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Copyright of National Academy of Sciences of Armenia Reports / Doklady Nacionalnaâ Akademiâ Nauk Armenii is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the present paper the plane problem of the gradient theory of elasticity in a thin rectangular region is considered. The asymptotic method is applied and an internal iterative process is constructed. Based on the results of the internal iterative process (respectively for the symmetric and inversely symmetric cases), the following is constructed: a) an applied model of the bending deformation of a thin beam according to the gradient theory of elasticity, b) an applied model of the compression tension of a beam according to the gradient theory of elasticity. The constructed applied models of bending deformation and compression tension of thin neams according to the gradient theory of elasticity will be further used to solve specific boundary value problems. [ABSTRACT FROM AUTHOR]

This paper is devoted to approximations of the product of two continuous functions on a finite segment by some special convolutions. The accuracy of the approximation depends on the length of the segment on which the functions are defined. These convolutions are generated by the Sturm-Liouville boundary value problems. The paper indicates that each boundary value problem for a second order differential equation generates its own individual convolution and its own individual Fourier transform. At that the Fourier transform of the convolution is equal to the product of the Fourier transforms. The latter property makes it possible to approximately solve nonlinear Burgers-type equations by first replacing the nonlinear term with a convolution of two functions. Similar methods of studying nonlinear partial differential equations can be found in the works of A. Y. Kolesov, N. H. Rozov, V. A. Sadovnichy. In this paper, we construct a concrete convolution generated by the Dirichlet boundary value problem for twofold differentiation. The properties of the constructed convolution and their connection with the corresponding Fourier transform are derived. In the final part of the paper, the convergence of convolution is proved (g(x) sin(x)) ∗ (f(x) sin(x)) defined on a segment C[0, b] to the product g(x)f(x) with b tending to zero for any two continuous functions f(x) and g(x).

This paper considers the stress state for a piecewise homogeneous plane, containing an absolutely rigid thin interfacial inclusion, one of the long sides of which is completely coupled to the plane, and the other side is in contact with it under dry friction conditions. On the basis of discontinuous solutions of the theory of elasticity for a piecewise homogeneous plane, a closed solution of the mixed boundary value problem is constructed. The regularities for the behavior of contact stresses near the end points of the inclusion are clarified, and simple formulas are obtained for determining both normal and tangential contact stresses acting on the long sides of the inclusion. [ABSTRACT FROM AUTHOR]

In present paper the applied (one-dimensional) model of micropolar elastic thin beam with a circular axis is constructed including the variation principles, on the basis of the two-dimensional equations of micropolar theory of elasticity with constrained rotation written in the polar coordinate system and previously developed hypotheses for thin bodies. Within the framework of this mathematical model, boundary problems with applied values are formulated, which are solved in the final form. Further, the scheme of application of finite element method (FEM) is developed for the boundary problems of micropolar elastic beam with a circular axis with constrained rotation. The boundary problems stated above are solved by using the FEM and on the basis of an analysis obtained through the numerical results, the specific property of micropolarity of the material is established, that it raises the rigidity of the beam as compared with the classical theory. [ABSTRACT FROM AUTHOR]

Copyright of Proceedings of the National Academy of Sciences of Armenia. Mechanics is the property of National Academy of Sciences of the Republic of Armenia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Radio Electronics, Computer Science, Control is the property of Zaporizhzhia National Technical University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper is an attempt to give the review of a part of our results in the area of singular partial differential equations. Using the results of the theory of complete generalized Jordan sets we consider the reduction of the PDE with the irreversible linear operator $B$ of finite index in the main differential expression to the regular problems. Earlier we and other authors applied similar methods to the development of Lyapunov alternative method in singular analysis and numerous applications in mechanics and mathematical physics. In this paper, we show how the problem of the choice of boundary conditions is connected with the $B$-Jordan structure of coefficients of PDE. The estimation of various approaches shows that the most efficient approach for solving this problem is the functional approach combined with the alternative Lyapunov method, Jordan structure coefficients and skeleton decomposition of irreversible linear operator in the main part of the equation. On this base, the problem of the correct choice of boundary conditions for a wide class of singular PDE can be solved. The aggregated theorems of existence and uniqueness of classical solutions can be proved with continuously depending of experimental definite function. The theory is illustrated by considering the solution of some integro - differential equations with partial derivatives.

