In this paper, we derive a numerical scheme for solving time-fractional reaction-diffusion problems with Robin boundary conditions, where the time derivative is in the Caputo sense of order \(\alpha \in (0,1)\) . The existence and uniqueness of the solution is proved. The proposed method is based on the spectral collocation method in space and Fractional Hamiltonian Boundary Value Methods in time. For the considered spectral collocation method, the basis functions used are not the standard polynomial basis functions, but rather are specifically constructed to inherently satisfy the Robin boundary conditions, and the exponential convergence property in space is established. The proposed procedure achieves spectral accuracy in both space and time. Some numerical examples are provided to support the theoretical results.
