In 1909 Volterra wrote (following the study of the modelling of hysteresis problems) that one is led ‘… ad equazioni che hanno tipo misto, cioé in parte quello delle equazioni differenziali a derivate parziale ed in parte quello delle equazioni integrali. Mi permetto perciò di chiamarle equazioni integro-differenziali.’ He then used such ‘equations of mixed type’, namely linear integro-differential equations involving Volterra integral operators, as models describing heredity effects (see Volterra (1913, pp. 138–162)). Related, but more general (nonlinear) versions became famous in Volterra's work, starting around 1926, on the growth of single-species or interacting populations. At the end of his 1909 paper (p. 174) he added, however, a cautionary note when he observed that ‘… il problema della risoluzione delle equazioni integro-differenziali costituisce in generale un problema essenzialmente distinto dai problemi delle equazioni differenziali e da quelli ordinarii delle equazioni integrali’ [his italics].

Although such functional equations may be viewed formally as ODEs perturbed by a ‘memory’ term given by a Volterra integral operator, the analysis of collocation methods will be more complex (perhaps not ‘essentially distinct’ – except when it comes to the analysis of qualitative properties) than simply a synthesis of the techniques employed in Chapters 1 and 2. The convergence results we establish in this chapter will of course yield those of Chapter 1 as special cases.

Our voyage through the preceding eight chapters has shown that we have certainly not yet reached the end of the story on collocation methods for Volterra functional integral and integro-differential equations. Many important questions remain unanswered. It is my belief that we have to find new mathematical approaches and tools (likely from very unexpected areas) if we are to make substantial progress towards finding complete solutions to these open problems.

It is the purpose of this brief final chapter to point to some possible, and seemingly very promising, new approaches for the numerical analysis of collocation solutions to Volterra functional equations.

Semigroups and abstract resolvent theory

The long-time integration of Volterra integral and integro-differential equations by collocation methods, in particular the asymptotic behaviour of collocation solutions, is not yet understood. As a number of papers and books have shown (see, e.g. Ito and Kappel (1989, 1991, 2002), Ito and Turi (1991), Brunner, Kauthen and Ostermann (1995), Bellen and Maset (1999), Maset (1999, 2003), and Bellen and Zennaro (2003, pp. 56–60)) the appropriate reformulation of the given equation as an abstract Cauchy problem and the exploitation of the underlying semigroup or abstract resolvent framework (integrability and asymptotic behaviour of resolvents) will often lead to deep insight into the qualitative properties of approximate solutions.

Summary: As we mentioned in the Preface the voyage through the previous seven chapters has now brought us in many ways to the ‘frontier’ of what is known about the analysis of collocation methods. Thus, in this chapter we will make this more precise, first by reviewing recent and current work on collocation methods for DAEs and Volterra-type integral-algebraic equations (IAEs) of index 1. This will be followed by an exploration of various directions for future research on IAEs in particular, and collocation methods in general, in more abstract settings that may contain the key to the solution of many of the open problems encountered earlier.

Basic theory of DAEs and IAEs

The purpose of this section, especially Section 8.1.1, is to present some of the modern tools that will be required in the analysis of collocation methods for integral-algebraic equations of Volterra type. Thus, we present a fairly detailed introduction to the basic theory of (index-1) DAEs: this will allow us better to appreciate the complexity behind the analysis of collocation methods for IAEs and, especially, IDAEs of higher index. As we have just said, much of the quantitative and qualitative analysis of collocation solutions to such problems remains to be carried out.

The principal aims of this monograph are (i) to serve as an introduction and a guide to the basic principles and the analysis of collocation methods for a broad range of functional equations, including initial-value problems for ordinary and delay differential equations, and Volterra integral and integro-differential equations; (ii) to describe the current ‘state of the art’ of the field; (iii) to make the reader aware of the many (often very challenging) problems that remain open and which represent a rich source for future research; and (iv) to show, by means of the annotated list of references and the Notes at the end of each chapter, that Volterra equations are not simply an ‘isolated’ small class of functional equations but that they play an (increasingly) important – and often unexpected! – role in time-dependent PDEs, boundary integral equations, and in many other areas of analysis and applications.

The book can be divided in a natural way into four parts:

• In Part I we focus on collocation methods, mostly in piecewise polynomial spaces, for first-kind and second-kind Volterra integral equations (VIEs, Chapter 2), and Volterra integro-differential equations (Chapter 3) possessing smooth solutions: here, the regularity of the solution on the interval of integration essentially coincides with that of the given data. […]