### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 49-63 |

Number of pages | 15 |

Journal | Mathematics of Computation |

Volume | 52 |

Issue number | 185 |

DOIs | |

State | Published - 1989 |

### Fingerprint

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*52*(185), 49-63. https://doi.org/10.1090/S0025-5718-1989-0971402-3

**First occurrence prime gaps.** / Bellen, A.; Jackiewicz, Zdzislaw; Vermiglio, R.; Zennaro, M.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 52, no. 185, pp. 49-63. https://doi.org/10.1090/S0025-5718-1989-0971402-3

}

TY - JOUR

T1 - First occurrence prime gaps

AU - Bellen, A.

AU - Jackiewicz, Zdzislaw

AU - Vermiglio, R.

AU - Zennaro, M.

PY - 1989

Y1 - 1989

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.

UR - http://www.scopus.com/inward/record.url?scp=84966230115&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966230115&partnerID=8YFLogxK

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

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

M3 - Article

AN - SCOPUS:84966230115

VL - 52

SP - 49

EP - 63

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 185

ER -