Differential Algebraic Equations, Indices, and Integral Algebraic Equations

Abstract

In a recent report, Hairer, Lubich, and Roche [Report CH-1211, Dept. de Mathematiques, Universite de Geneve, Geneve, Switzerland, 1988] define the index of differential algebraic equations (DAEs) by considering the effect of perturbations of the equations on the solutions. This index, which will be called the perturbation index $p_i $, is one more than the number of derivatives of the perturbation that must appear in any estimate of the bound of the change in the solution. An earlier form of index used by a number of authors including Gear and Petzold [SIAM J. Numer. Anal., 21 (1984), pp. 716–728] is determined by the number of differentiations of the DAEs that are required to generate an ordinary differential equation (ODE) satisfied by the solution. This will be called the differential index $d_i $. Hairer, Lubich, and Roche give an example whose differential index is one and perturbation index is two and other examples where they are identical. It will be shown that $d_i \leqq p_i \leqq d_i + 1$ and that $d_i = p_i $ if the derivative components of the DAE are total differentials. This means that the differential components have a first integral. The integrals are a special case of a new type of integral equation which will be called integral algebraic equations (IAEs). (An initial investigation of these equations indicates that they have properties very similar to DAEs.) The converse, namely that $p_i = d_i + 1$ if the differential components are not total differentials, is not true because the system could be composed of a combination of systems of different indices of which the highest index system is a total differential for which $p_i = d_i $, while lower index systems violate the total differential condition without changing the indices of the combined system.