Search

Contact problems for elastic stripes have been well studied and published in domestic scientific literature. This is partly due to the fact that normative documents on the foundation structure it is recommended to use this elastic foundation model for simulation of a “structure – foundation – soil foundation” system. Two variants of boundary conditions at the contact between a half-strip and a rigid non-deformable base are usually considered. The first boundary condition nullifies the vertical displacements and tangential stresses, the second one nullifies vertical and horizontal displacements. Contact problems for an elastic half-strip are much less investigated. The paper considers this contact problem when the first boundary condition for zeroing of vertical displacements and tangential stresses at the contact of a half-strip with a rigid, nondeformable base. When performing calculations in the traditional formulation without taking into account tangential stresses in the contact zone, the Zhemochkin method has been used, which reduces the solution of the contact problem of solid mechanics to the solution of a statically indeterminate problem by the mixed method of structural mechanics. Therefore, at first, we have found the displacements of the upper edge of the half-strip from the unit load uniformly distributed over the edge section. The resulting expression is used to compose a system of equations for the Zhemochkin method. The case of translational displacement of the die has been considered, and the graph of contact stress distribution under the die's sole has been given in the paper.

The paper considers one of the numerical methods to solve problems of scattering electromagnetic waves on the systems formed by parallel-oriented cylindrical elements – two-dimensional photonic crystals. The method is based on the classical partition approach used for solving the wave equation. Тhe method principle is to represent the field as the sum of the primary field and the unknown secondary field scattered on the medium elements. The mathematical expression for the latter is written as the infinite series according to elementary wave functions with unknown coefficients. In particular, the N elements-scattered field is found as the sum of N diffraction series in which one of the series is composed of the wave functions of one body and the wave functions in the remaining series are expressed in terms of the eigenfunctions of the first body using addition theorems. Further, to meet the boundary conditions, on the surface of each element, we obtain systems of linear algebraic equations with the infinite number of unknowns – the required expansion coefficients, which are solved by standard methods. A feature of the method is the use of analytical expressions to describe diffraction on a single element of the system. In contrast to most numerical methods, this approach allows one to obtain information on the amplitude-phase or spectral characteristics of the field only at the local points of the structure. The high efficiency of this method stems from the fact that there is no need to determine the field parameters in the entire area of space occupied by the multi-element system under consideration. The paper compares the calculated results of the transmission spectra of two-dimensional photonic crystals using the considered method with the experimental data and numerical results, obtained by other approaches, and demonstrates their good agreement.

The paper deals with determining a temperature field of an isotropic solid with inclusion represented as a spherical layer that absorbing penetrating radiation. A hierarchy of simplified analogues of the basic model of the heat transfer process in the system under study was developed, including a “refined model of concentrated capacity”, a “concentrated capacity” model, and a “truncated model of concentrated capacity”. Each of the mathematical models of the hierarchy is a mixed problem for a second-order partial differential equation of the parabolic type with a specific boundary condition that actually takes into account the spherical layer available in the system under study.The use of the Laplace integral transform and the well-known theorems of operational calculus in analytically closed form enabled us to find solutions to the corresponding problems of unsteady heat conduction. The “concentrated capacitance” model was in detail analysed with the object under study subjected to the radiation flux of constant density. This model is associated with a thermally thin absorbing inclusion in the form of a spherical layer. It is shown that it allows us to submit the problem solution of unsteady heat conduction in the analytical form, which is the most convenient in terms of both its practical implementation and a theoretical assessment of the influence, the spherical layer width has on the temperature field of the object under study.Sufficient conditions are determined under which the temperature field of the analysed system can be identified with a given accuracy through the simplified analogues of the basic mathematical model. For simplified analogues of the basic model, the paper presents theoretical estimates of the maximum possible error when determining the radiated temperature field.

