Differential-Algebraic Equation Index Transformations

In this paper we show an equivalence between semi-explicit and implicit differential-algebraic equations so that the theory for the former applies to the latter of one lower index. We also discuss techniques for the reduction by differentiation of higher index problems to lower index ones that can be solved numerically without introducing additional constants of integration. Furthermore, we will give transformations between general DAEs and ones in semi-explicit form, and, by these techniques, show how a problem can be reduced to an index 1, semi-explicit form that can be reduced to an implicit differential equation from whose solution the solution to the original problem can be retrieved by at most a differentiation and a linear transformation.