#### Term of Award

Spring 2004

#### Degree Name

Master of Science in Mathematics

#### Document Type and Release Option

Thesis (restricted to Georgia Southern)

#### Department

Department of Mathematical Sciences

#### Committee Chair

Laurene V. Fausett

#### Committee Member 1

Donald W. Fausett

#### Committee Member 2

Kanuri N. Murty

#### 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* k ^{th}* 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.

#### Recommended Citation

Tugcu, Gulcin, "First Order Linear Systems" (2004). *Legacy ETDs*. 35.

https://digitalcommons.georgiasouthern.edu/etd_legacy/35