Diierential Elimination-completion Algorithms for Dae and Pdae

dae and pdae are systems of ordinary and partial di erential-algebraic equations with constraints. They occur frequently in applications such as constrained multibody mechanics, space-craft control and incompressible uid dynamics. A dae has di erential index r if a minimum of r+1 di erentiations of it are required before no new constraints are obtained. While dae of low di erential index (0 or 1) are generally easier to solve numerically, higher index dae present severe di culties. Reich, Rabier and Rheinboldt have presented a geometric theory and an algorithm for reducing dae of high di erential index to dae of low di erential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for dae of low di erential index. We show that for analytic autonomous rst order dae this algorithm is equivalent to the Cartan-Kuranishi algorithm for completing a system of di erential equations to involutive form. The CartanKuranishi algorithm has the advantage that it also applies to pdae and delivers an existence and uniqueness theorem for systems in involutive form. We present an e ective algorithm for computing the di erential index of polynomially nonlinear dae. A framework for the algorithmic analysis of perturbed systems of pdae is introduced and related to the perturbation index of dae. Examples including singular solutions, the Pendulum and the Navier-Stokes equations are given. Discussion of computer algebra implementations is also given.