Solution of a Nonlinear Multi-Velocity Singularly Perturbed Inverse Transport Problem in an Unbounded Domain

In the theory of problems of mathematical physics associated with singularly perturbed equations, a substantial body of fundamental results has been accumulated, primarily devoted to direct problems and their analytical investigation. However, this cannot be fully attributed to singularly perturbed inverse transport problems in an unbounded domain, which determines the relevance of the research presented in this work. Additional difficulties arise in the study of multi-velocity inverse transport problems, since the implementation of most known solution methods is associated with the need to construct special coordinate systems that depend on the specific features of the problem under consideration. These systems may be either rectangular Cartesian coordinates, in which the problem is originally formulated, or curvilinear coordinates to which an appropriate transformation is performed. In this case, additional conditions are required for new functions, in which new variables are related to each other by certain relationships, depending on the given areas. In this paper, an n-velocity singularly perturbed inverse Kac-type transport problem in an unbounded domain is investigated. The proposed solution method makes it possible to remain within the Cartesian coordinate system regardless of the number of phase space variables, thereby significantly simplifying the analytical study and expanding the possibilities for practical application of the obtained results. For a singularly perturbed inverse transport problem of the Katz type in an unbounded domain, results are obtained in weighted space. The solution is constructed using a modified asymptotic method. Within the framework of this approach, estimates of the proximity of solutions of a singularly perturbed inverse problem and a degenerate inverse problem in a selected weight space are proven.