Differential algebraic equations, indices, and integral algebraic equations. (English) Zbl 0732.65061

Gear, C. W.

SIAM J. Numer. Anal. 27, No. 6, 1527-1534 (1990).

Author’s summary: In a recent report, E. Hairer, Chr. Lubich and M. Roche [The numerical solution of differential-algebraic systems by Runge-Kutta methods. Lect. Notes Math. 1409 (1989; Zbl 0683.65050)] 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 the present author and L. Petzold [SIAM J. Numer. Anal. 21, 716-728 (1984; Zbl 0557.65053)] is determined by the number of differentiations of the DAEs that are required to generate an ordinary differential equation satisfied by the solution. This will be called the differential index \(d_ i\). Hairer, Lubich and Roche [loc. cit.] 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\(d_ i\leq p_ i\leq d_ i+1\) 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. (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 differential for which \(p_ i=d_ i\) while lower index systems violate the total differential condition without changing the indices of the combined system.

Reviewer: Z.Jackiewicz (Tempe)

MSC:

65L05	Numerical methods for initial value problems involving ordinary differential equations
65R20	Numerical methods for integral equations
34A34	Nonlinear ordinary differential equations and systems
45D05	Volterra integral equations

Keywords:

differential algebraic equations; perturbation index; differential index; integral algebraic equations

Citations:

Zbl 0683.65050; Zbl 0557.65053