Thermo-technical installations consuming significant amounts of thermal energy are used in order to intensify precast and reinforced concrete production processes under industrial conditions. Despite significant progress in the study of concrete hardening in accelerated hydration devices, a prominent lack of reliable and cost-effective research and optimization methods of their operation is observed. The methods used in real production processes are mainly based on empirical dependences obtained for specific technological conditions. These methods can not always be applied for other modes and technologies. The present paper develops calculation methods based on fundamental laws that make it possible to obtain functions for evolution of concrete product hydration process. Methods of mathematical modeling permit to develop new ways directed on improvement of modes for heat treatment of concrete products and accelerated hydration technologies. The paper describes a mathematical model for calculating a hardening process of a concrete product that includes a transient three-dimensional heat conductivity equation, a function of internal heat release due to behavior of exothermic reactions of cement hydration and also a system of initial and boundary conditions. A numerical simulation for temperature and hydration coefficient of a concrete product having shape of a 0.1´0.1´0.1 m cube has been performed in the paper. Verification of the non-stationary mathematical model for calculating temperature fields and hydration degree while using experimental data on concrete product strength obtained under industrial conditions. Investigations on hydration degree function of time have shown that experimentally obtained values of compressive strength correlate with hydration coefficient and hydration rate functions of heat treatment time which are calculated on the basis of the proposed non-stationary mathematical model of concrete product hardening. Satisfactory agreement of experimental and calculated data confirms adequacy of the proposed non-stationary mathematical model for calculating temperature fields and hydration degree with accelerated heat treatment of concrete products.

Исследуется вопрос об однозначной разрешимости первой смешанной задачи для вырождающегося параболического уравнения 2го порядка в случае, когда граничная и начальная функции принадлежат пространствам типа L2. Данная тематика берет начало с классических работ Ф. Рисса 1 и Литтлвуда и Пэли 2, посвященных граничным значениям аналитических функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина 3 9. Условие гладкости границы (Q C2) можно ослабить (см. 7). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерий существования граничного значения установлен в 7 9. При этом, как показано в работе 9, все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса 1 и Литтлвуда Пэли 2, 3. В случае вырождения типа Келдыша ситуация более сложная. Постановка первой краевой задачи и поведение решения вблизи границы определяются порядком вырождения уравнения, а в случае сильного вырождения коэффициентами при младших членах. Вопросам разрешимости первой краевой задачи для вырождающихся эллиптических и параболических уравнений посвящено большое число работ. Достаточно отметить работы Ф. Трикоми 10, М. В. Келдыша11, А. В. Бицадзе 12, С. А. Терсенова 13, И. М. Петрушко 14, 15, О. А. Олейник, Е. В. Радкевича 16, Г. Фикеры 17 и др. Из недавних работ можно отметить 18. В настоящей работе рассматривается случай сильного вырождения параболического уравнения 2го порядка, когда соответствующая квадратичная форма убывает как r(x) и постановка первой смешанной задачи определяется коэффициентом при первой производной по нормали., The paper investigates the question of the unique solvability of the first mixed problem for a degenerate secondorder parabolic equation in the case when the boundary and initial functions belong to spaces of type L2. This topic originates from the classical works of F. Riesz 1 and Littlewood and Paley 2 devoted to the boundary values of analytic functions. Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criterion for the existence of a boundary value was established in 7 9. The boundary smoothness condition (Q C2) can be weakened (see 10). Moreover, as shown in 9, all directions of the adoption of boundary values for uniformly elliptic equations turn out to be equal the solution has the property similar to the property of continuity in the set of variables. In the case of degeneration of the equation at the boundary of the region, when the directions are not equal, the situation is more complicated. Moreover, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are nonzero (Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equation are very close to the properties of a uniformly elliptic equation. In particular, in this situation, analogs of the Riesz theorems 1 and Littlewood Paley 2, 3 are valid. In the case of degeneracy of the Keldysh type, the situation is more complicated. The statement of the first boundaryvalue problem and the behavior of the solution near the boundary are determined by the order of degeneracy of the equation, and in the case of strong degeneracy, by the coefficients of the lower terms. A large number of papers have been devoted to the solvability of the first boundaryvalue problem for degenerate elliptic and parabolic equations. Note the works of F. Tricomi 11, M. V. Keldysh 12, A. V. Bitsadze 13, S. A. Tersenov 14, I. M. Petrushko 15, O. A. Oleinik and E. V. Radkevich 16, G. Fichera 17, etc. From recent works it is worth noting 18. In this paper, we consider the case of strong degeneracy of a secondorder parabolic equation when the corresponding quadratic form decreases as r(x) and the formulation of the first mixed problem is determined by the coefficient of the first normal derivative., №3(103) (2019)

