Abstract :
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 infinitedimensional
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
difficult 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 specific 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 specific 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 specific 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).
