Optimal Superconvergence Of Some Volterra Equations With Delays

This paper is an overview of optimal superconvergence of some Volterra equations with delays. Delay Differential Equations of modern mathematics is an important topic. With the development of modern society Delay Differential Equations are extensively applied both in the natural sciences and social sciences fields. The optimal superconvergence of Volterra delay equations have been established in recent years.Firstly we introduce the Volterra functional integro-differential equations and (non-vanishing delay:the Volterra functional integro-differential equation (VFIDE) The Volterra functional integral equation (VFIE)The delay functionθisθ(t):t-τ(t)(t∈Ⅰ), the delayτ=τ(t) is non-negative on I. The delay functions include non-vanishing delay and vanishing delay: (i)θis non-vanishing delay, condition: (ND2)θis strictly increasing on I, VFIDE (1.1) is complemented by an initial condition thatΦis a initial function. (ⅱ)θis vanishing delay, condition:Then, a brief description of the Volterra functional integro-differential equations of the collocation method and error eh=u-uh. Point that in the analysis of superconvergence of uh, the representation of eh play a key role. The representation depends on the kind of the delays:for non-vanishing delay, it's attained by a standard'variation-of-constants formula' (more precisely, a'resolvent representation'), however, in the case of vanishing delay, the resolvent representation doesn't exist.Nextly, we discuss the optimal (global and local) superconvergence of Volterra func-tional integro-differential equations with non-vanishing delay or vanishing delays. As the global super-convergence is concerned, the Volterra functional integro-differential equations with non-vanishing delay agree with the classical ordinary integral equations. We can attain the optimal superconvergence estimate: The constant C depends on{ci} and on q but not on h. And the estimate is valid in the case of vanishing delay. In terms of local superconvergence, the Volterra functional integro-differential equations with non-vanishing delay, provided m-Gauss points, have the optimal local superconvergence estimate However, for m>2, the estimate is no longer valid for vanishing delay. The optimal local superconvergence of Volterra functional integro-differential equations with vanishing delay described by: Finally we intuoduce some open problems about the Volterra delay equations.