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Preprint 37-2014

A flow-on-manifold formulation of differential-algebraic-equations

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Author(s) : Ann-Kristin Baum

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 37-2014

MSC 2000

34A09 Implicit equations, differential-algebraic equations
37C10 Vector fields, flows, ordinary differential equations

Abstract :
In this paper, we derive a flow-on-manifold formulation for differential-algebraic equations (DAEs). DAEs are implicit differential equations whose dynamics are restricted by algebraic constraints. The framework of derivatives arrays and the trangeness-index allows to identify a class of DAEs that are uniquely solvable on a particular set of initial values [1]. For these systems, we can generalize the concept of the flow, which is the mapping that uniquely relates a given initial value with the solution through this point. Using a projection approach that has been derived in [2] for solving implicit algebraic equations, we give an explicit representation of the flow and its linearization. Interpreting the algebraic constraints of a DAE as a time-varying, embedded submanifold, then we obtain the proposed flow-on manifold formulation of DAEs. [1] P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich, CH, 2006. [2] A.K. Baum. A projection-based formulation of the implicit function theorem and its application to time-varying manifolds. Preprint 2014-15, Institut für Mathematik, TU Berlin, DE, Str. des 17. Juni 136, 10623 Berlin, DE, 2014.

Keywords : Differential-algebraic equations, flow, flow on surface, Dynamical systems

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