The paper is devoted to investigation of non-local boundary problem for a system of partial differential equations with the operator B = (B1, . . . , Bp), where Bj = zjδ/δzj, j = 1, . . . , p, are operators of the generalized differentiation, which operates on complex variable zj. Problem is incorrect in the Hadamard sense and the solvability of this problem depends on the small denominators which arising in the construction of the solution. By using of metric approach, the theorem about lower estimation of small denominators was proved, and also existence and uniqueness conditions of this solution in the scale of spaces of many complex variables functions are establish. [ABSTRACT FROM AUTHOR]

The article describes the boundary-value problem of the theory of optimal control for matrix differential equations in the critical case. This problem is urgent for wide range of applications like modeling of linear technological and economic processes. A problem of controlling movement is reduced to solutions of boundary-value problems of n x n systems of ordinary differential equations of the first order. Solution of these tasks is very complex in case of control of systems in which the number of processes exceeds the amount of information about the initial state. The paper is devoted to finding conditions for solvability and construction solutions of boundary-value problems in critical case of the theory of motion control, in which the number of boundary conditions does not coincide with the number of unknowns in the system. The condition of solvability of such problems is found. Author describes the approach for solution to the problem with the help of theory of pseudoinverse matrices. In addition this approach is applicable to the problem of analytic construction of regulators and the optimal stabilization. [ABSTRACT FROM AUTHOR]

The paper considers the previously proposed method of numerical integration using the matrix calculus in the study of boundary value problems for nonhomogeneous linear ordinary differential equations of the second order with variable coefficients. According to the indicated method, when compiling a system of difference equations, an arbitrary degree of the Taylor polynomial in expanding the unknown solution of the problem into a Taylor series can be chosen while neglecting the approximation of the derivatives by finite differences. Some aspects of the convergence of an unstable second-order difference boundary value problem are investigated. The concept of a pseudo-residual on a certain vector is introduced for an ordinary differential equation. On the basis of the exact solution of the difference boundary value problem, an approximate solution has been built, where the norm of pseudo-residuals is different from the trivial value. It has been established theoretically that the estimate of the pseudo-residual norm decreases with an increase in the used degree of the Taylor polynomial and with a decrease in the mesh discretization step. The definitions of conditional stability and conditional convergence are given; a theoretical connection between them is established. The perturbed solution has been built on the basis of the found vector of pseudo-residuals, the estimate of the norm of its deviation from the exact solution of the difference boundary value problem has been calculated, which allows one to identify the presence of conditional stability. A theoretical relationship between convergence and conditional convergence is established. The results of numerical experiments are presented.

Wind boundary layer flow over the mountain landscape and large structures located around runways (RWs) creates coherent vortex structures (CVSs) that can cross a glideslope and airspace in the vicinity of an airport. The aircraft, encountering a vortex structure, experiences significant changes of the aerodynamic forces and moments, what is especially hazardous due to proximity to terrain. From a mathematical point of view, the solution of this problem presents a challenge due to extremely large space – time scale of the phenomenon, the lack of relevant atmospheric models, as well as comprehensive initial – boundary conditions in numerical modeling. In this paper, a composite solution is constructed: the CVSs area generation is computed in sufficient details within the framework of the grid method. Based on the data obtained in the approximation of analytical functions, an initial vortex structure is formed, the evolution and stochastics of which are modeled within the potential approximation by means of Rankine vortices. The evaluation of the forces and moments increment from the impact of vortex structures on the aircraft was carried out by the panel method using the engineering approach. As an example, the CVSs, resulting from wind flow around the mountainous area of the Son Tra Peninsula, that is located short of RWs 35R-17L and 35L-17R of Da Nang airport, are investigated. To improve the computational grids quality and verify the method of solving the boundary value problem for the Reynolds-averaged Navier-Stokes equations, we used the criteria based on the principle of maximum pressure, requiring Q-parameter positivity property in the vortices cores and flow separation regions. A CVS related aviation event, involving a passenger aircraft MC-21, is studied. The aircraft, after takeoff from RW 35R-17L setting the course close to the direction of the vortex wind structure axis from the Son Tra Peninsula, encountered the mountainous area CVS.

