[en] A solution of a non-linear non-stationary equation for heat conductivity of a hollow cylinder with the first type boundary conditions on the inner and outer surfaces is considered. The solution is that the temperature dependencies of specific heat and heat conductivity factor are introduced into the respective derivatives with their expansions into the Fourier series in a finite area to follow. Following the elementary operations and a formal application of the reduction rule, the non-linear equation is transformed into a set of linear equations with the respective boundary conditions. On the basis of the obtained solutions of the system and their consecutive summation for a preheated hollow cylinder, analitical expressions are obtained. The expressions enable to determine the temperature fields in the cylinder thickness, and heat fluxes into the environment, the cylinder outer surface being flowed around by a cooling liquid