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.