This paper introduces two layer-adaptive computational approaches for solving a class of time-fractional parabolic problems involving delay in time. The fractional derivative is considered in the Caputo sense of order \(\bf{\alpha} \in (\bf0,1)\) . Due to a mild singularity in a narrow region near the initial time, the solution typically exhibits a layer, which leads to a reduction in convergence rate when standard polynomial interpolation is used on uniform time meshes. To address this issue, the tempered fractional derivative is discretized using the L1 scheme on a specially designed layer-adapted mesh that captures the singular behavior. For the spatial discretization, a uniform mesh with central difference approximation is employed. Under reasonable smoothness assumptions on the coefficients, a rigorous convergence and error analysis are provided in the discrete maximum norm. The theoretical results show that the scheme achieves a global convergence rate of \(\bf{ \min\{\gamma\alpha,2-\alpha\}}, \) where γ > 0 denotes the mesh grading parameter. To further enhance accuracy, a second scheme is developed using the recently introduced L1–2 method for time discretization. The corresponding analysis proves that this scheme attains the optimal convergence rate \(\bf{ \min\{\gamma\alpha,2\}}. \) Furthermore, both approaches are extended to handle the corresponding semi-linear problems. The proposed approaches are validated through numerical simulations, demonstrating their efficiency and applicability.
