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

摘要

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE) (∗)εy′(t)=q1(t)−q2(t)y(t)+∫0tK(t,s)y(s)ds,t∈I≔[0,T],y(0)=y0and Volterra integral equations (VIE) (∗∗)εy(t)=g(t)−∫0tK(t,s)y(s)ds,t∈Iby tension spline collocation methods in certain tension spline spaces, where ε is a small parameter satisfying 0<ε⪡1, and q1,q2,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<ε⪡1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for ε=1 to the singularly perturbed case.