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

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. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This research focuses on two types of inverse problems for diffusion equations with Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The study proves these inverse problems to be ill-posed in the sense of Hadamard when an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for stable solution reconstruction with error estimates obtained between exact and regularized solutions through a-priori and a-posteriori parameter choice rules, along with numerical examples provided for validation.