This paper presents the results of a search for the optimal design of a high-orbit gyro-TWT, which would make it possible to reduce the magnetostatic field when operating at high frequencies close to the millimeter wavelength range, increase the gain and gain bandwidth, and increase the efficiency of the gyro-TWT. To search for the optimal configuration of the high-orbit gyro-TWT, the Gyro-K program was used, in which the equations for the excitation of an irregular waveguide by an electron beam are constructed on the basis of the coordinate transformation method of A.G. Sveshnikov, which is based on replacing the problem of exciting an irregular waveguide with the problem of exciting a regular waveguide with a unit radius. This method allows one to search for the solution of wave equations in the form of expansions in terms of the system of basis functions of a regular cylindrical waveguide. To solve Maxwell's equations, the Galerkin method was used, which is also called the orthogonalization method. The coefficients of the expansion of the field in terms of eigenbasic functions are determined in this method from the condition of the orthogonality of the residuals of the equations for the eigenbasis functions of a regular waveguide. The boundary conditions at the open ends of the waveguide are determined for each mode of the regular waveguide separately, which eliminates the incorrectness of setting the boundary conditions for the full field, as is the case when using the “picˮ technology. As a result, we obtain a system of ordinary differential equations for the expansion coefficients, which now depend only on the longitudinal coordinate. This approach makes it possible to transform the threedimensional problem of excitation of an irregular waveguide into a one-dimensional problem. Ohmic losses in the walls of the waveguide are taken into account on the basis of the Shchukin – Leontovich boundary conditions. For a self-consistent solution of the problem of excitation of an irregular waveguide by an electron beam, the iterative method of sequential lower relaxation was used. An optimized version of a high-orbit gyroTWT has been obtained, which has an electronic efficiency of 28 %, a wave efficiency of 23 %, a gain of 34 dB and a gain band of 11 % at an operating frequency of more than 30 GHz. This was achieved by introducing an additional conducting section of the waveguide into the absorbing part of the waveguide, which led to an improvement in the azimuthal grouping of electrons in the Larmor orbit and, as a consequence, to an increase in the lamp efficiency. A twofold increase in the waveguide length made it possible to increase the lamp gain. Ohmic energy losses in the walls of the waveguide reach 5 % of the power of the electron beam. The implementation of such a powerful gyro-TWT (2 MW) in the millimeter wavelength range will significantly increase the capabilities of radar at long distances and increase the resolution of the radar.

Purpose. We have found exact solutions to boundary-value problems for the inhomogeneous Helmholtz equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyperplanes. Methodology and Approach. The paper considers Dirichlet and Dirichlet-Neumann boundary-value problems with polynomials in the right-hand sides of the boundary conditions. The Fourier transform of generalized functions of slow growth is applied. Results. It is shown that the Dirichlet and Dirichlet-Neumann boundary-value problems with polynomials in the right-hand sides of the boundary conditions for the inhomogeneous Helmholtz equation with the polynomial right-hand side have a solution that is a quasi-polynomial containing, in addition to power functions, hyperbolic or trigonometric functions. This solution is unique in the class of functions of slow growth if the parameter of the equation is not an eigenvalue. An algorithm for constructing this solution is presented and examples are considered. Theoretical and Practical Implications. Exact solutions to boundary-value problems for one of the well-known equations of mathematical physics have been obtained.

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.

