## Volume 25, # 1

### Mathematics

1. Makhmadrakhim G. Gadoev, Faridun S. IskhokovOn relative boundedness of a class of degenerate differential operators in the Lebesgue space
Abstract. In the space $L_p(\Omega)$, where $1<p<+\infty$ and $\Omega$ is an arbitrary (bounded or unbounded) domain in $R^n$, we investigate relative boundedness for a class of higher order partial differential operators in non-divergent form. These operators have nonpower degeneracy on the whole boundary of $\Omega$ and degeneracy with respect to each of independent variables is characterized by different functions.

In the earlier published papers in this direction, as a rule, firstly the operator is defined in $\Omega$ and then functions characterizing degeneracies of the operator’s coefficients are defined in this domain.

In contrast to that, here we define $\Omega$ and these functions related to each other while fulfilling the “immersion condition” introduced by P. I. Lizorkin in [19]. In addition, differentiability of the functions by which we define degeneracy of the investigated operator is not required. Study of relative boundedness of differential operators is one of the modern directions in such operators theory with results theory of differentiable functions of many variables, the separation theory of differential operators, the spectral theory of differential operators, etc.

Keywords: partial differential operator, non-power degeneracy, relative boundedness of operators, partition of the unit.

2. Ivan E. Egorov, Elena S. Efimova, Irina M. Tikhonova On Fredholm solvability of first boundary value problem for mixed-type second-order equation with spectral parameter
Abstract. We study the first boundary problem for the mixed-type second-order equation with a spectral parameter in a cylindrical domain in $R^{n+1}$. Previously, for the mixed-type second-order equations some results were received only in two-dimensional domains. V. N. Vragov was the first to propose a well-posed statement of a boundary problem for the mixed-type equations. One of the well-posedness conditions is non-negativity of the spectral parameter. Here we analyze the case of complex spectral parameter and receive a priori estimates under certain conditions, using which an existence and uniqueness theorem is proved for the first boundary problem in the energy class. Also, we obtain sufficient conditions for the Fredholm solvability in the energy class.

Keywords: mixed-type equation, a priori estimate, inequality, equality, orthogonality conditions.

3. Elena V. Karachanskaya, Alena P. Petrova Modeling of the programmed control with probability 1 for some financial tasks
Abstract. The description of the dynamics of some financial events can be related to Itô stochastic differential equations (SDE). In this paper, we consider a financial model affected by random disturbances which take the form of Wiener and Poisson perturbations. The construction of the programmed control with probability 1 (PCP1) is based on the concept of first integral for stochastic dynamic systems of diffusion type with jumps which are described by the Itˆo equations. Two types of financial models are considered as examples of the construction of PCP1: the investment portfolio model (diffusion model) and the interest rate model (diffusion with jumps). The given examples are accompanied by numerical modeling.

Keywords: programmed control with probability 1, stochastic Itô’s equation with jumps, first integral of system of the Itô equations, investment portfolio model, interest rate model.

4. Nyurgun P. Lazarev, Hiromichi Itou, Petr V. Sivtsev, Irina M. TikhonovaOn the solution regularity of an equilibrium problem for the Timoshenko plate having an inclined crack
Abstract. The equilibrium problem for an transversely isotropic elastic plate (Timoshenko model) with an inclined crack is studied. It is supposed that the crack does not touch the external boundary. For initial state, we assume that opposite crack faces are in contact with each other on a frictionless crack surface. Herewith, the crack is described with the use of a surface satisfying certain assumptions. On the crack curve defining the crack in the middle plane, we impose a nonlinear boundary condition as an inequality describing the nonpenetration of the opposite crack faces. It is assumed that on the exterior boundary of the cracked elastic plate the homogeneous Dirichlet boundary conditions are prescribed. We establish additional smoothness of the solution in comparison with that given in the variational statement. We prove that the solution functions are infinitely smooth under additional assumptions on the function of external loads and the functions of displacements near the curve describing the inclined crack.

Keywords: variational inequality, the Timoshenko plate, crack, nonpenetration condition, solution regularity.

5. Gennadii A. Rudykh, Eduard I. Semenov Research of compatibility of the redefined system for the multidimensional nonlinear heat equation (general case)
Abstract. We study the multidimensional parabolic second-order equation with the implicit degeneration and the finite velocity of propagation of perturbations. This equation is given in the form of an overdetermined system of the differential equations with partial derivatives (the number of the equations exceeds the number of the required functions). It is known that an overdetermined system of the differential equations may not be compatible as well as may not have any solutions. Therefore, in order to determine the existence of the solutions and the degree of their arbitrariness the analysis of this overdetermined system is carried out. As a result of the research, the sufficient and the necessary and sufficient compatibility conditions for the overdetermined system of the differential equations with partial derivatives are received. On the basis of these results with the use of the equation of Liouville and the theorem of the potential operators, the exact non-negative solutions of the multidimensional nonlinear heat equation with the finite velocity of propagation of perturbations are constructed. In addition, the new exact non-negative solutions of the nonlinear evolution of Hamilton–Jacobi equations are obtained; the solutions of the nonlinear heat equation and the solutions of Riemann wave equation are also found. Some solutions are not invariant fr om the point of view of the groups of the pointed transformations and Lie–Bäcklund’s groups. Finally, the transformations of Bäcklund linking the solutions of the multidimensional nonlinear heat equation with the related nonlinear evolution equations are obtained.

