Differential-Algebraic Equations: A Projector Based Analysis
Springer Science & Business Media, Jan 19, 2013 - Mathematics - 649 pages
Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980s, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to constraints, in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering, system biology.
DAEs and their more abstract versions in infinite-dimensional spaces comprise a great potential for future mathematical modeling of complex coupled processes.
The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and so to motivate further research to this versatile, extra-ordinary topic from a broader mathematical perspective.
The book elaborates a new general structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Numerical integration issues and computational aspects are treated also in this context.
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admissible matrix function admissible projector functions apply Assumption 3.16 Banach space C*-subspace characteristic values component compute constant coefficient constant rank continuous function continuously differentiable critical points DAEs with properly decomposition Denote Differential-Algebraic Equations DIII DIIID eigenvalues Euler method Example explicit ODE follows function space Gi+1 given hence higher index IERODE implies index-1 DAEs interval inverse kerA Lemma Let the DAE linear DAEs matrix function sequence matrix pencil nonlinear DAE nonsingular nonsingular matrix nullspace numerical open set partial derivatives perturbation projector based projector functions Q0 projectors Q0 Proof proper leading term properly stated leading Proposition quasi-proper leading term quasi-regular regular DAEs regular index-1 DAE regular with index regular with tractability regularity regions Runge–Kutta methods Section singular solvability standard form DAEs stepsize structure subspace Theorem tion tractability index unique yields