Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE) epsilony' (t) = q(1) (t) - q(2) (t) y (t) + integral(0)(1) K (t,s) y (s) ds, t is an element of I: = [0,T], y(0) = y(0) and Volterra integral equations (VIE) epsilony (t) = g (t) - integral(0)(1) K (t,s) y (s) ds, t is an element of I by tension spline collocation methods in certain tension spline spaces, where e is a small parameter satisfying 0 < F much less than< 1, and q(1), q(2), g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution. We give an analysis of the global convergence properties of a new tension spline collocation solution for 0 < e much less than< 1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for epsilon = 1 to the singularly perturbed case. (C) 2002 Elsevier Science B.V. All rights reserved.