The paper considers structures consisting of rods connected in one node.Longitudinal and transverse vibrations of such a structure are described by systems of linear differential equations on star graphs.The noted system of equations consists of three linear differential equations of different orders.Two equations correspond to two transverse vibrations, and the third equation describes the longitudinal vibrations of the bar.Moreover, the system of three linear differential equations in the general case does not decompose. In this work, a fundamental system of solutions of a homogeneous system is constructed when the conjugation conditions are satisfied at the point of connection of the rods. Also, by the method of variation of arbitrary constants, a particular solution of an inhomogeneous system is constructed, which is subject to the conjugation conditions at the point of connection of the rods. In subsequent works, the authors intend to investigate the natural frequencies of longitudinal and transverse vibrations of a structure consisting of many rods

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first - stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas. In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value problems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found. [ABSTRACT FROM AUTHOR]

The paper includes the well-known matrix method of numerical integration of boundary value problems for inhomogeneous linear ordinary differential equations with variable coefficients, which provides retaining an arbitrary number of Taylor series expansion members of the sought-for solution or, equally, using the Taylor polynomial of arbitrary degree.The difference boundary value problem approximating the differential boundary value problem is divided into two subtasks: the first subtask includes difference equations, in the construction of which the boundary conditions of the boundary value problem were not used. The second subtask includes difference equations, in the construction of which the boundary conditions of the problem were used.Based on the earlier results, the method of increasing the order of approximation of the second subtask per unit, and, consequently, of the entire difference boundary problem as a whole is obtained and tested. The earlier findings are as follows:a) the order of approximation of the first and second subtasks is proportional to the degree of the Taylor polynomial used;b) the order of approximation of the first subtask depends on the parity or oddness of the degree of the Taylor polynomial used. It turned out that when using the degrees of the Taylor polynomial which are equal to $2m{-}1$ and $2m$, the approximation orders of these two subtasks are the same;c) the order of approximation of the second subtask coincides with the order of approximation of the first subtask, if the second subtask does not contain the specified values of any derivatives included in the boundary conditions;d) the presence in the second subtask of at least one derivative value of varying degrees included in the boundary conditions leads to a decrease in the order of approximation per unit in both the second subtask and the entire difference boundary value problem in general.The theoretical conclusions have been confirmed by numerical experiments.

The use of the second degree Taylor polynomial in approximation of derivatives by finite difference method leads to the second order approximation of the traditional grid method for numerical integration of boundary value problems for non-homogeneous linear ordinary differential equations of the second order with variable coefficients.The study considers a previously proposed method of numerical integration using matrix calculus which didn’t include the approximation of derivatives by finite difference method for boundary value problems of non-homogeneous fourth-order linear ordinary differential equations with variable coefficients. According to this method, when creating a system of difference equations, an arbitrary degree of the Taylor polynomial can be chosen in the expansion of the sought-for solution of the problem into a Taylor series.In this paper, the possible boundary conditions of a differential boundary value problem are written both in the form of derived degrees from zero to three, and in the form of linear combinations of these degrees. The boundary problem is called symmetric if the numbers of the boundary conditions in the left and right boundaries coincide and are equal to two, otherwise it is asymmetric.For a differential boundary value problem, an approximate difference boundary value problem in the form of two subsystems has been built. The first subsystem includes equations for which the boundary conditions of the boundary value problem were not used; the second one includes four equations in the construction of which the boundary conditions of the problem were used.Theoretically, the patterns between the order of approximation and the degree of the Taylor polynomial were identified.The results are as follows:a) the approximation order of the first and second subsystems is proportional to the degree of the Taylor polynomial used;b) the approximation order of the first subsystem is two units less than the degree Taylor polynomial with its even value and three units less with its odd value;c) the approximation order of the second subsystem is three units less than the degree Taylor polynomial regardless of both even-parity or odd-parity of this degree, and the degree of the highest derivative in the boundary conditions of the boundary value problem.The approximation order of the difference boundary value problem with all possible combinations of boundary conditions is calculated.The theoretical conclusions are confirmed by numerical experiments.

