Open issues in devising software for the numerical solution of implicit delay differential equations

We consider initial value problems for systems of implicit delay differential equations of the form \begin{eqnarray} \begin{array}{rcl} M\, y'(t) &\! = \!& f\Bigl( t,y(t),y\bigl( \alpha_1 (t,y(t))\bigr), \ldots ,y\bigl( \alpha_p (t,y(t))\bigr)\Bigr), \end{array} \end{eqnarray} where $M$ is a constant square matrix (with arbitrary rank) and $alpha(t, y(t)) \le t$ for all $t$ and $i$. For a numerical treatment of this kind of problems, a software tool has been recently developed [6]; this code is called RADAR5 and is based on a suitable extension to delay equations of the 3-stage Radau IIA Runge–Kutta method. The aim of this work is that of illustrating some important topics which are being investigated in order to increase the efficiency of the code.They are mainly relevant to (i) the error control strategies in relation to derivative discontinuities arising in the solutions of delay equations; (ii) the integration of problems with unbounded delays (like the pantograph equation); (iii) the applications to problems with special structure (as those arising from spatial discretization of evolutions PDEs with delays). Several numerical examples will also be shown in order to illustrate some of the topics discussed in the paper.