В данной работе, являющейся продолжением [1], устанавливаются необходимые и достаточные условия того, чтобы решение параболического уравнения 2-го порядка с боковой границей, принадлежащей классу $C^{1+\lambda}$, $\lambda >0$, вырождающегося на границе области, имело предел в среднем на боковой поверхности цилиндрической области и предел в среднем на ее нижнем основании, и исследуется вопрос об однозначной разрешимости первой смешанной задачи для такого уравнения в случае, когда граничная и начальная функции принадлежат пространствам типа $L_2$. Наиболее близкими к рассматриваемому кругу вопросов являются теоремы Ф. Рисса и Ж. Литтлвуда и Р. Пэли, в которых даются критерии предельных значений в $L_p$, $p > 1$, аналитических в единичном круге функций. Дальнейшее развитие этой тематики для равномерно эллиптических уравнений получило в работах В. П. Михайлова, А. К. Гущина [2–4]. Условие гладкости границы ($\partial Q \in C^2$) можно ослабить (см. [5]). При наиболее слабых ограничениях на гладкость границы (и на коэффициенты уравнения) критерии существования граничного значения установлены в работах [4, 6–8]. При этом, как показано в работе [7], все направления принятия граничных значений для равномерно эллиптических уравнений оказываются равноправными, решение обладает свойством, аналогичным свойству непрерывности по совокупности переменных. В случае вырождения уравнения на границе области, когда направления не являются равноправными, ситуация более сложная. При этом постановка первой краевой задачи определяется типом вырождения. В случае, когда значения соответствующей квадратичной формы вырождающегося эллиптического уравнения на векторе нормали отличны от нуля (вырождение типа Трикоми), корректна задача Дирихле, и свойства такого вырождающегося уравнения весьма близки к свойствам равномерно эллиптического уравнения. В частности, в этой ситуации справедливы аналоги теорем Рисса [9] и Литтлвуда — Пэли [10, 11]., This work, being a continuation of [1], establishes the necessary and sufficient conditions for the solution of the second-order parabolic equations with a lateral boundary from the class $C^{1+\lambda}$, $\lambda >0$, degenerating on the boundary of the domain, to have an average limit on the lateral surface of the cylindrical domain and the limit in the mean on its lower base. Also, we study the question of the unique solvability of the first mixed problem for such equations in the case when the boundary and initial functions belong to spaces of the типа $L_2$ type. The closest to the questions under consideration are the theorems of F. Riesz and J. Littlewood and R. Paley, in which criteria are given for the limit values in $L_p$, $p > 1$, of the functions analytic in the unit disk. Further development of this topic for uniformly elliptic equations was obtained in the papers by V. P. Mikhailov and A. K. Gushchin [2–4]. The boundary smoothness condition ($\partial Q \in C^2$) can be weakened (see [5]). Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criteria for the existence of a boundary value were established in [4, 6–8]. In this case, as shown in [7], all directions of the acceptance of boundary values for uniformly elliptic equations turn out to be equal, while the solution has a property similar to the property of continuity with respect to the set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation is more complicated. In this case, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are different from zero (the Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equations are very close to the properties of uniformly elliptic equations; in particular, in this situation analogues of the Riesz [9] and Littlewood–Paley theorems [10, 11] are valid., Журнал «Математические заметки СВФУ», Выпуск 2 (106) 2020

Introduction. An efficient technique of tensor field scalarization is successfully used while investigating tensor elastic fields of displacements, stresses and deformations in the layered structures of different materials, including transversally isotropic composites. These fields can be expressed through the scalar potentials corresponding to the quasi-longitudinal, quasi-transverse, and transverse-only waves. Such scalarization is possible if the objects under consideration are tensors relating to the subgroup of general coordinate conversions, when the local affine basis has one invariant vector that coincides with the material symmetry axis of the material. At this, the known papers consider structures where this vector coincides with the normal to the boundary between layers. However, other cases of the mutual arrangement of the material symmetry axis of the material and the boundaries between layers are of interest on the practical side.Materials and Methods. The work objective is further development of the scalarization method application in the boundary value problems of the dynamic elasticity theory for the cases of an arbitrary arrangement of the material symmetry axis relative to the boundary between layers. The present research and methodological apparatus are developed through the general technique of scalarization of the dynamic elastic fields of displacements, stresses and strains in the transversally isotropic media.Research Results. New design ratios for the determination of the displacement fields, stresses and deformations in the transversally isotropic media are obtained for the cases of an arbitrary arrangement of the material symmetry axes of the layer materials with respect to the boundaries between layers. Discussion and Conclusions. The present research and methodological apparatus are successfully used in determining the stress-strain state in the layered structures of transversally isotropic materials, and in analyzing the diagnosis results of the state of the plane-layered and layered cylindrical structures under operation.

It is well known that the Dirichlet problem for the Laplace equation in a ball has a unique polynomial solution (harmonic polynomial) in the case if the given boundary value is the trace of an arbitrary polynomial on the sphere. S.M.Nikol'skii generalized this result in the case of a boundary value problem of the first kind for a linear differential self-adjoint operator of the order 2l with constant coefficients (in particular polyharmonic) and for a domain that is an ellipsoid in R n . For a polyharmonic equation in a ball (homogeneous and inhomogeneous), V.V. Karachik proposed the Almansi formula-based algorithm to construct a polynomial solution of the Dirichlet problem. The paper considers the Poisson equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyper-planes. Shows that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth, and this solution is a polynomial. Gives an algorithm for constructing this polynomial solution and considers examples. In particular, presents formulas to give exact values of certain integrals (including multi-dimensional ones) and sums of trigonometric series.

