A stable regularization approach for reconstructing the initial displacement or the initial velocity in the fractional two-dimensional diffusion-wave equation

Utilizing the fractional Caputo derivative, this paper investigates two complementary ISPs for the timespace FDWE posed in a certain bounded domain. In the first problem, the initial velocity [Formula: see text] is known, and the goal is to identify the initial displacement [Formula: see text]. In the second problem, the initial condition [Formula: see text] is known, and the goal is to reconstruct the initial velocity [Formula: see text]. Using the series representation of the solution to the direct problem, the ISPs are reformulated as integral equations. When an additional condition is imposed at the final time, the issues are shown to be ill-posed in the sense of the Hadamard approach. To address this, we propose a stable regularization approach based on the TRM. Convergence rates are established using both a priori and a posteriori parameter choice rules. Numerical examples with detailed descriptions are provided to validate the accuracy and stability of the nominated approach. Finally, various elevated considerations and future recommendations are provided.