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A two-dimensional mathematical model of the thermoelastic state has been built for a circular plate containing a curvilinear inclusion and a crack, under the action of a uniformly distributed temperature across the entire piece-homogeneous plate. Using the apparatus of singular integral equations (SIEs), the problem was reduced to a system of two singular integral equations of the first and second kind on the contours of the crack and inclusion, respectively. Numerical solutions to the system of integral equations have been obtained for certain cases of the circular disk with an elliptical inclusion and a crack in the disk outside the inclusion, as well as within the inclusion. These solutions were applied to determine the stress intensity coefficients (SICs) at the tops of the crack. Stress intensity coefficients could later be used to determine the critical temperature values in the disk at which a crack begins to grow. Therefore, such a model reflects, to some extent, the destruction mechanism of the elements of those engineering structures with cracks that are operated in the thermal power industry and, therefore, is relevant. Graphic dependences of stress intensity coefficients on the shape of an inclusion have been built, as well as on its mechanical and thermal-physical characteristics, and a distance to the crack. This would make it possible to analyze the intensity of stresses in the neighborhood of the crack vertices, depending on geometric and mechanical factors. The study's specific results, given in the form of plots, could prove useful in the development of rational modes of operation of structural elements in the form of circular plates with an inclusion hosting a crack. The reported mathematical model builds on the earlier models of two-dimensional stationary problems of thermal conductivity and thermoelasticity for piece-homogeneous bodies with cracks.

We study a nonlocal problem for ordinary differential equations of 2 ⁢ n {2n} -order with involution. Spectral properties of the operator of this problem are analyzed and conditions for the existence and uniqueness of its solution are established. It is also proved that the system of eigenfunctions of the analyzed problem forms a Riesz basis.

On the basis of results of experimental evaluation by 88 doctors and 100 pharmacists, the availability of information for them on probiotics has been investigated; new sources of such information have been detected, experts’ attitude to new probiotics has been studied. In the course of administration or recommendation of probiotics, experts are, mainly, guided by standards of medical aid and by their own experience. The main favorable effects of probiotics on human organism, in respondents’ opinion, are the normalization of the content of intestinal microflora and the normalization of functioning of digestive tract of human, as well as antialergetic and immunomodeling action. It is found that the main indication for application of probiotics are the syndrome of irritation of intestine and associated with antibiotic diarrhea. The averaged estimation (in points) of medicines has been calculated according to the following parameters: effectiveness, safety, frequency of prescription. With this, the competence of the experts was taken into account. The results of the carried out experts’ estimation can be used for optimization of the system of choice of probiotics

We examined a two-dimensional mathematical model for the problem of elasticity theory on welded dissimilar elastic half-planes containing rectilinear cracks under the action of mechanical efforts on the shores of a crack. As a consequence, the intensity of stresses in the vicinity of tops of the cracks increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks when the water, contained in them, freezes to ice. It creates normal pressure on the shores of the cracks. Based on the application of the apparatus of singular integral equations (SIE), the problem is reduced to the system of SIE of the first kind on the contours of cracks. We obtained numerical solutions to the corresponding integral equation in particular cases of two welded dissimilar half-planes with one randomly-oriented crack, as well as a two-link irregular crack, which crosses the line of junction when the crack’s shores are exposed to uniformly distributed normal pressure. By employing these solutions, we determined stress intensity coefficients (SIC) at the tops of the crack, which are subsequently used to determine critical values of the normal pressure on the shores of the crack. We built graphic dependences of SIC, which characterize distribution of the intensity of stresses at the tops of a crack, on the angle of crack inclination and elastic characteristics of half-planes. This makes it possible to analyze the intensity of stresses in the vicinity of a crack’s tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of normal pressure on the shores of the crack at which the growth of the crack starts, as well as the local destruction of the body. It is shown that the proper selection of elastic characteristics of the components of welded dissimilar half-planes can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIC at the crack’s tops

