A Fractional spectral method for weakly singular Volterra integro-differential equations with delays of the third-kind

In this paper, we present a fractional spectral collocation method for solving a class of weakly singular Volterra integro-differential equations (VDIEs) with proportional delays and cordial operators. Assuming the underlying solutions are in a specific function space, we derive error estimates in the $L_{{\omega }^{\alpha ,\beta ,\lambda }}^2$ and $L^{\mathrm{\infty }}$-norms. A rigorous proof reveals that the numerical errors decay exponentially with the appropriate selections of parameters $\lambda$. Subsequently, numerical experiments are conducted to validate the effectiveness of the method.