Special averaging method for solving some 3-D heat transfer problems

We consider the special conservative averaging method for solving the heat transfer boundary-value problem in the 3-D domain. Looking back to the history of mathematics, integral parabolic splines related to the conservative averaging method (CAM), in 1980s A. Buikis developed CAM for partial differential equations with discontinuous coefficients in environments with a layered structure. We consider newly designed for CAM the special integral hyperbolic type splines for solving some 3-D heat transfer initial-boundary value problems. Using these types of splines, the present problem is reduced to the 2-D problem with respect to one coordinate. This procedure also allows reducing the 2-D problem to 1-D initial-boundary value problem and to initial value problem for ordinary differential equations. Thus, the solution to the problem can be obtained analytically. The solution of corresponding 3-D initial-boundary value problem is obtained also numerically, using for approach differential equations of discretization in space applying the differences. This method can be considered as a generalization of the method of finite volumes, because it can solve BVP with convection, not only in z-direction, but also in y-direction and x-direction, based on the CAM, and, at the same time, through transformation, moving to the obtained BVP without convective terms. The approximation of the 3-D nonstationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the spline solution. The publication provides a wide-ranging insight into the practical uses of 3-D heat transfer boundary-value problems in general, as well as the effectiveness of the CAM application, in conjunction with the ADI method, is analyzed.