This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t)x(t), defined for t?[?1,k]t?[?1,k], (k?Nk?N), that satisfies this equation almost everywhere on [0,k?1][0,k?1] and assumes specified values on the intervals [?1,0][?1,0] and (k?1,k](k?1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.

CEMAT - Center for Computational and Stochastic Mathematics