Abstract
In this article, we study the problem of recovering an unknown function based on its known integrals over a specified family of curves, with consideration given to a weight function. The main objective is to establish a methodology for solving the integral geometry problem by transforming it into a boundary value problem involving a second-order partial differential equation of mixed type. The obtained results provide valuable insights into the solution of the finite difference problem, which encompasses both the boundary $$\Gamma$$ and a certain neighborhood in its vicinity, taking into account the presence of specific types of singularities. The estimation theorem for the solution is proven using a specialized factor-based technique. The implications of these findings extend to diverse applications, notably in fields such as seismic and medical imaging, as well as non-destructive material characterization. By presenting a comprehensive analysis and a detailed exploration of the problem at hand, this study contributes to the advancement of the theory and practical implementation of integral geometry in various scientific and technological domains.