Fractional Calculus

Abstract

A brief exposition of fractional order operators and their properties is given. After that, we introduce the notion of generalized fractional operators.Keywords Fractional derivatives and integrals · Generalized fractional derivatives and integrals · Fractional derivatives and integrals of variable order · RiemannLiouville, Hadamard and Caputo operators · Fractional integration by parts · Multidimensional generalized fractional calculus Fractional calculus was introduced on September 30, 1695. On that day, Leibniz wrote a letter to L'Hôpital, raising the possibility of generalizing the meaning of derivatives from integer order to noninteger order derivatives. L'Hôpital wanted to know the result for the derivative of order n = 1/2. Leibniz replied that "one day, useful consequences will be drawn" and, in fact, his vision became a reality. However, the study of noninteger order derivatives did not appear in the literature until 1819, when Lacroix presented a definition of fractional derivative based on the usual expression for the nth derivative of the power function (Lacroix 1819). Within years the fractional calculus became a very attractive subject to mathematicians, and many different forms of fractional (i.e., noninteger) differential operators were introduced: the Grunwald-Letnikow, Riemann-Liouville, Hadamard, Caputo, Riesz (Hilfer 2000;Kilbas et al. 2006;Podlubny 1999;) and the more recent notions of Cresson (2007), Katugampola (2011), Klimek (2005, Kilbas and Saigo (2004) or variable order fractional operators introduced by Samko and Ross (1993).In 2010, an interesting perspective to the subject, unifying all mentioned notions of fractional derivatives and integrals, was introduced in Agrawal (2010) and later studied in Bourdin et al. (2014), Klimek and Lupa (2013), Odzijewicz et al. (2012a, b, 2013a. Precisely, authors considered general operators, which by choosing special kernels, reduce to the standard fractional operators. However, other nonstandard kernels can also be considered as particular cases.This chapter presents preliminary definitions and facts of classical, variable order, and generalized fractional operators.

A numerical and analytical study of SE(Is)(Ih)AR epidemic fractional order COVID-19 model

“…The fractional calculus was first communicated between Leibnitz and L’Hospital for the nth derivative of y . Fractional derivative was first introduced by Lacroix [ 24 ]. Afterward, many of the researchers introduced fractional derivatives in different forms, among which the most valuable are Caputo fractional derivative [ 25 ], Riemann–Liouville fractional derivative [ 26 ], and Atangana–Baleanu derivative [ 27 ].…”