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. […]