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

• Published in: Advances in computational mathematics Vol. 49; no. 6
• Main Authors: Wen, Jin, Li, Zhi-Yuan, Wang, Yong-Ping
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Parameters; Regularization; Visualization; Wave equations; 35R11; 65N21; Backward problem; Quasi-reversibility regularization method; Uniqueness; Fractional wave equation
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-023-10080-w

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.