Performance and rigorous analysis of Jacobi weighted residual schemes for a class of singular Volterra integro-differential models containing dual delays

In this study, we analyze two types of Jacobi weighted residual techniques (namely, spectral and pseudo-spectral) for handling Volterra integro-differential models that feature two categories of state-dependent fading delays (linear as well as nonlinear) together with weakly singular kernels. In contrast, for the pseudo-spectral Jacobi Galerkin variant, those same integrals are evaluated numerically via reliable and efficient Gauss–Jacobi quadrature rules. A detailed convergence study is conducted for both formulations, under mild assumptions on the method parameters, particularly for the pseudo-spectral case. Due to the use of fast quadrature rules, the pseudo-spectral scheme offers substantially higher computational efficiency compared to the spectral approach. Several numerical examples are presented to validate the theoretical findings of the proposed pseudo-spectral method. Furthermore, a suitable change of variable eliminates the singularity of the solution in the target model, and an illustrative example is provided to demonstrate this approach.