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) = q_1(t) - q_2(t)y(t) + ∫_0~t K(t, s)y(s)ds, t ∈ I: = [0, T], y(0) = Y_0 (*) and Volterra integral equations (VIE) εy(t) = g(t) - ∫_0~t K(t, s)y(s)ds, t ∈ I (**) by tension spline collocation methods in certain tension spline spaces, where ε is a small parameter satisfying 0 < ε ≤ 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 < ε ≤ 1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for ε = 1 to the singularly perturbed case.