The paper presents a mathematical formulation of the problem formalization of the subject area related to aviation security in civil aviation. The formalization task is determined by the modern issue of providing aviation security. Aviationsecurity in modern systems is based upon organizational standard of security control. This standard doesn’t require calculating the security level. It allows solving the aviation security task without estimating the solution and evaluating the performance of security facilities. The issue of acceptable aviation security level stays unsolved, because its control lies in inspections that determine whether the object security facilities meet the requirements or not. The pending problem is also in whether the requirements are calculable and the evaluation is subjective.Lately, there has been determined quite a certain tendency to consider aviation security issues from the perspective of its level optimal control with the following identification, calculation and evaluation problems solving and decision making. The obtained results analysis in this direction shows that it’s strongly recommended to move to object formalization problem, which provides a mathematical modeling for aviation security control optimization.In this case, the authors assume to find the answer in the process of object formalization. Therefore aviation security is presented as some security environment condition, which defines the parameters associated with the object protec-tion system quality that depends on the use of protective equipment in conditions of counteraction to factors of external andinternal threats. It is shown that the proposed model belongs to a class of boundary value problems described by differential equations in partial derivatives. The classification of boundary value problems is presented.

Introduction. The work objective is to develop a new universal numerical method for solving boundary value problems for linear elliptic equations. Materials and Methods . The proposed method is based on the transformation of the original mathematical physics equation to a simpler inhomogeneous equation with the known fundamental solution. From this equation, the transition to an inhomogeneous integral equation with the kernel expressed by the known fundamental solution is carried out. The obtained integral equation with boundary conditions is solved numerically. An approximate solution, the field potential being in an analytical form, is resulted. That allows not only find an approximate value of the field potential at any point in the solutions domain, but also differentiate this potential, and all without perceptible loss of accuracy. This property of the developed numerical method sets it apart from the traditional numerical methods for solving boundary value problems, such as the finite element method. Research Results . To confirm the effectiveness of the proposed numerical method, the two-dimensional and three-dimensional boundary value problems with the known solutions are solved. The dependences of the numerical solution error on the number of linear equations in the resulting system are obtained. It is shown that even at a small number of equations in the system (some hundreds) the solution accuracy is achieved at the level of hundredths of a percent. Another major illustration of the proposed method effectiveness is the solution to quantum mechanical problems for the one-dimensional and two-dimensional quantum oscillators. It is shown that the given method allows finding the energy eigenvalues and eigenfunctions with an acceptable accuracy. The developed numerical technique allows greatly extend the application domain of the traditional point source method in solving applied problems for modeling fields of different physical nature, including the eigenvalue problems. Discussion and Conclusions. The results obtained confirm that a physical field described by any linear elliptic equation can be represented as a superposition of point source fields satisfying a simpler equation, the solution of which is obtained through the method of point source of the field. Therefore, the numerical method presented in this paper can be considered as a generalized point source method.

Stationary processes of different physical nature (oscillations, thermal conductivity, diffusion, electrostatics, etc.) are described by equations of elliptic type. In particular, some models, such as hydro and gas dynamics, consider elliptic equations. In this paper, we study a nonlocal boundary value problem with the Poincaré condition for an equation of the elliptic-hyperbolic type of the second kind, i.e. for an equation where the line of degeneration is a characteristic. Refs.: 8 titles.Keywords: nonlocal boundary value problem, Poincaré conditions, equations of elliptic - hyperbolic type, equations of the second kind., Стационарные процессы различной физической природы (колебания, теплопроводность, диффузия, электростатика и т.д.) описываются уравнениями эллиптического типа. В частности, некоторых моделях, таких, как гидро и газовой динамики рассматриваются эллиптические уравнения. В данной работе изучается нелокальная краевая задача с условием Пуанкаре для уравнения эллиптико-гиперболического типа второго рода, т.е. для уравнения, где линия вырождения является характеристикой. Библиогр.: 8 назв.Ключевые слова: нелокальная краевая задача, условия Пуанкаре, уравнения эллиптико – гиперболического типа, уравнения второго рода., Стаціонарні процеси різної фізичної природи (коливання, теплопровідність, дифузія, електростатика і т.д.) описуються рівняннями еліптичного типу. Зокрема, у деяких моделях, таких, як гідро і газової динаміки розглядаються еліптичні рівняння. У даній роботі вивчається нелокальна крайова задача з умовою Пуанкаре для рівняння елліптіко-гіперболічного типу другого роду, тобто для рівняння, де лінія виродження є характеристикою. Бібліогр.: 8 назв.Ключові слова: нелокальна крайова задача; умови Пуанкаре; рівняння елліптіко-гіперболічного типу другого роду.

