A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

• Published in: Chaos, solitons and fractals Vol. 150; p. 111127
• Main Authors: Djennadi, Smina, Shawagfeh, Nabil, Abu Arqub, Omar
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.09.2021
• Subjects: Caputo fractional derivative; Fractional Sturm-Liouville operator; Fractional Tikhonov regularization; Ill-posed problem; Inverse backward problem; Inverse source problem; Fractional Sturm-Liouville operator; Ill-posed problem; Inverse source problem; Fractional Tikhonov regularization; Caputo fractional derivative; Inverse backward problem
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2021.111127

•Two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space are given.•Two types of inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at a final time is given.•A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions.•An error estimates between the exact and it regularize solutions are obtained. In this research, we deal with two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The first one is to identify the source term and the second one is to identify the initial value along with the solution in both cases. These inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. Under the a-priori and the a-posteriori parameter choice rules, the error estimates between the exact and its regularized solutions are obtained. To illustrate the validity of our study, we give numerical examples. A final note is utilized in the ultimate section.