High-order collocation methods for singular Volterra functional equations of neutral type

doi:10.1016/j.apnum.2006.07.006

Applied Numerical Mathematics

Volume 57, Issues 5–7, May 2007, Pages 533-548

https://doi.org/10.1016/j.apnum.2006.07.006 Get rights and content

Abstract

In the first part of this paper we present a survey of recent results on the attainable order of (super-) convergence of collocation solutions for systems of Volterra functional integro-differential equations with weakly singular kernels $k_{α}$ and variable delay functions $θ (t) = t - τ (t)$ . Related functional equations and theoretical and computational aspects of collocation methods for their solution are also described. The paper concludes with comments on ongoing and possible future work.

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The analysis to the Volterra integral equations with proportional delays dates back to the works in [6] and [7]. Some more recent results on this subject can be found in [4,8–11]. During the past decade, numerical methods for (1) have attracted wide attention of many researchers.
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¹: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

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High-order collocation methods for singular Volterra functional equations of neutral type

Abstract

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Math. Anal. Appl.

Math. Biosci.

Appl. Math. Lett.

Appl. Numer. Math.

Appl. Numer. Math.

Numerical analysis of Volterra functional and integral equations

Population models with state-dependent delays

Constrained mesh methods for functional-differential equations

Numerical Methods for Delay Differential Equations

Existence, uniqueness, and extendibility of solutions to Volterra integral systems with multiple variable delays

Funkcial. Ekvac.

Some directions for mathematical epidemiology

The numerical solution of neutral Volterra integro-differential equations with delay arguments

Ann. Numer. Math.

The discretization of Volterra functional integral equations with proportional delays

The numerical analysis of functional integral and integro-differential equations of Volterra type

Acta Numerica

Collocation Methods for Volterra Integral and Related Functional Equations

The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays

Comm. Pure Appl. Anal.

Optimal superconvergence orders of iterated collocation solutions for Volterra integral equations with vanishing delays

SIAM J. Numer. Anal.

On the regularity of solutions to neutral-type Volterra functional-differential equations with weakly singular kernels

J. Integral Equations Appl.

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Math. Comp.

Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels

SIAM J. Numer. Anal.