Analytical study of the processes of heat conduction is one of the main topics of modern engineering research in engineering, energy, nuclear industry, process chemical, construction, textile, food, geological and other industries. Suffice to say that almost all processes in one degree or another are related to change in the temperature condition and the transfer of warmth. It should also be noted that engineering studies of the kinetics of a range of physical and chemical processes are similar to the problems of stationary and nonstationary heat transfer. These include the processes of diffusions, sedimentation, viscous flow, slowing down the neutrons, flow of fluids through a porous medium, electric fluctuations, adsorption, drying, burning, etc. There are various methods for solving the classical boundary value problems of nonstationary heat conduction and problems of the generalized type: the method of separation of variables (Fourier method) method; the continuation method; the works solutions; the Duhamel's method; the integral transformations method; the operating method; the method of green's functions (stationary and non-stationary thermal conductivity); the reflection method (method source). In this paper, based on the consistent application of the Laplace transform on the dimensionless time θ and finite sine integral transformation in the spatial coordinates X and Y solves the problem of unsteady temperature distribution on the mechanism of heat conduction in a parallelepiped with boundary conditions of first kind. As a result we have the analytical solution of the temperature distribution in the parallelepiped to a conductive mode free convection, when one of the side faces of the parallelepiped is maintained at a constant temperature, and the others with the another same constant temperature.

Предложен и реализован новый вариант метода коллокации и наименьших квадратов повышенной точности для численного решения уравнения Пуассона. Реализованный алгоритм применяется в неканонических областях, границы которых заданы дискретно. Для приближенного и однозначного задания границы области по ее дискретным данным в прямоугольной системе координат строится параметрический двойной сплайн, в качестве компонент которого взяты два кубических сплайна. Используется идея присоединения вытянутых несамостоятельных нерегулярных граничных ячеек к соседним самостоятельным с целью уменьшения обусловленности глобальной системы линейных алгебраических уравнений., The paper addresses a new version of the least squares collocation (LSC) method proposed and implemented for the numerical solution of boundary value problems for the Poisson’s equation in case of given discrete domain boundary. The computer program builds a continuous double spline when boundary is smooth or a piecewise smooth double spline if boundary has salient points. In this version of the method we apply the idea of using parts of the cells of a regular grid (outside the domain). These parts of the cells are cut off by the boundary to construct the LSC method. It is assumed that the solution has no singularities at the boundary and in a certain small neighborhood of it. The differential equation for the problem is correct not only in the computational domain but also in a small neighborhood of its boundary. Then the idea of attaching “small” irregular cells to neighboring ones is used in the work with the aim of reducing the number of conditionality of the global system of linear algebraic equations. It is shown that the approximate solutions obtained by the LSC method converge with an increased order and coincide with the analytical solutions of the test problems with high accuracy., №3(23) (2018)

Изучается разрешимость начально-краевой задачи с условиями сопряжения для двух неклассических дифференциальных уравнений составного типа. Описывается случай, когда коэффициенты каждого рассматриваемого уравнения имеют разрыв 1-го рода в точке нуль. Область исследований задана в виде полосы ввиду наличия точки разрыва, состоящей из двух подобластей. Таким образом, исследуемые уравнения рассматриваются в двух различных областях. Для доказательства существования и единственности регулярных решений (которые имеют все обобщенные производные, входящие в уравнения) начально-краевой задачи используется метод продолжения по параметру, имеющий широкое применение в теории краевых задач. С помощью принципа максимума устанавливается наличие всех необходимых априорных оценок для решений изучаемой задачи., In this paper we study the solvability of an initial-boundary value problem with conjugation conditions for two nonclassical differential equations of composite type. We describe the case when the coefficients of each equation under consideration have a discontinuity of the first kind at the point zero. The field of research is given in the form of a band, due to the presence of a discontinuity point consisting of two subregions. Thus, the investigated equations are considered in two different areas. To prove the existence and uniqueness of regular solutions (which have all the generalized derivatives entering into the equations) of the initial-boundary value problem, we use the method of continuation with respect to a parameter, which has a wide application in the theory of boundary value problems. Using the maximum principle, the presence of all necessary a priori estimates for the solutions of the problem being studied is established., №2(98) (2018)

