In this thesis, numerical solution techniques for solving second-kind mixed Volterra-Fredholm integral equations featuring algebraic endpoint singularities is presented. Building upon the Nystr\"{o}m method, we develop and analyze a discretization scheme that combines Gauss-Jacobi quadrature for the Fredholm part with product integration rules for the weakly singular Volterra component. The analysis is carried out in weighted Banach spaces $C_u$ equipped with Jacobi-type weights, which naturally accommodate the low regularity of solutions near the endpoints. Under mild smoothness assumptions on the kernels and right-hand side, we establish stability, convergence, and error estimates that are of the order of the best polynomial approximation in $C_u$. The theoretical framework is supported by existence and uniqueness results based on fixed-point theorems, and numerical experiments confirm the high accuracy of the proposed method.
