On the determination of points sources in inverse heat and mass transfer problems

Subject of research: problems of numerical determination of point sources in inverse heat and mass transfer problems. Purpose of research: description of theoretical results (theorem of existence and uniqueness of solutions of the inverse problem), creation of an algorithm for solving the problem of numerical determination of point sources, study of its properties, numerical implementation of the algorithm and its testing and verification of stability. Objects of research: problems of numerical determination of point sources (right-hand side of a special type) in inverse problems of heat and mass transfer. The sources are specified as a sum of Dirac delta functions with coefficients that depend on time and characterize the power of the corresponding source. They are unknown and are to be determined together with a solution of the equation. The values of the solution at a certain set of points lying inside the domain serve as the overdetermination data. Methods of research: the algorithm is based on the finite element method for spatial variables and the finite difference method for time. The unknown right-hand side is determined at each time layer using the overdetermination condition. Main results of research: description of the algorithm for solving the problem, its properties, results of numerical experiments. In particular, the conditions are described when the algebraic system to which the problem is reduced has a unique solution. The data obtained as a result of the calculations are compared with test examples. The calculations are also carried out with the addition of random noise of various levels to the measurement data. The results show that the solution is stable under random perturbations of the data.