of the heart of a mouse.