Elimination for Systems of Algebraic Differential Equations

<p>We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of the system of differential equations through the theory of differential kernels to provide a new upper bound for proving the consistency of the system. We then prove a new upper bound for the effective differential Nullstellensatz, which determines a sufficient number of times to differentiate the original system in order to prove its inconsistency. Finally, we study the Rosenfeld-Gröbner algorithm, which approaches differential elimination by decomposing the given system of differential equations into simpler systems. We analyze the complexity of the Rosenfeld-Gröbner algorithm by computing an upper bound for the orders of the derivatives in all intermediate steps and in the output of the algorithm.</p>

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

The Graduate School and University Center of The City University of New York

Year dc:date.available

2017

Author dc:creator

• Gustavson, Richard

Advisor dc:contributor.advisor

• Alexey Ovchinnikov

Committee members dc:contributor.committeemember

• Alexey Ovchinnikov
• Richard Churchill
• Benjamin Steinberg

• Algebra
• Partial Differential Equations
• algebraic differential equations
• differential elimination algorithms
• effective differential Nullstellensatz

Repository record dc:identifier

https://academicworks.cuny.edu/gc_etds/1970

OAI identifier oai:identifier

oai:academicworks.cuny.edu:gc_etds-2983