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

摘要

In this paper,we extend the recent results of H.Brunner in Bit (1997) for the DDE y (t)=by(qt), y(o)-1 and the DVIE y(t)=1+∫~t_by (qs) ds with proportional delay qt, o＜q≤1, to the.neutral functional-differential equation(NFDE) and the delay Volterra integro-differential equation(DVIDE) with proportional delays pit and qit, o＜pi,qi≤1 and complex numbers a,biand ci. 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), and investigate the existence of the collocation polynomials Mm(t) of v(th) or Mm(t) of Uit (th)t-(0.1) such that the rational approximant v(h) or Uit(h) is the (m.m)-pade approximant to y(h) and satisfies v(h)-y(h)=O(h~2m+1). If they exist, then we actually give the conditions of Mm(t) and Mm(t) respectively.