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On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays. (English) Zbl 0955.65098

The author extends the recent results of H. Brunner [On the discretization of differential and Volterra integral equations with variable delay, BIT 37, 1-12 (1997; Zbl 0873.65126)] for the y ' (t)=by(qt),y(0)=1, and the delay Volterra integral equations y(t)=1+ 0 t by(qs)ds with proportional delay qt,0<q1, to the neutral functional-differential equation (NFDE):

y ' (t)=ay(t)+ i=1 b i y(q i t)+ i=1 c i y ' (p i t),y(0)=1,

and the delay Volterra integro-differential equation (DVIDE):

y(t)=1+ 0 t ay(s)ds+ i=1 0 t b i y(q i s)ds+ i=1 0 t c i y ' (p i τ)dτ

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 · He analyzes 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 investigates 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)-Padé approximant to y(h) and satisfies |v(h)-y(h)|=O(h 2m+1 )· If they exist, then the conditions on M m (t) and M ^ m (t), respectively, actually are given.

65R20Integral equations (numerical methods)
45D05Volterra integral equations
34K28Numerical approximation of solutions of functional-differential equations
45G10Nonsingular nonlinear integral equations
65L05Initial value problems for ODE (numerical methods)
34K06Linear functional-differential equations