AN EXTENSION OF THE LANDWEBER REGULARIZATION FOR A BACKWARD TIME FRACTIONAL WAVE PROBLEM

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Automated summary

This paper investigates numerical methods for a backward problem of the time-fractional wave equation in bounded domains. Two fractional filter regularization methods are proposed, which efficiently overcome the well-known over-smoothing drawback caused by classical regularizations. Numerical examples confirm the theoretical results, showing that fractional regularization is more efficient for problems with low regularity.