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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 the time scale is the integers.