In this thesis, the Gauss--Legendre spectral collocation method is presented for the numerical solution of third-kind Volterra integral equations with proportional delay. These equations, characterized by weak singularities and delayed integration limits, arise in diverse applications including population dynamics, heat transfer with memory effects, and control systems. A rigorous theoretical framework based on cordial Volterra integral operators is established to analyze the existence and uniqueness of solutions for both compact and non-compact operator cases. The spectral method is systematically compared with the conventional hat function method--a piecewise linear approximation technique widely employed for integral equations. Error analysis demonstrates that the spectral approach achieves exponential convergence under appropriate regularity conditions, whereas the hat function method exhibits only low-order algebraic convergence. Numerical experiments on benchmark problems with known exact solutions confirm the theoretical predictions. The stability and accuracy of the spectral scheme are further validated for equations involving non-compact operators, where solvability depends on spectral conditions rather than compactness. All computations are performed with controlled precision to eliminate round-off effects. The results conclusively demonstrate that the Gauss--Legendre spectral collocation method constitutes a highly accurate and efficient tool for solving singular third-kind Volterra delay integral equations, exhibiting remarkable superiority over conventional piecewise polynomial approaches when high precision is required.
