Error Estimates of a Symmetric Spectral Method for a Linear Volterra Integral Equation

A symmetric spectral method is applied to investigate the two-dimensional Volterra integral equation with weakly singular kernels and delays. In this work, the solution of the equation we considered is assumed to be sufficiently smooth so that the spectral method can be applied naturally. Employing three couples of variable transformations, we apply the two-dimensional Gauss quadrature rule to approximate the weakly singular integral with delays and obtain the spectral discretization. Then we derive the convergence results of the proposed approximation scheme. We show that the errors of solution decay exponentially in both the infinity norm and weighted square norm. In the end, we carry out numerical experiments to verify the theoretical results.