Fractional Vector Calculus

Abstract

The calculus of derivatives and integrals of non-integer order go back to Leibniz, Liouville, Grünwald, Letnikov and Riemann. The fractional calculus has a long history from 1695, when the derivative of order α = 0.5 was described by Leibniz (Oldham and Spanier, 1974; Samko et al., 1993; Ross, 1975). The history of fractional vector calculus (FVC) is not so long. It has only 10 years and can be reduced to the papers (Ben Adda, 1997, 1998a, b, 2001; Engheta, 1998; Veliev and Engheta, 2004; Ivakhnychenko and Veliev, 2004; Naqvi and Abbas, 2004; Naqvi et al., 2006; Hussain and Naqvi, 2006; Hussain et al., 2006; Meerschaert et al., 2006; Yong et al., 2003; Kazbekov, 2005) and (Tarasov, 2005a, b, d, e, 2006a, b, c, 2007, 2008). There are some fundamental problems of consistent formulations of FVC that can be solved by using a fractional generalization of the fundamental theorem of calculus (Tarasov, 2008). We define the fractional differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’ theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. A consistent FVC can be used in fractional statistical mechanics (Tarasov, 2006c, 2007), fractional electrodynamics (Engheta, 1998; Veliev and Engheta, 2004; Ivakhnychenko and Veliev, 2004; Naqvi and Abbas, 2004; Naqvi et al., 2006; Hussain and Naqvi, 2006; Hussain et al., 2006; Tarasov, 2005d, 2006a, b, 2005e) and fractional hydrodynamics (Meerschaert et al., 2006; Tarasov, 2005c). Fractional vector calculus is very important to describe processes in complex media (Carpinteri and Mainardi, 1997).KeywordsFractional DerivativeFractional CalculusFundamental TheoremFractional GeneralizationVector CalculusThese 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.