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Iterated collocation methods for Volterra integral equations with delay arguments. (English) Zbl 0803.65139
The paper is concerned with the analysis of the convergence and local superconvergence properties of the continuous collocation method and the iterated collocation method applied to solving Volterra integral equations with constant delay. The author continues his previous research work [SIAM J. Numer. Anal 21, 1132-1145 (1984; Zbl 0575.65134) and J. Comput. Math. 10, No. 4, 348-357 (1992; Zbl 0758.65083)] in the area of numerical treatment of Volterra integral equations. The main result of this paper (Theorem 4.1) is that $O (h\sp{2m})$- convergence at the mesh points $\Pi\sb N$ can be attained using the iterated collocation solution corresponding to collocation in $S\sp{(- 1)}\sb{m-1} (\Pi\sb N)$ (piecewise polynomials of degree $m-1$ possessing jump discontinuities on $\Pi\sb N)$. On the whole interval I, the order of convergence will turn out to be $O(h\sp m)$ (Theorem 3.1 and Theorem 3.2). Finally, the local convergence result expressed by Theorem 4.1 is extended (Theorem 4.4) to the discretized collocation solution and to the corresponding iterated collocation solution if the delay integrals are approximated by a given quadrature.

##### MSC:
 65R20 Integral equations (numerical methods) 45G10 Nonsingular nonlinear integral equations
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