Theory and Application of a Class of Abstract Differential-Algebraic Equations

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.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2005

Author dc:creator

• Pierson, Mark A.

Chairs dc:contributor.committeechair

• Herdman, Terry L.
• Cliff, Eugene M.

Committee members dc:contributor.committeemember

• Borggaard, Jeffrey T.
• Burns, John A.
• Russell, David L.

• well-posedness
• systems of partial differential equations
• existence and uniqueness
• hybrid systems
• partial differential-algebraic equations (PDAE)
• abstract differential-algebraic equations (DAE)

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-04282005-163231

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/27416