Reducing the index of differential-algebraic equations by exploiting underlying structures

Differential-algebraic equations arise from the equation based modelling of physical systems, such as those found for example in engineering or physics. This thesis is concerned with square, sufficiently smooth, potentially non-linear differential-algebraic equations. Differential-algebraic equations can be classified by their index. This is a measure of how far a differential-algebraic equation is from an equivalent ordinary differential equation. To solve a differential-algebraic equation one usually transforms the problem to an ordinary differential equation, or something close to one, via an index reduction algorithm. This thesis examines how the index reduction (using dummy derivatives) of differential-algebraic equations can be improved via structural analysis, specifically the Signature Matrix method. Improved and alternative algorithms for finding dummy derivatives are presented and then a new algorithm for finding globally valid universal dummy derivatives is presented. It is also shown that the structural index of a differential-algebraic equation is invariant under order reduction.