TY - JOUR

T1 - Natural continuous extensions of Runge-Kutta methods for Volterra integral equations of the second kind and their applications

AU - Bellen, A.

AU - Jackiewicz, Zdzislaw

AU - Vermiglio, R.

AU - Zennaro, M.

PY - 1989/1

Y1 - 1989/1

N2 - We consider a very general class of Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCE’s) of Runge-Kutta methods, i.e., piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration. The particular properties required of the NCE’s allow us to construct the tail approximations, which are quite efficient in terms of kernel evaluations.

AB - We consider a very general class of Runge-Kutta methods for the numerical solution of Volterra integral equations of the second kind, which includes as special cases all the more important methods which have been considered in the literature. The main purpose of this paper is to define and prove the existence of the Natural Continuous Extensions (NCE’s) of Runge-Kutta methods, i.e., piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration. The particular properties required of the NCE’s allow us to construct the tail approximations, which are quite efficient in terms of kernel evaluations.

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U2 - 10.1090/S0025-5718-1989-0971402-3

DO - 10.1090/S0025-5718-1989-0971402-3

M3 - Article

AN - SCOPUS:84966230115

SN - 0025-5718

VL - 52

SP - 49

EP - 63

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 185

ER -