**
**### Mathematics

1. Igor V. Bubyakin

About the structure of ﬁve-dimensional complexes of two-dimensional planes in projective space P^{5} 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 P

^{5} 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 P

^{5} to 9-dimensional algebraic manifold Ω(2, 5) of space P

^{19}. 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.