In this thesis, a numerical method is introduced for solving second-kind fractional Volterra integro-differential equations (FVIDEs). The proposed approach is a spectral Galerkin method based on shifted Jacobi polynomials, with a weight function specifically chosen to accommodate the inherent weak singularity of the problem. The weight parameters are selected so as to incorporate the singular behavior of the kernel directly into the approximation space. Furthermore, an iterated version of the spectral Galerkin method is considered, which enhances accuracy and yields superconvergence under appropriate regularity assumptions. Additionally, a smoothing transformation is employed to mitigate the lack of smoothness of the exact solution near the origin, a typical characteristic of fractional-order problems, thereby recovering high-order convergence rates. The theoretical framework includes rigorous convergence analysis in both the supremum norm and the weighted \(L^2\)-norm, supported by regularity results for the solution and the kernel of the equivalent weakly singular Volterra integral equation. The proposed method is compared with the homotopy analysis transform method (HATM) as described in \cite{compare}, and both methods are evaluated.