The paper considers the problem of choosing phase operators corresponding to cosine and sine of the angle of harmonic waves in the classical approximation. According to the principle of correspondence of classical evolution equations for variable phase sine and cosine to their quantum analogues, we introduce phase operators. This definition is quite ambiguous, because there is a whole class of operators classically equivalent to phase variables, but essentially different in a quantum area. We show that a set of operators can be reduced to the unique pair of operators under general physical requirements. We also demonstrate that operators are unambiguously defined by the density of phase variable distribution in the vacuum state. We also specify the method for constructing the phase operator by using moments of the density distribution of the phase variable in the vacuum state. This method provides a convenient tool for calculating some phase characteristics. We show that the infinite sequence of the moments of density uniquely defines the phase operator. Additionally, we obtain matrices of phase operators in the basis of occupation numbers. We also find explicit expressions for calculating arbitrary moments of the operators and analyze their behavior. [ABSTRACT FROM AUTHOR]

This paper considers the issues of optimal control for singular perturbed by the spatial value linear parabolic equations with nonlocal boundary value conditions and the quadratic performance criterion. We construct the complete asymptotes of optimal solutions for the initial problem under optimal conditions by boundary functions method. Unlike similar problems for parabolic equations with local boundary value conditions, the iterative problems for boundary functions do not "decompose". We prove that their solutions belong to the class of boundary functions, and complete decompositions are asymptotes of solutions of the corresponding order for the initial problem. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the nonviscous model of compact vortex close to the classical point vortex. The obtained vortex differs from the classical point vortex flow; its azimuthal velocity field is not potential. In addition to the classical flow solution, there is a term that makes the vortex compensated. The overall vorticity in it equals zero. Unlike the classical point vortex, the obtained one is compact and its velocity field meets the non-slip condition at the outer boundary like the (Taylor-Couette) flow between two coaxial cylinders. Furthermore, we introduce the nondimensional parameter, the function of radial coordinate indicating locally how the quasi-point vortex differs from the point vortex. The proposed model is an alternative to the point vortex one. The quasi-point vortex can be used in all relevant problems, where its counterpart (the point vortex) was used before. Crucially, its flow domain will always have finite scales just like it's in the nature. [ABSTRACT FROM AUTHOR]

Using the first three terms of Taylor expansion of the required function in the approximate derivative by finite differences leads to the second order approximation of the traditional numerical quadrature method of boundary value problems for linear ordinary second order differential equations with variable coefficients. The paper shows previously proposed numerical quadrature method using tools of matrix calculus where the approximate derivative by finite differences was not used. Agreeing to above method the arbitrary number of terms of Taylor expansion for the required solution may be used when compiling the difference equation system. When using the three first terms of expansion the difference equation system coincided with the traditional system. The estimation of residuals and the order of approximation depending on the number of the used terms of Taylor expansion is given. It is theoretically shown that for the boundary value problem with boundary conditions of the first kind the approximation method order increases in direct proportion with the increasing in the number of members used in Taylor series expansion only for odd values of this number. For even values of this number the order of approximation coincides with the order of approximation for the number less by unit of the odd values. For boundary value problems with boundary conditions of the second and third kinds the order of approximation was directly proportional to the number of used terms in the Taylor series expansion of the required solution of the problem regardless of evenness. In these cases the order of approximation of the boundary points and therefore the whole problem turned out to be one unit less than the order for the inner points of the grid for the interval of integration. The method of approximation order increase at the boundary points up to the approximation order in the inner points of the grid is presented. The theoretical conclusions are confirmed by a numerical experiment for a boundary value problem with boundary conditions of the first and third kinds.

In this paper the nonlocal boundary value problem for the pseudo-parabolic equation of the third-order in a multidimensional domain is considered. Using an iterative method, the solving process of the nonlocal boundary value problem is reduced to solving the series of some local problems. An a priori estimate for the convergence of the iterative method in the norm $W_2^1(G)$ is obtained.

The basic boundary value problems for semilinear equations of elliptic type with a spectral parameter and discontinuous nonlinearity are considered in a bounded domain with a sufficiently smooth boundary. The parameter values for which the corresponding problem has the nonzero solution are called eigenvalues. The existence of eigenvalue problem solutions for equations of elliptic type with discontinuous nonlinearities is considered in this paper. Estimations of the differential operator are obtained for these problems.

The eigenvalue problems for equations of elliptic type with discontinuous by the phase variable nonlinearities are considered. The character of nonlinearity discontinuities is investigated. In this paper the restrictions on discontinuity points of nonlinearity are weaker than in works of other authors.