65L80 Methods for differential-algebraic equations

65J10 Equations with linear operators

Abstract

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