METHOD OF OBTAINING EXACT ANALYTICAL SOLUTIONS OF TASKS OF HEAT CONDUCTIVITY WITH WARMTH SOURCES

By application of additional required function and additional boundary conditions to the integral method of heat balance, the exact analytical decision of the heat conductivity task for a semiinfinite plate was received in case of the symmetric boundary conditions of the first kind with uniformly distributed warmth source. Introduction of the additional required function representing change of temperature in time in plate center is based on the heat conduction of the infinite speed of warmth distribution described by the parabolic equation according to which temperature in any point of a plate begins to change right after application of a boundary condition of the first kind on its surface. Additional boundary conditions are so that their execution, by the required decision, was equivalent to execution of the equation of a boundary value problem in boundary points. In case of their finding the differential equation and the given boundary conditions is used. The general formulas given in article allow to find additional boundary conditions for any number of approaches. It is shown that execution of the equation in boundary points leads to its execution also in the area with an accuracy depending on number of approaches (number of additional boundary conditions). Use of an integral method of a heat balance allows to consolidate the solution of a partial equation to integration of the ordinary equation of rather additional required function. Absence of need of integration of an input equation on space variable allows to use this method in case of the solution of many difficult boundary value problems (non-linear, with float factors, etc.) for which it is difficult to receive the decision by means of classical exact analytical methods. Using the found analytical solution, and also results of temperature change in time in one of plate points received by method of finite differences, the solution of the reverse task of heat conductivity regenerated the power of an internal source of warmth. Results of operation can be used for identification of the sources of warmth arising in case of influence of electromagnetic waves, high-frequency oscillations and so forth, and also in case of melting or crystallization of the alloys which are followed by origin of internal sources of warmth.

non-stationary heat conductivity, semi-infinite plate, warmth source, infinite speed of warmth distribution, integrated method of thermal balance, exact analytical decision, additional required function, additional boundary conditions

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