Keywords: multidimensional nonlinear heat equation, nonlinear evolution equations, finite velocity of propagation of perturbation, exact nonnegative solutions, Bäcklund transformation.

6. Elizaveta M. Streletskaya, Vladimir E. Fedorov, Amar DebboucheThe Cauchy problem for distributed order equations in Banach spaces
Abstract. The Cauchy problem for a distributed order equation in a Banach space with the fractional Gerasimov–Caputo derivative and a linear bounded operator in the right-hand side is studied. Existence and uniqueness conditions for the problem solution in the space of exponentially growing functions are found by the methods of the Laplace transformation theory. The solution is presented in the form of a contour integral of the bounded operator resolvent with a complex argument determined by the form of the distributed derivative. The analyticity of the solution in the right half-plane of the complex plane is proved. The general result is applied to the research of the Cauchy problem for an integro-differential system of equations with right-hand side in the form of composition of an integral operator with respect to the spatial variables and the linear transformation of the unknown vector-function.

Keywords: evolution equation, fractional Gerasimov-Caputo derivative, Cauchy problem, distributed order equation.

7. Alexander M. Khludnev, Tatyana S. Popova On junction problem for elastic Timoshenko inclusion and semi-rigid inclusion
Abstract. An equilibrium problem for elastic bodies with a thin elastic inclusion and a thin semi-rigid inclusion is investigated. The inclusions are assumed to be delaminated fr om the elastic bodies, forming therefore a crack between the inclusions and the elastic matrix. Nonlinear boundary conditions are considered at the crack faces to prevent mutual penetration between the crack faces. The inclusions have a joint point. We present both differential formulation in the form of a boundary value problem and a variational formulation in the form of a minimization problem for an energy functional on a convex set of admissible displacements. The unique solvability of the problem is substantiated. Equivalence of differential and variational statements is shown. Passage to the limit is investigated as the rigidity parameter of the elastic inclusion goes to infinity. The limit model is analyzed. Junction boundary conditions are found at the joint point for the considered problem as well as for the lim it problem.

Keywords: Timoshenko inclusion, semi-rigid inclusion, elastic body, crack, nonlinear boundary conditions.

### Mathematical modeling

8. Sardaana R. Krylatova, Andrey I. Matveev, Ivan F. Lebedev, Boris V. YakovlevModeling of particle motion in spiral pneumoseparator by statistical methods
Abstract. In mathematical modeling of mineral processing, there arise problems of determining the probability of the particle presence on the working surfaces of devices. In the paper, we propose a statistical approach to solving such problem, i. e., the idea of the Gibbs method is used. We consider problems of modeling processes in an air spiral separator. A mathematical model of the spiral surface of a pneumoseparator, a model of particle motion, a flux of noninteracting particles along the separator working surface, and an algorithm for determining the particle flux concentration are developed. The calculated distribution of the noninteracting particles concentration on the working surface of the device is identified with the probability distribution of the location of one particle. The developed algorithm for determining the probability of position of a particle on the working surface of the pneumoseparator can be used as an element of a more complex mathematical model, for example, a model wh ere interactions between particles are taken into account.

Keywords: spiral separator, concentration, statistical method, motion equation, particle flow, enrichment, mathematical model.

9. Vladislav N. Khankhasaev , Erdeni V. Darmakheev On some applications of the hyperbolic heat equation and the methods for solving it
Abstract. Creation of new technological processes based on the use of high-intensity energy fluxes makes it necessary to take into account the final rate of heat propagation when determining the temperature state. This account can be realized with the help of the hyperbolic heat equation obtained by A. V. Lykov in the framework of nonequilibrium phenomenological thermodynamics as a consequence of the generalization of the Fourier law for flows and the heat balance equation. In the previous works by V. N. Khankhasaev, the process of switching off the electric arc in a spiral gas flow was simulated using this equation. In this paper, a mathematical model of this process is developed with the addition of a period of steady burning of the arc until the moment of disconnection and replacement of the strictly hyperbolic heat conduction equation by a hyperbolic-parabolic equation. For the resulting mixed heat conduction equation, a number of boundary value problems in the Fortran and Matcad software environments are correctly posed and numerically solved, obtaining temperature fields that are in good agreement with the available experimental data.

Keywords: hyperbolic-parabolic equations, hyperbolic heat equation, alternating direction scheme, Navier-Stokes equations, heat balance.