## Volume 24, # 4

### Mathematics

1. Igor V. Bubyakin About the structure of complexes of m-dimensional planes in projective space Pn containing a ﬁnite number of developable surfaces
Abstract. We consider the projective differential geometry of m-dimensional plane submanifolds of manifolds G(m, n) in projective space $P^n$ containing a finite number of developable surfaces. To study such submanifolds we use the Grassmann mapping of manifolds G(m, n) of m-dimensional planes in projective space $P^n$ to $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m, n)$ of space $P^N$, where $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. This mapping combined with the method of external Cartan’s forms and moving frame method made it possible to determine the dependence of considered manifolds structure and the configuration of the (m − 1)-dimensional characteristic planes and (m + 1)-dimensional tangential planes of developable surfaces that belong to considered manifolds.

Keywords: Grassmann manifold, complexes of multidimensional planes, Grassmann mapping, Segre manifold.

2. Sirozhiddin Z. Jamalov On correctness of nonlocal edge problem with constant coeﬃcient for multidimensional second order equation of mixed type
Abstract. We formulate a nonlocal boundary-value problem for a second order multidimensional equation of mixed type covering classical elliptic, hyperbolic, and parabolic equations. We prove regular solvability of the posed nonlocal boundary-value problem in Sobolev spaces.

Keywords: second order multidimensional equation of mixed type, nonlocal boundary value problem, generalized solution, regular solution, uniqueness, existence, smoothness of solution, method of $\varepsilon$-regularization, Galerkin method, a priori estimates.

3. Ivan E. Egorov and Elena S. Efimova A boundary value problem for the third-order equation not solvable with respect to the highest-order derivative
Abstract. We consider a boundary value problem for the third-order equation not solvable with respect to the highest-order derivative. Equations of this type, often called Sobolev type equations, occur in many applied problems. The nonstationary Galerkin method and regularization method are applied to prove the existence and uniqueness theorem for a regular solution of the boundary value problem. Also we obtain an error estimate via regularization parameter and in terms of eigenvalues of the spectral problem for the Laplace operator.
Keywords: pseudoparabolic equation, boundary value problem, solvability, a priori estimate, approximate solution, error estimate.

4. Nyurgun P. Lazarev and Vladimir V. Everstov An optimal size of an external rigid thin inclusion for a nonlinear problem describing equilibrium of a three-dimensional cracked cylindrical body
Abstract. A mathematical model describing equilibrium of cracked three-dimensional bodies with rigid thin stiffener on the outer boundary is studied. Inequality type boundary condition is imposed at the crack faces providing a mutual non-penetration between crack faces. We analyze the dependence of solutions on the size of the thin rigid stiffener reinforcing the cracked body on the outer edge. Existence of the solution to the optimal control problem is proved. For this problem the cost functional is defined by an arbitrary continuous functional, while the size parameter of the thin rigid stiffener is chosen as a control parameter.
Keywords: variational inequality, optimal control problem, nonpenetration, non-linear boundary conditions, crack.

5. Viktor G. Markov and Sergey V. Popov Parabolic equations of the fourth order with changing time direction with complete matrix of gluing conditions
Abstract. We study solvability of boundary value problems for the fourth order parabolic equations with changing time direction in case of complete matrix of gluing conditions. For boundary value problems for equations with changing time direction, the smoothness of the initial and boundary data does not guarantee that the solution belongs to a Holder space. In the simplest cases, S. A. Tersenov obtained necessary and sufficient conditions for solvability of such problems for second order parabolic equations in the spaces $H^{p,p/2}_{x\,t}$ for p > 2. Moreover, the solvability (orthogonality) condition was written in an explicit form. Note that in the one-dimensional case the number of orthogonality conditions is finite, while in the multidimensional case the number of orthogonality conditions of the integral character is infinite.

We show that the Holder solution classes of boundary value problems for the fourth order parabolic equations with changing time direction, as well as the number of solvability conditions, depend on the form of the matrix of gluing conditions with real coefficients.

Keywords: solvability, boundary value problems, parabolic equations with changing time direction, matrix of gluing conditions, singular equations, Holder space.

6. Valery E. Fedorov The stationary Galerkin method applied to the ﬁrst boundary value problem for a higher order equation with changing time direction
Abstract. We prove the existence of the unique regular solution to the first boundary value problem for the higher order equation with changing time direction in the Sobolev space. The stationary Galerkin method is applied for which the estimate of the rate of convergence is obtained in the terms of the eigenvalues to the self-adjoint spectral problem for the quasielliptic equation.
Keywords: higher-order equation, changing time direction, first boundary value problem, regular solvability, Sobolev space, stationary Galerkin method, convergence rate estimate.

7. Abdukomil R. Khashimov On the second boundary value problem for nonstationary third-order equations of composite type
Abstract. We consider the second boundary value problem for nonstationary third-order equations of mixed type. and study asymptotic characteristics of fundamental solutions of the equations, which are used for constructing of regular solutions to the boundary value problems.
Keywords: third order equation, nonstationary PDE, uniqueness of solution, regular solution, boundary value problem, mixed type equation, method of potentials, fundamental solution.

8. Polina A. Shaikhullina On solution to the simplest functional equation in curvilinear-band-type domain
Abstract. We consider the functional equation $u(\xi + 1) − u(\xi) = d(\xi), \xi\in\Pi,$, in the region $\Pi\subset C$  of “curvilinear band” type. For sufficiently fast decreasing at infinity and holomorphic within the domain $\Pi$  functions $d(\xi)$ the existence of a holomorphic and bounded solution is shown, the uniqueness of the solutions is investigated. We also obtained precise estimates of the constructed solutions and its asymptotic behavior.
Keywords: semihyperbolic maps, functional equations, analytic classification.

### Mathematical modeling

9. Viktor A. Ivanov Numerical research on eﬀect of thermal insulation on a gas pipeline’s performance in Far North environment
Abstract. A research on gas ﬂow parameters related to pressure drop in a pipeline is conducted. Temperature and pressure distributions along the pipeline are established in accordance with the equations of gas dynamics and the state equation of a real gas with known parameters of incoming gas. Dependence of the mass ﬂow on the known outlet pressure is evaluated. Thermal interaction of the moving gas and the frozen soil is simulated in a conjugate form and thus the real physical process is described more accurately. By means of numerical modeling with the finite elements method various thermal insulation options are examined.
Keywords: gas pipeline, conjugate problem, permafrost, mathematical modeling.