Obtaining exact analytical decisions of tasks heat conductions with variables in time boundary conditions

Abstract

On the basis use of additional required function and additional boundary conditions in an integral method of heat balance, the exact analytical decision of the nonstationary task of heat conduction for the infinite plate with a variable boundary condition of the first kind in time is received. Use of time-dependent additional required function is based on the heat conduction of the infinite speed of distribution of warmth described by the parabolic equation according to which temperature in any point of a plate begins to change right after the application of a boundary condition of the first kind on its surface. The additional boundary conditions used when obtaining the solution are found in such look that their execution by the required decision was equivalent to execution of a differential equation of boundary value problem in boundary points. It is shown that execution of the equation in boundary points, leads to its execution and in the considered area. Execution of integral of a heat balance, that is input differential equation, average within thickness of a plate, allows to consolidate the solution of a partial equation to integration of an ordinary differential equation of rather additional required function. Absence of need of integration of the input differential equation on space variable, being restricted only to execution of integral of a heat balance, allows to apply this method to the tasks including difficult differential equations (non-linear, with variable physical properties, etc.) which obtaining exact decisions by means of classical analytical methods isn't possible.