Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation

• Published in: Journal of inverse and ill-posed problems Vol. 33; no. 3; pp. 369 - 399
• Main Authors: Yang, Fan, Li, Ruo-Hong, Gao, Yin-Xia, Li, Xiao-Xiao
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.06.2025; Walter de Gruyter GmbH
• Subjects: 35R25; 35R30; 47A52; Caputo–Hadamard type; Crank–Nicolson format; error estimations; Finite difference method; formula; fractional Landweber iterative regularization method; fractional Tikhonov regularization method; Inverse problems; Parameters; Regularization; Stability; Wave equations
• ISSN: 0928-0219; 1569-3945
• DOI: 10.1515/jiip-2024-0051

In this paper, the inverse problem of source term identification for Caputo–Hadamard type time-fractional diffusion-wave equation is studied. Firstly, we prove that the problem is ill-posed, and give the optimal error bound and the conditional stability results. Secondly, we apply fractional Tikhonov regularization method and fractional Landweber iterative regularization method to solve the problem. Based on the conditional stability results, we give error estimates under the a priori regularization parameter selection rule and the a posteriori regularization parameter selection rule respectively. In addition, we give three numerical examples to prove the validity and feasibility of the selected regularization method. What is novel is that we apply L2 formula, Crank–Nicolson format and the finite difference method to discrete equation.