Composite quadrature rules for a class of weakly singular Volterra integral equations with noncompact kernels

A special class of weakly singular Volterra integral equations with noncompact kernels is considered. We consider a representation of the unique smooth solution of the equation and present a novel class of numerical approximations based on Gaussian quadrature rules. It is shown that the method of this type has a(n) (nearly) optimal rate of convergence under a specific condition which is very practical and easy to check. In some cases, the superconvergence property is also achieved. A stability analysis of the method is also provided. The method may be preferred to the iterated collocation method which is superconvergent under many conditions on the unknown solution. Some numerical examples are presented which are in accordance with the theoretical results.