(untitled) (Journal of Mathematics Mechanics and Computer Science, 2026)
The problems where the coefficients or the right-hand side of a differential equation are determined simultaneously with its solution are called inverse problems of mathematical physics. Such problems frequently arise in a wide variety of fields, which makes them one of the most pressing issues in modern mathematics. This paper considers a class of problems modeling the process of determining the temperature and density of heat sources with given initial and final temperatures. Their mathematical formulation includes inverse problems for the heat equation, where it is necessary not only to solve the equation, but also to find an unknown right-hand side depending only on the spatial variable. In such inverse problems for the heat equation with a discontinuous coefficient, the existence and uniqueness of a classical and generalized solution can be established. The problem considered in this paper can arise in describing the diffusion of particles in a turbulent plasma as well as in modeling temperature field of heat propagation in a thin rod of finite length consisting of two sections with different thermophysical properties. In such problems, at the interface between two media with different thermophysical properties, it is necessary to specify not only boundary conditions but also conjugation conditions.