Geometric meshes in collocation methods for Volterra integral equations with proportional time delays

In this paper we introduce new kind of nonuniform mesh, the so-called geometric mesh, and discuss the corresponding collocation method for Volterra integral equations of the second kind with proportional delay of the form $qt$ ($0 < q < 1$). It will be shown that, in contrast to the uniform mesh, the iterated collocation solution associated with such a mesh exhibits almost optimal superconvergence at the mesh points, provided that collocation parameters are chosen as the Gauss points in $(0,1)$