Fractional Differentiation

Abstract

The fractional calculus is a theory of integrals and derivatives of arbitrary (i.e., non-integer) order. And it is considered as a natural extension of classical calculus. Thus there are many preserved basic properties between them. This thesis, consisting of four chapters, explores the concept and definition of fractional calculus. In this thesis, a brief history and definition of fractional calculus are given. Two definitions of fractional derivative are considered, namely the Riemann-Liouville and the Caputo definitions of the fractional derivative. Some illustrative examples are included. Further we present some basic properties with proofs. Finally, present some fractional differential equations with an emphasis on the Laplace transform of the fractional derivative.