При распараллеливании решения трехмерных краевых задач, особенно в областях со сложной геометрией, важными являются технологии проведения вычислений и структуры данных. От них зависит объем хранимой информации и время решения. В статье предлагаются технологии распараллеливания метода декомпозиции расчетной области на подобласти, сопрягаемые без наложения, на квазиструктурированных сетках. Разработаны параллельные сеточные структуры данных, ориентированные преимущественно на работу со структурированными массивами данных. Приведен иллюстративный пример, показывающий основные положения предлагаемого подхода., When parallelizing the solution of three-dimensional boundary value problems, especially in domains with complex geometry, the сomputational technologies and data structureы are important. The amount of stored information and the computational time depend on them. In this paper we propose the technologies for parallelizing the method of decomposition of the computational domain into subdomains conjugated without overlapping on a quasistructured grid. Parallel grid data structures oriented mainly to work with structured data arrays are developed. An illustrative example clarifying the fundamentals of the proposed approach is discussed., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 4 2018

Статья посвящена ускорению параллельного решения трехмерных краевых задач методом декомпозиции расчетной области на подобласти, сопрягаемые без наложения. Декомпозиция проводится равномерной параллелепипедальной макросеткой. В каждой подобласти и на границе сопряжения (интерфейсе) строятся свои структурированные подсетки. Объединение этих подсеток образует квазиструктурированную сетку, на которой решается поставленная задача. Распараллеливание решения осуществляется при помощи MPI-технологий. Предложен и экспериментально исследован алгоритм ускорения внешнего итерационного процесса по подобластям для решения системы линейных алгебраических уравнений, аппроксимирующих уравнение Пуанкаре-Стеклова на интерфейсе. Проведены серии численных экспериментов на различных квазиструктурированных сетках и при различных параметрах вычислительных алгоритмов, показывающих ускорение вычислений., This paper is devoted to the acceleration of the parallel solution of three-dimensional boundary value problems by the computational domain decomposition method into subdomains that are conjugated without overlapping. The decomposition is performed by a uniform parallelepipedal macrogrid. In each subdomain and on the interface, some structured subgrids are constructed. The union of these subgrids forms a quasi-structured grid on which the problem is solved. The parallelization is carried out using the MPI-technology. We propose and experimentally study the acceleration algorithm for an external iterative process on subdomains to solve a system of linear algebraic equations approximating the Poincare-Steklov equation on the interface. A number of numerical experiments are carried out on various quasi-structured grids and with various parameters of computational algorithms showing the acceleration of computations., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 2 2018

Изучается влияние ширины канала на гидроупругие волны в канале, покрытом льдом, вызванные движением нагрузки вдоль ледового покрова. Внешняя нагрузка моделируется гладким локально распределенным давлением. За основу математической модели берутся дифференциальное уравнение колебаний вязкоупругой ледовой пластины и уравнение Лапласа для потенциала скорости течения жидкости под ледовым покровом. Данные уравнения замыкаются граничными условиями непротекания на стенках и дне канала, условиями жесткого защемления льда на стенках канала, кинематическим и динамическим условием на границе раздела лед – жидкость. Исследуется решение в виде бегущей волны, которое не зависит от времени в системе координат, движущейся вместе с внешней нагрузкой. С помощью преобразования Фурье по переменной, направленной вдоль канала, рассматриваемая задача сводится к двумерной задаче относительно профиля волны поперек канала, которая решается методом разложения профиля волны на нормальные моды колебаний закрепленной балки. Задача о колебаниях бесконечного ледового покрова решается с помощью двойного преобразования Фурье. Проведен анализ численных результатов при увеличении ширины канала, а также сравнение полученного решения с решением для бесконечной пластины.DOI 10.14258/izvasu(2016)1-35, In this paper, the channel width influence on hydroelastic waves in an ice-covered channel is investigated. Waves are generated by a load moving along the ice sheet. External load is modeled by a localized smooth pressure distribution. The differential equation of oscillations of viscoelastic ice plate and the Laplace equation for the velocity potential of fluid flow under the ice cover are used as a mathematical model. These equations are supplemented with impermeability boundary conditions on channel walls and bottom, clamped conditions for ice on the channel walls, and the kinematic and the dynamic condition on the ice – liquid interface. The time-independent travelling wave solution in a coordinate system moving with the external load is studied. Applying the Fourier transformation allows the initial problem to be reduced to the two-dimensional problem of wave profile across the channel, which is solved by the normal mode method for a fixed beam. The problem of deflections in the infinite ice sheet is solved by double Fourier transformation. Numerical results of the channel width influence are discussed. The ice defection and strains in the ice sheet are calculated and compared with the results of problem solution for infinite ice plate.DOI 10.14258/izvasu(2016)1-35

The paper considers the problem of increasing the approximation order of transparent boundary conditions for the wave equation while using finite difference schemes up to the sixth order of accuracy in space. As an example, the problem of wave propagation in a semi-infinite rectangular waveguide is formulated. Computationally efficient and highly accurate formulas for discretizing operator of transparent boundary conditions are proposed. Numerical results confirm the accuracy and stability of the obtained difference algorithms.

