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.KeywordsFractional CalculusHausdorff DimensionFractional IntegrationHausdorff MeasureFractal DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.