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Vol 63, No 12 (2019)

Article

Solvability of Nonlocal Problems for Systems of Sobolev-Type Differential Equations with a Multipoint Condition

Assanova A.T., Imanchiyev A.E., Kadirbayeva Z.M.

Abstract

We consider a nonlocal problem for a system of loaded differential equations of the Sobolev type with a multipoint constraint. By introducing additional unknown functions, we reduce the problem under consideration to an equivalent problem consisting of a nonlocal multipoint problem for a system of loaded hyperbolic equations of the second order with functional parameters and integral correlations. We propose algorithms for solving the equivalent problem. Moreover, we establish conditions for the well-posedness of the nonlocal multipoint problem for the system of loaded hyperbolic equations of the second order and conditions for the existence of a unique classical solution to the nonlocal problem for the system of differential equations of the Sobolev type with a multipoint constraint.

Russian Mathematics. 2019;63(12):1-12
pages 1-12 views

Bifurcation of the Birth of a Closed Invariant Curve in a One-Parameter Family of Quadratic Mappings of the Plane

Bel’mesova S.S.

Abstract

We give an example of a one-parameter family of quadratic endomorphisms of the plane, in which a closed invariant curve is born from an elliptic fixed point.

Russian Mathematics. 2019;63(12):13-20
pages 13-20 views

Generalized Solutions of the Linear Boundary Value Problems

Burskii V.P.

Abstract

We use constructions from the general theory of boundary value problems to build a theory of generalized boundary value problems for the generalized Poisson equation. Namely, generalized solutions of various boundary value problems are introduced and studied for the matrix generalization of the Poisson equation, a description of the set of all such well-posed problems is given. The obtained results are used for advancements in the original general theory of boundary value problems.

Russian Mathematics. 2019;63(12):21-31
pages 21-31 views

Approximation by Classical Orthogonal Polynomials with Weight in Spaces L2,γ(a,b) and Widths of Some Functional Classes

Vakarchuk S.B.

Abstract

We investigate approximations of functions of classes W2r(Dγ;(a,b)), r = 2, 3, …, by classical orthogonal polynomials with a weight γ in the spaces L2(a,b). We obtain upper and lower estimates for different widths on the classes W2rm,γ, Ψ; (a,b)), where r ∈ ℤ+, m ∈ ℕ, Ψ is a majorant, Ωm,γ is a generalized modulus of continuity of m-th order. We find the condition on majorant, which enable us to compute the exact values of widths, and give certain examples of these values. In all mentioned above classes we obtain bounds (including the least upper bounds) for the Fourier coefficients.

Russian Mathematics. 2019;63(12):32-44
pages 32-44 views

Generalization of the Crocco Invariant for 3D Gas Flows Behind Detached Bow Shock Wave

Golubkin V.N., Sizykh G.B.

Abstract

We study a steady 3D flow of the ideal gas. In the flow between the bow shock wave and the nose part of the body streamlined by the uniform supersonic flow, we consider isoentropic stream surfaces that originate at closed lines on the shock and envelope its leading point. We show that each vortex line is closed and once envelopes the isoentropic stream surface. We obtain the integral invariant of isoentropic stream surfaces, namely, the circulation of the scaled (in a certain way) vorticity vector over (closed) vortex lines. This result is a 3D generalization of the streamline invariant obtained by L. Crocco for axisymmetric flows.

Russian Mathematics. 2019;63(12):45-48
pages 45-48 views

Nehari Type Theorems and Uniform Local Univalence of Harmonic Mappings

Graf S.Y.

Abstract

The paper is devoted to finding conditions, sufficient for uniform local univalence of sense-preserving mappings, harmonic in the unit disc of the complex plane; the conditions are given in terms of the generalized Schwarzian derivative introduced by R. Hernández and M. J. Martín. The main section contains proofs of the conditions of univalence and uniform local univalence. In the proofs, the methods of the theory of linear-invariant families and generalized Schwarzian derivatives are used. The proved criteria are effective in the case of quasiconformal harmonic mappings; this is confirmed by examples. In the final section, some related methods are applied to harmonic mappings associated with non-parametric minimal surfaces. An estimation of the Gaussian curvature of minimal surfaces is obtained; it is given in the terms of the order of the associated harmonic mapping.

Russian Mathematics. 2019;63(12):49-60
pages 49-60 views

On Differentiation with Respect to Parameter of a Hypergeometric Function of a Special Type

Ivankov P.L.

Abstract

We propose an effective construction of Padé approximation for the hypergeometric function of special type and its derivative with respect to parameter and this parameter is included both in numerator and denominator of the general term of the corresponding series. The aforementioned construction has been realized by means of a modification of the method that was applied earlier in simpler situations. Thus obtained approximation is made use of afterwards to establish the lower estimates of the numerical linear forms in the values of the functions under consideration. One could obtain similar estimates by the known in the theory of transcendental numbers Siegel’s method, however the estimates obtained by this method would be of less exactness.

Russian Mathematics. 2019;63(12):61-70
pages 61-70 views

Boundary Control of the Heat Transfer Process in the Space

Fayazova Z.K.

Abstract

We consider the simplest mathematical model of the following problem. On a part of the border of a region D ⊂ ℝ2 there is a heater having the adjustable temperature. It is required to find a mode of operation of the heater such that the average temperature in a certain subregion of region D takes the specified value. The existence of the control parameter proved under certain restrictions on the values of the function defined by the integral constraint.

Russian Mathematics. 2019;63(12):71-79
pages 71-79 views

Solvability of Cauchy Problem for a Differential-Algebraic System with Concentrated Delay

Chuiko S.M.

Abstract

We study the Cauchy problem for a linear differential-algebraic system of equations with concentrated delay. Our research continues investigation of solvability of linear Noether boundary value problems for systems of functional-differential equations given in the monographs by A.D. Myshkis, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko, and A.A. Boichuk; meanwhile, we use essentially the tool of Moore-Penrose inverse matrices. For a linear differential-algebraic system with concentrated delay, we find sufficient conditions for its solvability and give a construction of generalized Green’s operator for Cauchy’s problem. We also give some examples which illustrate in detail the solvability conditions and the suggested construction.

Russian Mathematics. 2019;63(12):80-95
pages 80-95 views

Brief Communications

On Strong Solutions of a Fractional Nonlinear Viscoelastic Voigt-Type Model

Zvyagin V.G., Orlov V.P.

Abstract

Existence and uniqueness of a strong solution of the initial-boundary value problem for a system of the motion equations of a nonlinear viscoelastic fluid being a fractional analogue of the Voigt model are established in the plane case.

Russian Mathematics. 2019;63(12):96-100
pages 96-100 views

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