Recent trends in the numerical solution of retarded functional differential equations

Retarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the solution depends on a discrete or distributed set of values attained by the solution itself in the past. Thus the initial problem for RFDEs is an infinite-dimensional problem, taking its theoretical and numerical analysis beyond the classical schemes developed for differential equations with no functional elements. In particular, numerically solving initial problems for RFDEs is a diffcult task that cannot be founded on the mere adaptation of well-known methods for ordinary, partial or integro-differential equations to the presence of retarded arguments. Indeed, efficient codes for their numerical integration need speciffc approaches designed according to the nature of the equation and the behaviour of the solution. By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described. Despite being apparently similar, they are intrinsically different. Indeed, in the presence of speciffc types of functionals on the right-hand side, only one of them can have an explicit character, whereas the other gives rise to an overall procedure which is implicit in any case, even for non-stiff problems. In the panorama of numerical RFDEs, some critical situations have been recently investigated in connection to speciffc classes of equations, such as the accurate location of discontinuity points, the termination and bifurcation of the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible termination points, and the treatment of equations stated in the implicit form, which include singularly perturbed problems and delay differential-algebraic equations as well. All these issues are tackled in the last three sections. In this paper we have not considered the important issue of stability, for which we refer the interested reader to the comprehensive book by Bellen and Zennaro (2003).