Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Abstract

Summary. In this paper we present an approach for the numerical solution of delay differential equations \begin{equation} \left\{ \begin{array}{l} y^{\prime }\left( t\right) =Ly\left( t\right) +My\left( t-\tau \right) \;\;t\geq 0 y\left( t\right) =\varphi \left( t\right) \;\;-\tau \leq t\leq 0, \end{array} \right. \end{equation} where $\tau >0$ , $L,M\in \mathbb{C}^{m\times m}$ and $\varphi \in C\left( \left[ -\tau ,0\right] ,\mathbb{C}^m\right) $ , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1).