ANALYSIS OF THE MODIFIED TIKHONOV METHOD FOR SOLVING A LINEAR ILL-POSED PROBLEM WITH A SOLUTION CONTAINING CONTINUOUS an‎d DISCONTINUOUS COMPONENTS

In the paper, a linear operator equation that does not satisfy the Hadamard well-posed conditions is considered. It is assumed that the solution is representable as the sum of smooth and discontinuous components in different regions of its domain. For the stable separate reconstruction of a solution, a modified Tikhonov method is used. In this method, the stabilizer is chosen as a sum of the functions: Lebesgue norm and its variation. In the sum, every stabilizing functional depends on one component only. Convergence theorem in one-dimentional case is proved for the regularized solution. A scheme of finite-dimensional approximations of the regularized problem in two-dimentional case is investigated, and the results of numerical experiments in one-dimentional case are presented.