Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

A spectral Jacobi-collocation approximation is proposed for Volterra delayintegro-differential equations with weakly singular kernels. In this paper, we considerthe special case that the underlying solutions of equations are sufficiently smooth. Weprovide a rigorous error analysis for the proposed method, which shows that boththe errors of approximate solutions and the errors of approximate derivatives decayexponentially in $L^∞$ norm and weighted $L^2$ norm. Finally, two numerical examples arepresented to demonstrate our error analysis.