NUMERICS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

For a delay differential equation of neutral type (in short NDDE) the usual approach for solving delay equations, based on the use of a continuous ODE method, must take care of the additional term involving the derivative of the solution at some previous instant. Therefore, besides the continuous extension for approximating the solution, the method must provide a continuous approximation for the derivative of the solution as well. Starting from this point we present a general class of numerical methods for solving NDDEs and stress the peculiarities and the differences with respect to the standard case of DDEs. First we analyze some basic aspects of convergence theory. Then we focus our attention on the regularity of the solutions; since for NDDEs the smoothing of solutions does not take place, taking care of jump discontinuities is much more important with respect to DDEs. Finally we explain why uniqueness and existence of the solution is not anymore guaranteed for general state-dependent problems and how to detect these occurrences automatically.