A super-attainable order in collocation methods for differential equations with proportional delay

We are concerned with the pantograph differential equation y^′ (t) = ay (t) + by (qt) + f (t), t > 0, y (0) = y₀, with proportional delay qt, 0 < q < 1. In the literatures, it is known that if we choose some proper m collocation points for m ≥ 2, then collocation leads to a superconvergence result of order p^* = 2 m + 1 at the first mesh point t = h. In this paper, in such collocation solution to the above equation, we show that there are cases for some 0 < q < 1 such that the attainable order at the first mesh point t = h, becomes a super-attainable order, just O (h^{2 m + 2}). Numerical experiments of such 0 < q < 1 are also presented.