A constructive scheme for the construction of a solution of a mixed problem for the heat conduction equation with piecewise-continuous coeffi cients coordinate-dependent in the fi nal interval is suggested and validated in the present work. The boundary conditions are assumed to be most general. The scheme is based on: the reduction method, the concept of quasi-derivatives, the currently accepted theory of the systems of linear differential equations, the Fourier method, and the modifi ed method of eigenfunctions. The method based on this scheme should be related to direct exact methods of solving mixed problems that do not employ the procedures of constructing Green′s functions or integral transformations. Here the theorem of eigenfunction expansion is adapted for the case of coeffi cients that have discontinuity points of the 1st kind. The results obtained can be used, for example, in investigating the process of heat transfer in a multilayer slab under conditions of ideal thermal contact between the layers. A particular case of piecewise-continuous coeffi cients is considered. A numerical example of calculation of a temperature fi eld in a real four-layer building slab under boundary conditions of the 3rd kind (conditions of convective heat transfer) that model the phenomenon of fi re near one of the external surfaces is given. Introduction.There is now an extensive literature devoted to the contemporary analytical methods for calculating temperature fi elds in lamellar inhomogeneous systems. Such methods are conventionally divided into two types: a) direct or classical, i.e., those based on the Fourier method of separation of variables, and b) operational using various integral transformations [1,2].In classical monographs [1, 3], the problems of heat transfer for one-layer structures are solved methodically by direct and operational methods, for comparison. This allows one to make a choice between these two approaches in solving a specifi c problem.Monograph [1] has evidently been the fi rst to demonstrate the Laplace transform method in solving the problem on temperature fi eld distribution over the thickness of a system of two infi nite plates, where the problem in images was solved by a conjugation method. This seems altogether reasonable that it spurred the rapid development of such kind of approach in solving similar problems not only for parabolic-type equations.Another approach whose idea seems to evolve from work [4] affords the possibility to consider mathematical models

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Purpose. To develop an algorithm for calculating the problem of determining the nonstationary temperature field through the thickness of a multilayered structure, taking into account changes in the thermophysical characteristics and geometric dimensions (fluctuations) of the applied fire protection coating. Methodology. Application of the direct method for solving the differential equation of heat conduction using the method of reduction, the concept of quasiderivatives, the method of separation of variables and the modified method of eigenfunctions of Fourier. findings. An algorithm for determining the nonstationary temperature field in a multilayered flat structure is proposed, taking into account changes in the thermophysical characteristics and geometric dimensions (bursting expansion process) of the fire protection coating. This is achieved by solving a sequence of two tasks (the temperature field before the swelling and after the swell ing of the coating). originality. For the first time, using the direct method, in solving the problem of nonstationary heat conductivity, an algo rithm for determining the temperature field in multilayer elements with variable thickness of a layer on the example of building structures with flame retardant systems based on intumescent coatings is proposed. Practical value. Further, this approach can be implemented for approximation of solutions of heat conduction problems and it will allow catalyzing studies on fire retardant properties of intumescent coatings.

Introduction. The current urgent task is to find the temperature field distribution in cylindrical structures such as "solid cylinder inside a multilayer cylindrical shell". A characteristic feature of such structures is different mechanical and thermophysical characteristics of the layers combination, which makes them more perfect. However, this approach causes significant difficulties in developing analytical methods for their study. Therefore, new research methods development for multilayer, in particular, cylindrical structures is an urgent task today.Purpose. Direct method is used to study the heat transfer processes in the system "one-piece cylinder inside a multilayer cylindrical shell".Methods. To solve the initial parallel, the auxiliary problem of determining the distribution of a nonstationary temperature field in a multilayer hollow cylindrical structure with a "removed" cylinder of a sufficiently small radius is set. The solution of the auxiliary problem is realized by applying the method of reduction using the concept of quasi-derivatives. The Fourier schemeis used by using a modified method of eigenfunctions.Results. To find the solution to the problem, we used the idea of a boundary transition by directing the radius of the removedcylinder to zero. It is established that in this approach, all eigenfunctions of the corresponding problem have no singularities atzero, which means that the solutions of the original problem are limited in the whole structure. To illustrate the proposed method,a model example of finding the temperature field distribution in a four-layer column of circular cross-section (tubular concretecolumn) under the influence of the standard temperature of the fire. The results of the calculations are presented in the form of athree-dimensional graph of temperature changes depending on time and spatial coordinates.Conclusions. A direct method was used to solve the initial problem, using the idea of a boundary transition for the first time.In the general formulation (the function of changing the temperature of the environment is considered arbitrary, no restrictionsare imposed on the thickness of the shell and the number of layers) such a problem is solved for the first time.The structure of the obtained explicit exact formulas allows creating an algorithm for calculating the temperature field inthe form of automated programs, where it is enough to enter the initial data. Note that such algorithms include: a) calculating theroots of the characteristic equation; b) multiplication of a finite number of known matrices; c) calculation of definite integrals; d)summation of the required number of members of the series to obtain a given accuracy of the calculation.