A Note on Jacobi Spectral-Collocation Methods for Weakly Singular Volterra Integral Equations with Smooth Solutions

Abstract

This work is concerned with spectral Jacobi-collocation methods for Volterra integral equations of the second kind with a weakly singular of the form $(t-s)^{-\alpha}$. When the underlying solutions are sufficiently smooth, the convergence analysis was carried out in [Chen & Tang, J. Comput. Appl. Math., 233 (2009), pp. 938-950]; due to technical reasons the results are restricted to $0<\mu<\frac{1}{2}$. In this work, we will improve the results to the general case $0<\mu<1$ and demonstrate that the numerical errors decay exponentially in the infinity and weighted norms when the smooth solution is involved.