Finding analytical solutions to the problems of thermal conductivity with variable physical properties of the medium by classical analytical methods is very complicated mathematically. The known expressions repre-senting complex infinite series including two types of Bessel functions and gamma-functions are, in fact, numerical as they require a numerical solution to complex transcendental equations with eigenvalues of the boundary problem. Such solutions can hardly be used in engineering applications, especially in cases when a solution to a certain problem is only an intermediate stage in other problems (such as thermoelasticity and control problems, inverse problems, etc.) which can be solved effectively only by finding analytical solutions to the initial problems. Therefore, an urgent problem now is to develop new methods of obtaining analytical solutions to the abovementioned problems, at least approximate ones. The study employed methods of additional boundary conditions and additional unknown functions in the integral method of heat balance. High-precision approximate analytical solutions to the transient heat conduction problem with nonhomogeneous physical properties of the medium for an infinite plate under symmetric boundary conditions of the first type have been obtained. The initial problem for partial differential equations is reduced to two problems in which ordinary differential equations are integrated. Additional boundary conditions are defined in such a way that their fulfillment in accordance with the new method is equivalent to the result of solving the initial partial differential equation at the boundary points and at the temperature perturbation front (for the first stage of the process). By combining methods with finite and infinite heat propagation rate we have been able to obtain high-precision analytical solutions for the whole time range of the unsteady process including its small and ultra small values. The solutions look like simple algebraical polynomials not including special functions (Bessel, Legendre, gamma-functions and others). Since it is not necessary to directly integrate the initial equations by the space variable and to reduce them to ordinary differential equations with additional unknown functions, the considered method can be used for solving complex boundary problems in which differential equations do not allow distinguishing between the variables (into nonlinear, with linear boundary conditions and heat sources, etc.).

The high-precision approximate analytic solution of the nonlinear quasi-static problem of thermoelasticity for an infinite hollow cylinder with variable along the radial coordinate physical properties is obtained using the orthogonal Bubnov-Galyorkin method developed by the construction of systems of coordinate functions exactly satisfying inhomogeneous boundary conditions in any approximation. The mathematical formulation includes non-linear equations for the unknown function of displacement and inhomogeneous boundary conditions. The desired solution is supposed to precisely satisfy the boundary conditions in advance. The exact fulfillment of the boundary conditions is achieved using the coordinate functions of special design. The unknown coefficients are found by constructing the disparity of original differential equation, that should be orthogonal to all the coordinate functions. Hence, the unknown coefficients of solution yields a system of linear algebraic equations, which number is equal to the number ofapproximations of the solution. It is shown that the solution accuracy increases substantially with increasing the number of approximations. Thus, already in the ninth approximation the disparity of original differential equation is zero almost the entire range of the spatial variable. The maximum disparity in the sixth approximation is $\varepsilon = 5\cdot 10^{-4}$.

An exact analytical solution of the thermoelasticity problem is presented for a multilayer shallow cylinder with physical properties of the medium that are constant in the limits of each layer. The effect of thermal shock of different intensity as well as the penetration depth of thermal disturbance on the distribution of radial and circumferential thermal stresses has been studied as applied to the one- and two-layer cylinder. It has been shown that the circumferential compressive stresses appear on the external surface of the cylinder under thermal shock while the tensile stresses appear on the internal surface.

Based on the mathematical apparatus of the generalized function theory and the Kantorovich orthogonal method, approximate analytical solutions of the thermal conductivity problem for a multilayered plate are obtained that allow for evaluating the thermal state of a construction practically in the entire time range of nonstationary process, including its small and minute values.

Theoretical properties of a method for construction of analytical solutions pertaining to heat conduction boundary value problem for multilayer structures have been given on the basis of the Kantarovich orthogonal method.Methodology for construction of coordinate function systems has been developed that satisfy boundary conditions in any approximation and satisfy matching conditions at any number of contacting bodies. Analytical solution of the heat conduction problem has been obtained on the basis of temperature perturbation front introduction and additional boundary conditions with regard to the last layer of the multilayer system. The solution makes it possible to evaluate temperature state of a structure for small and very small time values.

The technics for the construction of approximate analytical solutions for the quasistatic problems of thermoelasticity (plane-stressed state, plane deformation) for the multilayered bodies with variable within limits of each layer physical properties of medium. The recursive method is 'used for the construction of systems of coordinate functions, satisfying the boundary matching conditions, given as the equality of radial (normal) stresses and displacements in the layer-contact points.

Using double integral Laplace-Carson transformation and orthogonal method of Bubnov-Galyorkin, the analytical solution of the non-stationary problem of heat transfer in a cylindrical channel in the laminar flow of fluids was obtained. It has two components: stationary and non-stationary, each part has application only in a certain range of temporal and spatial coordinates. For the stationary Graetz-Nusselt problem on the basis of introduction of the temperature perturbation front and additional boundary conditions it was managed to find an analytical solution that allows the assessment of liquid thermal state with small values of spatial variable, directed along the stream flow. It is not possible to obtain such results using the well-known exact analytical methods because of the poor convergence of infinite series of received solutions.