Differential elimination-completion algorithms for DAE and PDAE

Differential-algebraic equations (DAE) and partial differential-algebraic e quations (PDAE) are systems of ordinary equations and PDAEs with constraint s. They occur frequently in such applications as constrained multibody mech anics, spacecraft control, and incompressible fluid dynamics. A DAE has differential index r if a minimum of r+1 differentiations of it a re required before no new constraints are obtained. Although DAE of low dif ferential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties. Reich et al. have presented a geometric theory and an algorithm for reducin g DAE of high differential index to DAE of low differential index. Rabier a nd Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index, We show that for analytic autonomous first-order DA E, this algorithm is equivalent to the Cartan-Kuranishi algorithm for compl eting a system of differential equations to involutive form. The Cartan-Kur anishi algorithm has the advantage that it also applies to PDAE and deliver s an existence and uniqueness theorem for systems in involutive form. We pr esent an effective algorithm for computing the differential index of polyno mially 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-Stoke s equations are given. Discussion of computer algebra implementations is al so provided.