Iterative weighted Tikhonov regularization technique for inverse problems in time-fractional diffusion-wave equations within cylindrical domains

This study investigates the backward problem of a time-fractional diffusion-wave equation (TFDWE) within a cylindrical domain, confirming the inherent ill-posedness and conditional stability of this inverse problem. To address these challenges, we introduce an iterative weighted Tikhonov regularization technique (IWTRT), which stabilizes the solution process by incorporating constraints and weights into the framework. We rigorously derive the convergence rates for the regularized solution using both a-priori and a-posteriori strategies for selecting the regularization parameter, ensuring a comprehensive understanding of the method's performance in various scenarios. Numerical experiments validate the method's effectiveness, demonstrating its ability to accurately reconstruct the initial condition from noisy data. The a-posteriori strategy of the IWTRT, in particular, outperforms the a-priori approach in terms of accuracy and noise resilience, as evidenced by the experimental results. Our findings contribute to the field of fractional calculus, offering a robust framework for addressing inverse problems characterized by fractional dynamics. This work also provides a systematic approach for parameter selection, paving the way for future research and practical applications in engineering, physics, and applied mathematics.