Solution of Cauchy problem to stationary heat conduction equation by modified method of elementary balances with interpolation of the solution in physical plane

Abstract

This article presents a solution of stationary direct and inverse problems (Cauchy problem) of cooling a circular ring with the modified method of elementary balances. The idea of the method itself relies on the division of the considered range into elements and interpolation of the solution within the elements with the help of linear combination of base functions. In the vicinity of every mesh node, the control regions are created in which the energy is balanced. The regions include or interpenetrate with each other. The numerical calculation has been carried out with the use of a quadrilateral mesh with four nodes and a triangular mesh with six nodes. The presented solution to the inverse problem with randomly perturbed boundary conditions of the flux and temperature at the outer boundary of the ring with maximal error reaching up to 10% gave very good results. The solution of the inverse problem has been obtained in the sense of singular value decomposition algorithm.