The studied problem for the essentially loaded heat equation is connected with mathematical modeling of thermophysical processes in the electric arc of high-current disconnecting devices. Experimental studies of such phenomena are difficult due to their transience, and in some cases only a mathematical model is able to provide adequate information about their dynamics. The study of the mathematical model is carried out when the order of the derivative in the loaded summand is less than, equal to and greater than the order of the differential part of the heat equation, at a fixed point of the load and in the case when the load point moves at a variable speed. The article is focused mainly on scientific researchers engaged in practical applications of loaded differential equations.

On the calculation of the rectangular finite element of the plate The article is devoted to the study of the thin plate bending by the finite element method. The application of the finite element method to solving the problem of the plate bending leads to the necessity of studying the rectangular finite element of the plate. All deformation and statics characteristics of the plate are functions of the displacement in the direction of the normal to the middle surface of the plate, which is determined by the deflection function. In the article, the formation of the plate deflection function in explicit form is carried out. The ways for finding the deflection function by division of the variables in the equilibrium equation of the plate, through an incomplete fourth-degree polynomial and in the form of Hermite polynomials are presented. The article is focused mainly on mechanics, engineers and scientific employees of technical specialties.

On the application of mathematical methods for the research of vibration processes in mechanics Article represents the study of applied problems of mathematics, whose mathematical modeling leads to boundary problems for equations in partial derivatives. Mathematical methods, applied to these models, enable to obtain exact analytical results. Detailed result is represented for boundary problem of oscillations of thin structures with boundary conditions in general terms. Application of spectral decomposition for sufficiently smooth function, characterizing the membrane deviation from equilibrium state, enables to define exact analytic representation of inflection function for studied problem. To calculate multilayer plates, method of finite elements is applied.

On the calculation of rectangular plates by the collocation method The article is devoted to the application of the collocation method to solving differential equations, which are the basis for calculating many problems of mechanics. In this article the structure of this method is presented, its main components are highlighted; its types are characterized, as well as its classical approaches. The research of the problem of rectangular plates bending is carried out by the method of collocations in this article. The collocation method, like all numerical-analytical approximate methods, has a number of advantages and disadvantages, which are also noted in this article. The article is focused mainly on mechanics, engineers and technical specialists.

The article is devoted to the study of plate bending problems, which are of great applied importance and are found everywhere in various branches of science and technology. In this article the structure of the calculation methods is described, their main components are highlighted; the classical approach of calculating rectangular plates hinged supported on two parallel sides and with arbitrary boundary conditions on each of the other two sides is characterized. The mathematical apparatus of the method of trigonometric series is presented in the volume necessary for calculating the plates. Special cases of the calculation for the bending of a rectangular plate by the Levi method are given. This article is focused mainly on mechanics, physicists, engineers and technical specialists.