The Numerical Integration of Neutral Functional‐Differential Equations by Fully Implicit One‐Step Methods

Abstract

An algorithm for the numerical solution of neutral functional‐differential equations is described. This algorithm is based on divided difference representation of fully implicit one‐step methods. The resulting systems of nonlinear equations are solved using the predictor‐corrector approach for nonstiff equations and by the modified Newton method for stiff equations. The step control strategy is based on local error estimation by comparing two approximations of successive orders. The details of implementations are described for systems of neutral delay‐differential equations with state dependent delays, for Volterra integro‐differential equations and for stiff delay‐differential equations. The results of some numerical experiments on four test examples are presented and discussed.

Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays

“…We refer to [22] for a recent survey on this topic with more than 220 references. In spite of that one of the first model appeared in the literature with state-dependent delays, the mathematical model for a two-body problem of classical electrodynamics introduced by Driver [8][9][10] involves NFDEs with state-dependent delays, much less work is devoted to SD-NFDEs [1,3,4,6,12,21,26,34,35]. Most of the above papers deal with SD-NFDEs of the form…”