Abstract
This paper investigates a time-fractional parabolic equation with Zaremba-type boundary conditions. The main objective of the present work is to construct reliable and efficient numerical approximations for such problems. To this end, stable finite difference schemes are developed within a consistent analytical framework. A key result is obtaining a coercive stability estimate for the first-order scheme, which guarantees its consistency and supports its practical use in computations. In addition, both first- and second-order schemes are implemented in the one-dimensional case using a modified Gaussian elimination approach. This implementation simplifies the solution process and improves computational reliability when handling the resulting systems. The behavior of the proposed methods is examined through several numerical experiments designed to reflect different parameter choices and settings. The results demonstrate that the schemes achieve the expected levels of accuracy, consistency, and efficiency. An accompanying error analysis explains the observed outcomes and supports the theoretical findings. The numerical results, presented in tables, show strong agreement with the theoretical predictions, thereby confirming the validity and effectiveness of the proposed approach. These conclusions highlight the practical applicability of the proposed numerical schemes for solving this fractional parabolic problem with mixed boundary conditions.