In 1 we give a general introduction to differential-algebraic equations (DAE), i.e. ODE where the solution is subject to an (algebraic) constraint. It can be seen as a limiting case of certain singularly perturbed ODE. In 2 a theory for linear DAE is developed. Here the notion of the matrix pencil turns out to be crucial. It induces an index concept, which relates to a degree of complexity of the problem. Next, more general DAE are introduced with a corresponding notion of (differential) index in 3. An important class of DAE, viz. semi-explicit DAE, is the subject of 4. As far as numerical methods are concerned, this chapter mainly deals with (implicit) multisteps. Therefore, we consider BDF methods for DAE in 5. Since higher index problems cause (numerical) difficulties, one is often interested in lowering the index, which can always be achieved by differentiation. However, this may cause so-called drift, and that is why regularization methods for alleviating this problem are considered in 6.

1. Introduction

The type of equations we have discussed so far always contained an explicit derivative, i.e. could be written as

We may consider this as a special case of the implicit ODE

Only if ?k/ ?x is nonsingular may we rewrite (1.2) as an explicit ODE. If this is not the case, we have an essentially more complicated problem. In order to understand this, consider the following set of equations:

Clearly (1.3a) looks like an ODE for x ₁,...