Fractional Integration and Fractals

Abstract

The theory of integration and differentiation of non-integer order has a long history from 30 September 1695, when the derivatives of order α = 1/2 was described by Leibniz in a letter to L’Hospital (Oldham and Spanier, 1974; Samko et al., 1993; Ross, 1975). The earliest theory of integrals and derivatives of non-integer order goes back to Liouville and Riemann (Ross, 1975). There are many interesting books about fractional calculus and fractional differential equations (Oldham and Spanier, 1974; Samko et al., 1993; Miller and Ross, 1993; Podlubny, 1999; Kilbas et al., 2006; Nahushev, 2003; Pshu, 2005); see also (McBride, 1976, 1986; Srivastava and Owa, 1989; Nishimoto, 1989; Kiryakova, 1994; Rubin, 1996). Derivatives and integrals of non-integer order, and fractional integro-differential equations have found many applications in recent studies in physics (for example, books (West et al., 2003; Zaslavsky, 2005; Uchaikin, 2008; Mainardi, 2010), edited volumes (Carpinteri and Mainardi, 1997; Hilfer, 2000; Sabatier et al., 2007), and reviews (Metzler and Klafter, 2000; Zaslavsky, 2002; Montroll and Shlesinger, 1984; Metzler and Klafter, 2004)). Now we can state that fractional dynamics form a new paradigm in science.