A fundamental occupation of a mathematician is to describe a physical situation by a set of equations in order to solve real life problems. Most natural events can be expressed as differential and difference equations. In this respect, Ordinary Differential Equations (ODE) are one of the most useful parts of mathematics for theory and applications.
The objective of this project is to study systems of linear differential and difference equations. First, we compare two solution forms for the first order matrix differential equation Y'=AY+YB. The first form, due to Neudecker, utilizes the Kronecker products of matrices to convert an n x n matrix differential equation into an n x n vector ODE. The second form, due to Murty, finds the solution in terms of the fundamental matrix solutions of two n x u vector ODE. This allows dealing with relatively much larger matrices than in Neudecker's solution.
Second, we present some basic results on the relation between the kth order difference equation and the companion matrix equation; these results are not available in the literature. Then, we present a set of necessary and sulfirient conditions for the complete controllability and observability of the general first order difference system. Examples are provided to illustrate many of the theoretical results.
","abstract_html":"<p>A fundamental occupation of a mathematician is to describe a physical situation by a set of equations in order to solve real life problems. Most natural events can be expressed as differential and difference equations. In this respect, <em>Ordinary Differential Equations (ODE)</em> are one of the most useful parts of mathematics for theory and applications.</p> <p>The objective of this project is to study systems of linear differential and difference equations. First, we compare two solution forms for the first order matrix differential equation Y'=AY+YB. The first form, due to Neudecker, utilizes the Kronecker products of matrices to convert an <em>n x n</em> matrix differential equation into an <em>n x n</em> vector ODE. The second form, due to Murty, finds the solution in terms of the fundamental matrix solutions of two n x u vector ODE. This allows dealing with relatively much larger matrices than in Neudecker's solution.</p> <p>Second, we present some basic results on the relation between the<em> k<sup>th</sup></em> order difference equation and the companion matrix equation; these results are not available in the literature. Then, we present a set of necessary and sulfirient conditions for the complete controllability and observability of the general first order difference system. Examples are provided to illustrate many of the theoretical results.</p>","abstract_has_math":false,"creators":["Tugcu, Gulcin"],"institution":null,"degree_name":"Master of Science in Mathematics","degree_level":"Thesis (restricted to Georgia Southern)","degree_discipline":"Department of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Donald W. Fausett","Kanuri N. Murty"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004-01-01T08:00:00Z","date_published":"2004-01-01T08:00:00Z","updated_at":"2026-07-24T02:28:15Z","subjects":["ETD","Ordinary differential equations","ODE","Linear differential equations","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd_legacy/35","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Donald W. Fausett","Kanuri N. Murty"]},{"key":"dc:creator","label":"Author","values":["Tugcu, Gulcin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["1970-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (restricted to Georgia Southern)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","Ordinary differential equations","ODE","Linear differential equations","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd_legacy/35"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["A fundamental occupation of a mathematician is to describe a physical situation by a set of equations in order to solve real life problems. Most natural events can be expressed as differential and difference equations. In this respect, Ordinary Differential Equations (ODE) are one of the most useful parts of mathematics for theory and applications.
The objective of this project is to study systems of linear differential and difference equations. First, we compare two solution forms for the first order matrix differential equation Y'=AY+YB. The first form, due to Neudecker, utilizes the Kronecker products of matrices to convert an n x n matrix differential equation into an n x n vector ODE. The second form, due to Murty, finds the solution in terms of the fundamental matrix solutions of two n x u vector ODE. This allows dealing with relatively much larger matrices than in Neudecker's solution.
Second, we present some basic results on the relation between the kth order difference equation and the companion matrix equation; these results are not available in the literature. Then, we present a set of necessary and sulfirient conditions for the complete controllability and observability of the general first order difference system. Examples are provided to illustrate many of the theoretical results.
"]},{"key":"dc:title","label":"Title","values":["First Order Linear Systems"]}]}],"canonical_facts":{"dc:contributor":["Donald W. Fausett","Kanuri N. Murty"],"dc:creator":["Tugcu, Gulcin"],"dc:date.available":["1970-01-01T08:00:00Z"],"dc:description.abstract":["A fundamental occupation of a mathematician is to describe a physical situation by a set of equations in order to solve real life problems. Most natural events can be expressed as differential and difference equations. In this respect, Ordinary Differential Equations (ODE) are one of the most useful parts of mathematics for theory and applications.
The objective of this project is to study systems of linear differential and difference equations. First, we compare two solution forms for the first order matrix differential equation Y'=AY+YB. The first form, due to Neudecker, utilizes the Kronecker products of matrices to convert an n x n matrix differential equation into an n x n vector ODE. The second form, due to Murty, finds the solution in terms of the fundamental matrix solutions of two n x u vector ODE. This allows dealing with relatively much larger matrices than in Neudecker's solution.
Second, we present some basic results on the relation between the kth order difference equation and the companion matrix equation; these results are not available in the literature. Then, we present a set of necessary and sulfirient conditions for the complete controllability and observability of the general first order difference system. Examples are provided to illustrate many of the theoretical results.
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