In this paper, we study the fractional order Rayleigh-Stokes problem, where the time-fractional derivative is considered in the sense of Caputo with order \(\alpha \in (0,1)\) . Using the Faedo-Galerkin method, we first discuss the existence and uniqueness of weak solutions. Subsequently, we analyze the problem through the semi-discrete and fully discrete scheme, utilizing finite differences in time and the local discontinuous Galerkin (LDG) method in space. We approximate the Caputo time-fractional derivative on the uniform mesh using the L1 scheme. We examine the well-posedness of both semi-discrete and fully discrete numerical schemes using Banach-Nečas-Babuška theorem and establish the optimal order of convergence rate. Moreover, we also discuss the non-uniform L1 scheme to handle the weak singularity of the solution. Finally, we present a few numerical tests that demonstrate the effectiveness of the approach and validate the validity of the underlying science.
