In this thesis, a mesh-free collocation method based on radial basis functions (RBFs) is investigated for solving third-kind Volterra integral equations with nonlinear vanishing delays. In this approach, approximations are constructed using scattered nodes and exploit the exponential convergence of infinitely smooth kernels, such as Gaussian and inverse multiquadric (IMQ) functions. The convergence of the proposed method is rigorously analyzed within the framework of native spaces, accompanied by explicit error estimates in both maximum and Sobolev norms. Specifically, by employing the theory of reproducing kernel Hilbert spaces and key concepts such as fill distance, the convergence rate is carefully examined as a function of node density and the kernel shape parameter. The RBF-based method is compared against an $hp$-version Legendre spectral collocation scheme. The reference method utilizes a specially designed graded mesh to resolve the singularity induced by the vanishing coefficient %$t^\beta$ ($0 < \beta < 1$), and its error behavior is analyzed in weighted Sobolev spaces. For both methods, existence and uniqueness of the solution in appropriate Banach and Hilbert spaces, numerical stability, and error estimates are thoroughly established. Numerical comparisons between the two approaches not only validate the theoretical analyses but also highlight their relative strengths in terms of accuracy, implementation simplicity, computational efficiency, and sensitivity to design parameters. The comparative results, including error tables and convergence plots, are comprehensively reported.
