Volume 24, # 2

Mathematics

1. Igor V. Bubyakin About the structure of five-dimensional complexes of two-dimensional planes in projective space P5 with a single developable surface
Abstract. This article focuses on projective differential geometry of submanifolds of 2-dimensional planes manifolds G(2, 5) in projective space P5 containing single developable surface. To study such submanifolds we use the Grassmann mapping of manifold G(2, 5) of 2-dimensional planes in projective space P5 to 9-dimensional algebraic manifold Ω(2, 5) of space P19. This mapping combined with the method of external Cartan’s forms and moving frame method made possible to determine the structure of considered manifolds.
Keywords: Grassmann manifold, complexes of multidimensional planes, Grassmann mapping, Segre manifold.

2. Semen V. Mestnikov and Nikolay V. Petrov Numerical construction of the information set and sufficient conditions for k-detection in the simple search game on the plane
Abstract. The antagonistic differential simple search game on the plane is considered. The searcher is moving along straight lines. Using the auxiliary game with a detail of pursuers the estimates for the detection probability are determined. The information sets are built in the example with two and three pursuers.
Keywords: differential search game, information set, mixed strategies.

3. Alexander O. Petrikov Inner extensions of partial operations on a partial semigroup
Abstract. We analyse inner extensions of partial operations on a partial semigroup. The problem of extension of a partial operation internally to a full one with preservation of associativity is studied. The possibilities of continuing a partial operation on a partial semigroup of non-zero elements of a completely 0-simple semigroup by standard and non-standard methods are considered. A negative answer is obtained in relation to the question about whether any extension of a partial operation on a partial semigroup of non-zero elements is a completely simple semigroup, and whether any extension is standard. However, in certain cases the answers are positive. The article deduces the necessary and sufficient conditions of extendibility of a partial operation on a semigroup of residue modulo n, and also of a partial operation on a semigroup of non-zero elements of (2×2)-matrices over the field. The uniqueness of the extension of a partial operation on the semigroup of non-zero (2 × 2)-matrices over a field is shown.
Keywords: inner extension, partial semigroup, completely 0-simple semigroup, semigroup of residue modulo n.

Mathematical modeling

4. Maria V. Vasilyeva and Grigorii A. Prokopiev Numerical solution to the problem of two-phase filtration with heterogeneous coefficients by the finite element method
Abstract. We consider the process of filtration of a two-phase fluid in a porous, heterogeneous medium. This process is described by a coupled system of equations for saturation, filtration rate, and pore pressure. We consider mathematical models with and without capillary forces, in the presence of which, for saturation, we have a nonstationary convection-diffusion equation. Since this process is characterized by a significant predominance of the convective term in the equation for saturation, countercurrent approximations are used by adding non-uniform artificial diffusion. Speed and pressure are approximated using a mixed finite element method. The results of numerical calculations for a two-dimensional case with strongly heterogeneous permeability coefficients of a porous medium are presented. Several cases of relative fluid permeability associated with linear and nonlinear coefficients and the presence of capillary forces are
considered.
Keywords: porous medium, two fasefiltration, finite elements method, Galerkin method, numerical simulation.
  
5. Aima M. Efimova Computational identification of the boundary condition in the heat transfer problems
Abstract. The inverse boundary-value problems of heat transfer are of great practical importance, and the work of many authors is devoted to the numerical methods of their solution. We consider a direct method for solving inverse boundary-value problems for a one-dimensional parabolic equation that decomposes a finite-difference analogue of the problem at each time layer. With the help of the proposed numerical solution, we solve the inverse boundary-value problems with a fixed boundary, with a moving boundary, and the Stefan problem. The results of numerical calculations are discussed.
Keywords: inverse boundary problem, inverse Stefan problem, finite difference method, marching method.

6. Petr V. Sivtsev and Petr E. Zakharov Numerical calculation of the effective coefficient in the problem of linear elasticity of a composite material
Abstract. We consider the numerical calculation of the effective coefficient of the linear elasticity problem on a representative volume element having a similar volume fraction of fibers. The obtained effective coefficient is used to solve the complete problem on a coarse grid. As an example, the problem of calculating the deformation of a concrete block with the inclusion of steel fibers under the action of a three-point bending is considered. The computational implementation of the problem is carried out by the finite element method using the FEniCS computing platform.
Keywords: numerical homogenization, linear elasticity, composite material, mathematical modeling, effective coefficient.

7. Vladislav V. Popov Mathematical model of soil freezing
Abstract. We compare two mathematical models of the freezing process of a moist soil saturated with an aqueous solute of salt. For the mathematical model with constant coefficients in the heat equation in a two-phase domain, the possibility of a continuous transition of water saturation to zero at the interface between two-phase and frozen domains is shown.
Keywords: phase transition, thermodynamic equilibrium, self-similar solution, diffusion, thermal diffusivity, water saturation, moisture.

8. Georgii G. Tsypkin A mathematical model of freezing of unsaturated soils in the presence of capillary pressure
Abstract. A mathematical model of freezing of soils saturated with a heterogeneous mixture of water and air in the presence of capillary pressure is proposed. The derivation of the diffusion equation for the redistribution of moisture from the laws of conservation of mass and momentum is given. This allows us to define the diffusion coefficient through the parameters of the porous medium and fluids. Balance relations through the water crystallization front is derived. A self-similar solution of the problem in the linear approximation is obtained. It is shown that the growth of capillary forces reduces the amount of ice formed, and a more intensive freezing regime leads to an increase in the saturation with ice.
Keywords: unsaturated soil, capillary pressure, filtration, freezing, ice.

9. V. I. Vasil’ev and Ling-De Su Numerical method for solving boundary inverse problem for one-dimensional parabolic equation
Abstract. We consider a numerical method for solving boundary inverse problem using the implicit difference scheme for approximation by time and finite difference method for the boundary inverse problem. A numerical solution to the boundary inverse problem is determined by special decomposition which transforms the problem into two standard problems. We present the results of numerical experiments, including those with random errors in the input data, which confirm the capabilities of the proposed computational algorithms for solving this boundary inverse problem.
Keywords: boundary inverse problem, finite difference method, numerical solution, parabolic partial differential equation.