Identification of unknown sources in time-space fractional parabolic equation

Abstract

This study considers the issue for recognizing unknown source within a time-space fractional parabolic equation. This particular issue is characterized by severe ill-posedness, where the solution does not depend continuously on the data. To tackle this issue, the wavelet dual least squares method is extended to handle ill-posedness of the issue under priori rule. Additionally, a posterior wavelet regularization parameter selection rule is introduced to address the Cauchy problem. Furthermore, error estimates between the exact solution and its approximation are established using both a priori and a posteriori methodologies. Numerical examples are given to validate the effectiveness and stability of the proposed method.