Collocation methods for Volterra functional integral equations with non-vanishing delays

In this paper the existence, uniqueness, regularity properties, and in particular, the local representation of solutions for general Volterra functional integral equations with non-vanishing delays, are investigated. Based on the solution representation, we detailedly analyze the attainable (global and local) convergence order of (iterated) collocation solutions on θ-invariant meshes. It turns out that collocation at the m   Gauss (-Legendre) points neither leads to the optimal global convergence order m+1,m+1, nor yields the local convergence order 2m on the whole interval, which is in sharp contrast to the case of the classical Volterra delay integral equations. However, if the collocation is based on the m   Radau II points, the local superconvergence order 2m−12m−1 will exhibit at all mesh points. Finally, some numerical experiments are performed to confirm our theoretical findings.