Mathematical simulation of three-dimensional temperature fields by projection-structural method using the S-functions

Abstract

The article suggests a new approximate analytical approach to the mathematical simulation of three-dimensional temperature fields in bodies, limited by cylindrical surface of complex shape and two parallel planes, whose generators are perpendicular to the planes. The application of the final integral transformations on the coordinate, parallel to the generators of the cylindrical surface, allowed reducing the solution of three-dimensional heat conduction problems to the solution of the corresponding two-dimensional problems with a parameter of the Sturm-Liouville problem. Built analytical structures of approximate solutions of the heat conduction problems in the image area exactly satisfy the corresponding boundary conditions. Geometrical information is taken into account by the S-functions. The basic functions, contained in the approximate analytical structures of solutions of the heat conduction problems in the images area, are continuously differentiable. These qualities of the basic functions are possible to obtain due to the fact that the S-functions permitted to build the equations of smooth surfaces of bodies and the equations of smooth boundaries of areas of complex shape. According to the theorem of Kantorovich-Krylov, in these cases the basic functions, built using complete systems of functions in a particular coordinate system, also form complete systems of continuously differentiable functions. Undetermined coefficients of analytical structures of solutions in the image area are discovered from the corresponding algebraic systems with a parameter of the Sturm-Liouville problem, obtained by the Bubnov-Galerkin method.