Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

This paper is devoted to the backward problem of determining the initial value and initial velocity simultaneously in a time-fractional wave equation, with the aid of extra measurement data at two fixed times. Uniqueness results are obtained by using the analyticity and the asymptotics of the Mittag-Leffler functions provided that the two fixed measurement times are sufficiently close. Since this problem is ill-posed, we propose a quasi-reversibility method whose regularization parameters are given by the a priori parameter choice rule. Finally, several one- and two-dimensional numerical examples are presented to show the accuracy and efficiency of the proposed regularization method.

Automated summary

This paper focuses on solving the backward problem of determining the initial value and initial velocity simultaneously in a time-fractional wave equation, with the help of extra measurement data at two fixed times. The uniqueness of solution is achieved by utilizing the analyticity and asymptotics of the Mittag-Leffler functions under the condition that the two fixed measurement times are sufficiently close. As the problem is ill-posed, a quasi-reversibility method is proposed, with regularization parameters determined by an a priori parameter choice rule. The accuracy and efficiency of the proposed regularization method are demonstrated through several numerical examples in one and two dimensions.