О ЕДИНСТВЕННОСТИ В ЗАДАЧАХ ОПРЕДЕЛЕНИЯ ТОЧЕЧНЫХ ИСТОЧНИКОВ В МАТЕМАТИЧЕСКИХ МОДЕЛЯХ ТЕПЛОМАССОПЕРЕНОСА

Abstract

We consider the problem of determining point sources for mathematical models of heat and mass transfer. The values of a solution (concentrations) at some points lying inside the domain are taken as overdetermination conditions. A second-order parabolic equation is considered, on the right side of which there is a linear combination of the Dirac delta functions δ(x–xi) with coefficients that depend on time and characterize the intensities of sources. Several different problems are considered, including the problem of determining the intensities of sources if their locations are given. In this case, we present the theorem of uniqueness of solutions, the proof of which is based on the Phragmén–Lindelöf theorem. Next, in the model case, we consider the problem of simultaneous determining the intensities of sources and their locations. The conditions on the number of measurements (the ovedetermination conditions) are described which ensure that a solution is uniquely determined. Examples are given to show the accuracy of the results. This problem arises when solving environmental problems, first of all, the problems of determining the sources of pollution in a water basin or atmosphere. The results are important when developing numerical algorithms for solving the problem. In the literature, such problems are solved numerically by reducing the problem to an optimal control problem and minimizing the corresponding objective functional. The examples show that this method is not always correct since the objective functional can have a significant number of minima.