Volume 24, # 1

1. Petrushko Igor Meletievich (on the occasion of his 75th birthday)

Mathematics

2. Stepan I. Bashmakov, Mikhail I. Golovanov Axiomatization of intuitionistic logics defined by small frames
Abstract. In this paper, we study the tabular intuitionistic logics semantically characterized by the Kripke frames of the depths no greater than 3 and widths no greater than 2. The axiomatization of such basic logics is given; the lattice generated by them is constructed. Known methods make it possible, using given axiomatization, to specify the axiomatic of the remaining logics from the lattice.
Keywords: superintuitionistic logic, Kripke frame, axiomatization of logic.

3. Elena S. Efimova Stationary Galerkin method for the semilinear nonclassical equation of higher order with alternating time direction
Abstract. In a cylindrical domain Q ⊆ Rn, we study a boundary value problem for the semilinear parabolic equation of odd order with alternating time direction. The theorem about the unique solvability of the boundary value problem is proved in the weighted Sobolev space. The stationary Galerkin method is applied to solve the problem and the error estimation for this method is obtained.
Keywords:  stationary Galerkin method, approximate solution, inequality, estimation.

4. Sergey I. Mitrokhin On the spectrum of the multipoint boundary value problem for an odd order differential operator with summable potential
Abstract. A boundary value problem for an odd order differential operator with multipoint boundary conditions is studied. The interior points in which boundary conditions are set can divide the segment on which the operator is considered into the incommensurable parts. The potential of the differential operator is a function Lebesgue integrable at the segment on which the operator is considered. We study the asymptotic behaviour of the solutions to the corresponding differential equation for large values of the spectral parameter. The equation on the eigenvalues of the operator is received. We obtain the indicator diagram of that equation. The asymptotic behavior of the eigenvalues in all sectors of the indicator diagram is studied.
Keywords:  differential operator, summable potential, boundary value problem, multipoint boundary conditions, indicator diagram, the asymptotics of the eigenvalues.

5. Sergey V. Popov The Gevrey boundary value problem for a third order equation
Abstract. We consider the Gevrey and Cauchy problems for a third order equation with multiple characteristics with weighted gluing conditions. In the case of continuous gluing conditions, the solvability of the Gevrey problem is reduced to the theory of homogeneous integral equations of degree -1 with a kernel. In the case of weighted gluing conditions, the solvability is reduced to the theory of singular integral equations with a singular kernel. The solvability of boundary value problems is established in Holder spaces. It is shown that the Holder classes of solutions of the Gevrey problem in the case of weighted gluing functions depend both on the non-integer Holder exponent and on the weight coefficients of the gluing conditions when necessary and sufficient conditions are satisfied for the input data of the problem. 
Keywords:  the Gevrey problem, the Cauchy problem, equations with changing time direction, gluing conditions, correctness, H¨older space, singular integral equation. 
    
6. Ivan V. Tikhonov,  Vu Nguyen Son Tung Formulas for an explicit solution of the model nonlocal problem associated with the ordinary transport equation
Abstract. A specific nonlocal problem for the multidimesional ordinary transport equation is studied. An additional condition for the time averages is given. The theorem of existence and uniqueness of solution is obtained. We show that the solution could be found constructively, explicitly, and in a finite number of iterations.
Keywords:  transport equation, nonlocal problem, resolving formula.

7. Boir M. Tsybikov The inverse problem of recovering a leading coefficient in the two-dimensional heat equation
Abstract. We consider the inverse problem of recovering a leading coefficient independent of one of the spatial variable y in the two-dimensional heat equation. The overdetermination data is the values of the solution on the cross-section of the domain by the hyperplane y = 0. The solution is sought in the class of functions whose Fourier image in the variable y is compactly supported in the dual variable. Existence and uniqueness conditions of the solution to this problem in this class are established.
Keywords:  inverse problem, overdetermination condition, second order parabolic equation, initial-boundary value problem.

Mathematical modeling

8. Vasiliy V. Grigoriev and Petr E. Zakharov Numerical modeling of the two-dimensional Rayleigh-Benard convection
Abstract. We study Rayleigh-Bénard convection which is a type of natural convection occurring in a plane horizontal layer of viscous fluid heated from below and cooled from above, in which the fluid develops a regular pattern of convection cells known as Benard cells. The process of rotation is described by a system of nonlinear differential Oberbeck-Boussinesq equations. As convection parameters, the Rayleigh number and the Prandtl number are taken. The system is solved using the finite element method by FEniCS. We obtain numerical results for varying Rayleigh numbers and study the dependence of the Nusselt number on the Rayleigh number. 
Keywords:  natural convection, Oberbeck–Boussinesq approximation, finite element method, convection cells.
   
9. Andrey I. Matveev, Dulustan A. Osipov,  Dulustan R. Osipov, Boris V. Yakovlev Modeling of dynamics of the shape of a flat body of malleable metal in case of isotropic bombing by sand particles
Abstract. The initial form of the grains of gold found in the nature in most cases is a flat plate (a scaly form). However, during pneumoseparation, the toroidal shape of pieces of gold is often found and considered to be the most effective. Thus, the task of estimating time of formation of a toroidal piece of gold is important. In the paper, we consider the evolution of the surface of a flat disk of malleable metal deformed by isotropic bombing with fine particles and develop a mathematical model of this evolution. We obtain a differential equation describing the change of the deformed surface of a round disk which is solved then by a Runge-Kutta method. Studying the solution of the equation, we found that the body rather quickly reaches the most stable toroidal form when the deformed surface gets its maximal value and then a slower transformation of the surface into the sphere follows. We estimate the time of formation of a toroid from a disk with certain parameters of the considered system. The received results could be used for developing more exact models of evolution of flat bodies bombed with fine particles.
Keywords: mathematical model, differential equation, deformed surface, toroid, enrichment, separation, minerals, kinetic energy of particles, evolution of a surface.