On the Attainable Order of Collocation Methods for the Neutral Functional-Differential Equations with Proportional Delays

Abstract

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y′(t)= by(qt), y(0)=1 and the DVIE y(t)=1+∫0 t by(qs)ds with proportional delay qt, 0<q≤1, to the neutral functional-differential equation (NFDE): and the delay Volterra integro-differential equation (DVIDE) : with proportional delays p i t and q i t, 0<p i ,q i ≤1 and complex numbers a,b i and c i . We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t=h for the collocation solution v(t) of the NFDE and the `iterated collocation solution u it (t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M m (t) of v(th) or M^ m (t) of u it (th), t∈[0,1] such that the rational approximant v(h) or u it (h) is the (m,m)-Pade approximant to y(h) and satisfies |v(h)−y(h)|=O(h 2 m +1). If they exist, then we actually give the conditions of M m (t) and M^ m (t), respectively.