Решена задача Неймана для обыкновенного дифференциального уравнения дробного порядка с постоянными коэффициентами. Построена функция Грина, доказана конечность числа вещественных собственных значений., A linear ordinary differential equation of fractional order with constant coefficients is considered in the paper. Such equation should be subsumed into the class of discretely distributed order, or multi-term differential equations. The fractional differentiation is given by the Caputo derivative. We solve The Nuemann problem for the equation under study, prove the existence and uniqueness of the solution, find an explicit representation for solution in terms of the Wright function, and construct the respective Green function. It is also proved that the real part of the spectrum of the problem may consist at most of a finite number of eigenvalues.

Purpose. The work proposed for the modeling of transients in Lehera line uses a modified Hamilton-Ostrogradskiy principle. The above approach makes it possible to avoid the decomposition of a single dynamic system that allows you to take into account some subtle hidden movements. This is true for systems with distributed parameters, which in the current work we are considering. Methodology. Based on our developed new interdisciplinary method of mathematical modeling of dynamic systems, based on the principle of modified Hamilton-Ostrogradskiy and expansion of the latter on the non-conservative dissipative systems, build mathematical model Lehera line. The model allows to analyze transient electromagnetic processes in power lines. Results. In this work the model used for the study of transients in the non-working condition Lehera line. Analyzing the results shows that our proposed approach and developed based on a mathematical model is appropriate, certifying physical principles regarding electrodynamics of wave processes in long power lines. Presented in 3D format, time-space distribution function of current and voltage that gives the most information about wave processes in Lehera line at non-working condition go. Originality. The originality of the paper is that the method of finding the boundary conditions of the third kind (Poincare conditions) taking into account all differential equations of electric power system, i.e. to find the boundary conditions at the end of the line involves all object equation. This approach enables the analysis of any electric systems. Practical value. Practical application is that the wave processes in lines affect the various kinds of electrical devices, proper investigation of wave processes is the theme of the present work., В работе, на основе обобщенного междисциплинарного (интердисциплинарного) метода математического моделирования, основанного на модификации интегрального вариационного принципа Гамильтона-Остроградского, предложена математическая модель двухпроводной длинной линии электропередач, которая работает на холостом ходу. Представлены результаты компьютерной симуляции переходных процессов в виде рисунков, которые анализируются.

In the paper we construct the adjoint problem for the explicit and implicit parabolic quazi-linear grid boundary-value problems with one spatial variable; the coefficients of the problems depend on the solution at the same time and earlier times. Dependence on the history of the solution is via the state vector; its evolution is described by the differential equation. Many models of diffusion mass transport are reduced to such boundary-value problems. Having solutions to the direct and adjoint problems, one can obtain the exact value of the gradient of a functional in the space of parameters the problem also depends on. We present solving algorithms, including the parallel one.

In this paper, we study quantum star graphs with time-dependent bond lengths. Quantum dynamics are treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. The time-dependence of the average kinetic energy is analyzed. The space-time evolution of a Gaussian wave packet is treated for an harmonically breathing star graph.

The prediction of the laminar/turbulent transition location in supersonic boundary layers plays an important role to accurately compute aerodynamic forces and heating rates for the aerodynamic design and control of hypersonic vehicles. The stability characteristics of supersonic boundary layers depend e.g. on nose bluntness, transverse curvature, wall temperature, shock waves, etc. Most parameters can be theoretically investigated by performing conventional stability calculations with vanishing or asymptotic perturbation conditions at the far field. In this approach the formation of a shock in front of the leading edge of a blunt body is ignored. However, to improve the understanding of the interaction between instability waves originating inside supersonic boundary layer with those coming from the inviscid entropy layer, the presence of the shock has to be taken into account. This paper presents a method, how shock effects can be physically consistently included in stability calculations. The outer free-stream boundary conditions are replaced by linebreak appropriate shock conditions. The required perturbation equations can be derived from the linearized unsteady Rankine,-,Hugoniot equations, accounting for the effect of shock oscillations due to perturbated waves which originate from the flow field windward of the shock.