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 thesolution depends on a discrete or distributed set of values attained by thesolution itself in the past. Thus the initial problem for RFDEs is an infinitedimensional problem, taking its theoretical and numerical analysis beyondthe classical schemes developed for differential equations with no functionalelements. In particular, numerically solving initial problems for RFDEs is adifficult task that cannot be founded on the mere adaptation of well-knownmethods for ordinary, partial or integro-differential equations to the presenceof retarded arguments. Indeed, efficient codes for their numerical integrationneed specific approaches designed according to the nature of the equation andthe 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 forthe 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, inthe presence of specific types of functionals on the right-hand side, only oneof them can have an explicit character, whereas the other gives rise to anoverall procedure which is implicit in any case, even for non-stiff problems.In the panorama of numerical RFDEs, some critical situations have beenrecently investigated in connection to specific classes of equations, such as theaccurate location of discontinuity points, the termination and bifurcation ofthe solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possibletermination points, and the treatment of equations stated in the implicit form,which include singularly perturbed problems and delay differential-algebraicequations as well. All these issues are tackled in the last three sections.In this paper we have not considered the important issue of stability, forwhich we refer the interested reader to the comprehensive book by Bellen andZennaro (2003).