Modification of the tikhonov method under separate reconstruction of components of solution with various properties

In this paper a linear ill-posed is considered. Its solution is given in the form of a sum of two components: one contains breaks and the other is continuous, but admits breaks of derivative. For stable separate reconstruction of a solution, a modified Tikhonov method is applied. In this method, the stabilizer is chosen as a sum of two functionals with using total variation of function and its derivative, where every stabilizing functional depends on one component only. The convergence of the sum of the regularized components to a solution of the initial problem is proved. A scheme of finite-dimensional approximations of the regularize problem is investigated and the results of numerical experiments are presented.