### Abstract

To analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods for the delay differential equation (DDE) y′(t) = by(qt), 0 < q ≤ 1 with y(0) = 1, and the delay Volterra integral equation (DVIE) y(t) = 1 + b/q ∫^{qt}_{0} y(s) ds with proportional delay qt, 0 < q ≤ 1, our particular interest lies in the approximations (and their orders) at the first mesh point t = h for the collocation solution v(t) of the DDE and the iterated collocation solution u_{it}(t) of the DVIE to the solution y(t). Recently, H. Brunner proposed the following open problem: "For m ≥ 3, do there exist collocation points c_{i} = c_{i}(q), i = 1, 2, . . ., m in [0,1] such that the rational approximant v(h)is the (m, m)-Padé approximant to y(h)? If these exist, then |v(h) - y(h)| = O(h^{2m+1}) but what is the collocation polynomial M_{m}(t;q) = K Π^{m}_{i-1} (t - c_{i}) of v(th), t ∈ [0, 1]?" In this paper, we solve this question affirmatively, and give the related results between the collocation solution v(t) of the DDE and the iterated collocation solution u_{it}(t) of the DVIE. We also answer to Brunner's second open question in the case that one collocation point is fixed at the right end point of the interval.

Original language | English |
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Pages (from-to) | 374-394 |

Number of pages | 21 |

Journal | BIT Numerical Mathematics |

Volume | 40 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2000 |

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### Keywords

- Attainable order
- Collocation and iterated collocation method
- Delay differential and integral equation
- Padé approximant
- Proportional delay

### Cite this

*BIT Numerical Mathematics*,

*40*(2), 374-394. https://doi.org/10.1023/A:1022351309662