Applications of fractional calculus in Laplace transform theory

In the study of Laplace transforms, the need sometimes arises to calculate a transform or inverse transform involving variables of non-integer order. The integral definitions of the Laplace transform can always be used directly but the required integration is sometimes rather difficult to carry out. When the variables involved are of integer order, a number of convenient operational techniques are available to aid in calculation. The purpose of this thesis is to generalize certain of these techniques so that application is possible to non-integer order variables. Before extending the methods of handling Laplace transforms, it is necessary to introduce the concept of fractional calculus. Since the subject is not widely known, a part of this thesis is concerned with providing the background material necessary to develop the Laplace transform extensions. Included is a historical review of fractional calculus along with a presentation of certain methods which can be used to calculate fractional derivatives and integrals for several specific functions. Finally, certain theorems of Laplace transform theory are modified to encompass the concepts of fractional calculus. These theorems are then applied to several example problems to demonstrate their usefulness.