The Numerical Solution of Two-Dimensional Volterra Integral Equations by Collocation and Iterated Collocation

While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is now well understood there exist no systematic studies of the approximate solution of their two-dimensional counterparts. In the present paper we analyse the numerical solution of such equations by methods based on collocation and iterated collocation techniques in certain polynomial spline spaces. The analysis focuses on the global convergence and local superconvergence properties of the approximating spline function

Continuous collocation approximations to solutions of first kind Volterra equations

Abstract. In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach. 1