Regularization Methods Based on a Priori Information for Backward Inhomogeneous Caputo Time‐Fractional Parabolic Equation

ABSTRACT In this work, we address the inverse problem of reconstructing the initial condition for an inhomogeneous Caputo time‐fractional parabolic equation in a bounded domain of , based on noisy final‐time data. Since the problem is severely ill‐posed, we introduce two Tikhonov‐type regularization frameworks, formulated with respect to the norms and , to ensure the stability of the approximation. For the corresponding regularized solutions, we establish sharp error estimates of both Hölder‐type and logarithmictypes. In particular, the Hölder‐type estimate improves upon earlier results in certain parameter regimes. Furthermore, we provide a set of numerical experiments on different domains, employing a finite element discretization combined with an efficient minimization algorithm, which confirms the validity of the theoretical error estimates.