Direct and inverse problems for kinetic equations (Q2712491)

The authors discuss a large class of processes described by kinetic equations, namely direct and especially inverse problems for these equations. Main results are given for one-velocity, non-stationary, multidimensional, and linear radiation transfer. The inverse problem of a simultaneous determination of the solution and the dispersion index or source function from given boundary values of the solution is investigated in detail. The approach is based on spherical harmonics representation of the solution, the dispersion index, and the source function. Substituting these series into the equation and comparing the coefficients of harmonics, an infinite system of linear differential equations is obtained. For finite expression of given information, in particular, for the finite Fourier transform, existence and uniqueness theorems are proved in admissible classes of dispersion indices or source functions.