On the attainable order of collocation methods for delay differential equations with proportional delay

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 + $$\tfrac{b}{q}\int {_0^{qt} }$$ 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 i=1 m (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.