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 ∫qt0 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 uit(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 ci = ci(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(h2m+1) but what is the collocation polynomial Mm(t;q) = K Πmi-1 (t - ci) 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 uit(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.
- Attainable order
- Collocation and iterated collocation method
- Delay differential and integral equation
- Padé approximant
- Proportional delay