PDAEs and Further Mixed Systems as Abstract Differential Algebraic Systems

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 22

MSC 2000

34A09 Implicit equations, differential-algebraic equations

35G05 General theory of linear higher-order PDE

35K40 General theory of parabolic systems of PDE

49K22 Problems involving integral equations

65L80 Methods for differential-algebraic equations

65J10 Equations with linear operators

Abstract differential algebraic systems (ADASs), i.e., differential algebraic systems with operators acting in real Hilbert spaces are introduced for a systematical treatment of coupled systems of PDEs, DAEs and integral equations. Using the finite-dimensional decoupling theory for DAEs as motivation, this paper will examine what one appropriate analogue is for infinite-dimensional systems. This leads to an index definition for ADASs. Thereby, instead of the inherent regular ODE one obtains an explicit (abstract) differential equation. In particular, when discussing PDAEs, the inherent regular differential equation is actually a parabolic PDE. The decoupling procedure provides, additionally, appropriate initial and boundary conditions for unique solvability of the coupled systems. The concept to handle ADASs is explained in different case studies.

Keywords:differential-algebraic equations; partial differential equations, integral equations, partial differential algebraic equations, implicit differential equation, coupled systems, index