Properties of the Generalized Laplace Transform and Transport Partial Dynamic Equation on Time Scales

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for periodic functions on a periodic time scale. Regressivity and its relationship to the Laplace transform is examined, and the Laplace transform for several functions is explicitly computed. Finally, we explore some inversion formulas for the Laplace transform via contour integration. In Chapter 4, we develop two recursive representations for the unique solution of the transport partial dynamic equation on an isolated time scale. We then use these representations to explicitly find the solution of the transport equation in several specific cases. Finally, we compare and contrast the behavior with that of the well-known behavior of the solution to the transport partial difference equation in the case where T = Z. iii DEDICATION This dissertation is dedicated to my wife Carol and my daughter Michelle. iv ACKNOWLEDGMENTS I would like to thank my advisors, Dr. innumerable suggestions, insights and corrections have been extremely helpful in not only preparing this dissertation, but in helping me to mature in my mathematical ability. Further, your ability to motivate and encourage me throughout this entire process has made this work possible. Dr. Peterson has given me the opportunity to work with his REU students as a graduate mentor for the last three summers, and I would like to thank you especially for that fantastic opportunity to learn what undergraduate research is all about. To the other members of my committee, Dr. Pardy, thank you. To all the professors who I have had the opportunity to work with here at UNL, I truly appreciate the support you have given me and for never once making me feel like I was " just a lowly grad student. " And finally, to my friends and my entire family, especially my wife, I thank you for your constant support and prayers. Not only has my wife Carol put up …