Title page for ETD etd-04282005-163231


Type of Document Dissertation
Author Pierson, Mark A
Author's Email Address mark.pierson@vt.edu
URN etd-04282005-163231
Title Theory and Application of a Class of Abstract Differential-Algebraic Equations
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Cliff, Eugene M. Committee Co-Chair
Herdman, Terry L. Committee Co-Chair
Borggaard, Jeffrey T. Committee Member
Burns, John A. Committee Member
Russell, David L. Committee Member
Keywords
  • well-posedness
  • systems of partial differential equations
  • existence and uniqueness
  • hybrid systems
  • partial differential-algebraic equations (PDAE)
  • abstract differential-algebraic equations (DAE)
Date of Defense 2005-04-25
Availability unrestricted
Abstract
We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed.
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