Abstract
The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.
Highlights
Fractional calculus, a popular name used to denote the calculus of non integer order, is as old as the calculus of integer order as created independently by Newton and Leibniz
Our main result appears in section three, in which we present and demonstrate the many faces of the fundamental theorem of fractional calculus (FTFC), in all different versions and which are interpreted as a generalization of the fundamental theorem of calculus
After a brief introduction about the calculus of non-integer order, popularly known as fractional calculus, we presented the concept of fractional integral in the Riemann-Liouville sense
Summary
Fractional calculus, a popular name used to denote the calculus of non integer order, is as old as the calculus of integer order as created independently by Newton and Leibniz. In contrast with the calculus of integer order, fractional calculus has been granted a specific area of mathematics only in 1974, after the first international congress dedicated exclusively to it Before this congress there were only sporadic independent papers, without a consolidated line [1,2]. The main objective of this paper is to explain what is meant by calculus of non integer order and collect any different versions of the fractional derivatives associated with a particular fractional integral. The paper is written as follows: in section two, we first review the concept of fractional integral in the Riemann-Liouville sense, which can be interpreted as a generalization of the integral of integer order and in the Liouville sense, which is a particular case of the Riemann-Liouville one.
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