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) ɛy'(t)=q1(t)-q2(t)y(t) + ∫0t K(t,s)y(s) ds, t ɛ I:=[0, T], y(0) = y0 and Volterra integral equations (VIE) ɛy(t) = g(t) - ∫0t K(t,s)y(s) ds, t ɛ I by tension spline collocation methods in certain tension spline spaces, where ɛ is a small parameter satisfying 0 1, and ql, 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 1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for ɛ = 1 to the singularly perturbed case.

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