Approximation of nonsmooth solutions of a retrospective problem for an advection-diffusion model

A retrospective problem, which consists in recovering an a priori unknown initial state of a dynamical system from its known final state, is investigated. The time evolution of the system is described by a nonlinear boundary value problem for the inhomogeneous Burgers equation. This problem, as well as other similar problems, is ill- posed. We propose to solve the problem by Tikhonov's variational method, which consists in minimizing some suitable residual functional on the set of admissible solutions of the problem. The case of a discontinuous solutions is covered by employing stabilizers with the norm of the Sobolev space $W^\gamma_p([0,l])$ with fractional derivatives. For solving the extremal problems, iterative methods are proposed and justified, which reduce the initial unstable problem to a series of well-posed problems. A numerical investigation of the effectiveness of various stabilizers is carried out and the results of numerical calculations are presented.