A growth of harmonic functions in the whole space ℝn is examined. We found the estimate for a uniform norm of spherical harmonics in terms of the best approximation of harmonic function in the ball by harmonic polynomials. An approximation error of harmonic function in the ball is estimated by the maximum modulus of an entire harmonic function in space, as well as the maximum modulus of an entire harmonic function in space in terms of the maximum modulus of some entire function of one complex variable or the maximal term of its power series. These results allowed us to obtain the necessary and sufficient conditions under which a harmonic function in the ball of an n-dimensional space, n≥3, can be continued to the entire harmonic one. This result is formulated in terms of the best approximation of the given function by harmonic polynomials. In order to characterize growth of an entire harmonic function, we used the generalized and the lower generalized orders. Formulae for the generalized and the lower generalized orders of an entire harmonic function in space are expressed in terms of the approximation error by harmonic polynomials of the function that continues. We also investigated the growth of functions of slow increase. The obtained results are analogues to classical results, which are known for the entire functions of one complex variable.The conducted research is important due to the fact that the harmonic functions occupy a special place not only in many mathematical studies, but also when applying mathematical analysis to physics and mechanics, where these functions are often employed to describe various stationary processes

The paper considers linear and non-linear mathematical models for the thermal conductivity process in designs that are described by a plate and a layered plate with a foreign parallelepiped-shaped through-inclusion on whose one boundary surface heat flux is concentrated. Classic methods do not make it possible to solve the boundary problems of mathematical physics that match these models in a closed form. Given this, in the present work we propose an approach that is based on the fact that the thermal-physical parameters for the piecewise uniform environments are described using generalized functions as a single entity for the whole system. As a result, we obtained one equation of thermal conductivity with generalized derivatives for the entire system with boundary conditions at the boundary surfaces of non-uniform environments. In a classic case, the process of thermal conductivity would be described by a system of equations of thermal conductivity for each of the elements of a non-uniform environment with conditions for an ideal thermal contact at the interface surfaces of non-uniform elements and boundary conditions on boundary surfaces of non-uniform environments. For the case of non-linear models, the condition of temperature equality at the interface surfaces of non-uniform elements of the designs is not applicable. With regard to the aforementioned, this work proposed yet another approach, which is in the introduction of linearizing functions that make it possible to linearize corresponding nonlinear boundary problems for these designs, which, as a result, allows us to solve this kind of boundary problems in mathematical physics. We received calculation formulas for determining the temperature field in the examined thermosensitive systems in the case of linearly variable coefficient of thermal conductivity of design materials. By using the obtained analytical-numerical solutions of linear and nonlinear boundary problems for the given piecewise-uniform structures, we created computational programs that make it possible to obtain the numerical values of temperature distribution and analyze the structures in terms of thermostability. As a result, it becomes possible to improve thermal stability of these designs and thus protect them from overheating, which can cause destruction of separate elements and even entire systems.

Context. The non-linear boundary value problem of heat conduction for a thermosensitive non-homogeneous strip-shaped element of a radio-electronic system with a through inclusion has been solved whose analytical-numerical solution enables us to analyze temperature regimes in the element. Objective. Is to develop such a method of linearization of mathematical model of heat conduction which enables us to obtain analytical numerical solution of the corresponding non-linear boundary value problem for determination of temperature field in elements of radio electronic devices, which are geometrically represented by a thermosensitive plate with a through inclusion. Method. A linearizing function which enables us to partially linearize the initial non-linear mathematical model of heat conduction for a thermosensitive non-homogeneous element of a radio electronic system in the form of “plate-inclusion” structure has been suggested. The introduced piece-wise linear approximation of temperature on plate-inclusion interfaces has enabled us to completely linearize the corresponding partially linearized boundary value problem relative to the linearizing function. After this, it became possible to apply Fourier’s integral transformation to the obtained linear problem with respect to one of the spatial coordinates, as well as to determine the linearizing function. The linear dependence of the coefficient of heat conductivity on temperature for structure materials with the use of the linearizing function has been considered. By solving the boundary value problem, the formulae for determination of temperature field in the “plate-inclusion” thermosensetive structure have been obtained. Results. The obtained formulae for determination of temperature field in a thermosensitive non-homogeneous element of radio electronic system were used to create the software which enables us to obtain distribution of value of temperature and to analyze temperature regimes. Conclusions. A mathematical model for the calculation for the temperature field in a “plate-inclusion” thermosensitive structure is adequate to the actual physical process, because no jump of temperature at “plate-inclusion” interfaces is observed. The numerical results for the chosen materials under linear dependence of the coefficient of thermoconductivity on temperature differ by 7% from the results which are obtained for constant coefficient of heat conductivity. Prospect of further investigation will consider more complicated geometric representation of elements of radio electronic systems.