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).
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References
F. Ben Adda, 1997, Geometric interpretation of the fractional derivative, Journal of Fractional Calculus, 11, 21–52.
F. Ben Adda, 1998a, Geometric interpretation of the differentiability and gradient of real order, Comptes Rendus de 1’Academie des Sciences. Series I: Mathematics, 326, 931–934. In Frcnch.
F. Ben Adda, 1998b, The differentiability in the fractional calculus, Comptes Rendus de 1’Academie des Sciences. Series I: Mathematics, 326, 787–791.
F. Ben Adda, 2001, The differentiability in the fractional calculus, Nonlinear Analysis, 47, 5423–5428.
G.D. Birkhoff, 1927, Dynamical Systems, American Mathematical Society.
A. Campa, T. Dauxois, S. Ruffo, 2009, Statistical mechanics and dynamics of solvable models with long-range interactions, Physics Reports, 480, 57–159.
A. Carpinteri, F. Mainardi (Eds.), 1997, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York.
N. Engheta, 1998, Fractional curl operator in electromagnetics, Microwave and Optical Technology Letters, 17, 86–91.
I.M. Gelfand, G.E. Shilov, 1964, Generalized Functions I: Properties and Operations, Academic Press, New York and London.
R. Gilmor, 1981, Catastrophe theory for Scientists and Engineers, Wiley, New York.
J. Guckenheimer, P. Holmes, 2002, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin.
A. Hussain, S. Ishfaq, Q.A. Naqvi, 2006, Fractional curl operator and fractional waveguides, Progress In Electromagnetics Research, 63, 319–335.
A. Hussain, Q.A. Naqvi, 2006, Fractional curl operator in chiral medium and fractional non-symmetric transmission line, Progress In Electromagnetics Research, 59, 199–213.
M.V. Ivakhnychenko, E.I. Veliev, 2004, Fractional curl operator in radiation problems, 10th International Conference on Mathematical Methods in Electromagnetic Theory, Sept. 14–17, Ukraine, IEEE, 231–233.
K.K. Kazbekov, 2005, Fractional differential forms in Euclidean space, Vladikavkaz Mathematical Journal, 7, 41–54. In Russian, http://www.vmj.ru/articles/20052-5.pdf
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Factional Differential Equations, Elsevier, Amsterdam.
N. Laskin, G.M. Zaslavsky, 2006, Nonlinear fractional dynamics on a lattice with long-range interactions, Physica A, 368, 38–54.
A.C.J. Luo, 2006, Singularity And Dynamics on Discontinuous Vector Fields, Elsevier, Amsterdam.
A.C.J. Luo, 2010, Discontinuous Dynamical Systems in Time-Varying Domains, Springer, Berlin.
M.M. Meerschaert, J. Mortensen, S.W. Wheatcraft, 2006, Fractional vector calculus for fractional advection-dispersion, Physica A, 367, 181–190; and New Zealand Mathematics Colloquium, Massey University, Palmerston North, New Zealand, December 2005, http://www.stt.msu.edu/mcubed/MathsColloq05.pdf
K.S. Miller, B. Ross, 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Q.A. Naqvi, M. Abbas, 2004, Complex and higher order fractional curl operator in electromagnetics, Optics Communications, 241, 349–355.
S.A. Naqvi, Q.A. Naqvi, A. Hussain, 2006, Modelling of transmission through a chiral slab using fractional curl operator, Optics Communications, 266, 404–406.
K.B. Oldham, J. Spanier, 1974, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York.
J. Palis, W. De Melo, 1982, Geometric Theory of Dynamical Systems. An Introduction, Springer, New York.
I. Podlubny, 1999, Fractional Differential Equations, Academic Press, New York.
B. Ross, 1975, A brief history and exposition of the fundamental theory of fractional calculus, Lecture Notes in Mathematics, 457, 1–36.
S.G. Samko, A.A. Kilbas, O.I. Marichev, 1993, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russian; and Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.
V.E. Tarasov, 2005a, Fractional generalization of gradient systems, Letters in Mathematical Physics, 73, 49–58.
V.E. Tarasov, 2005b, Fractional generalization of gradient and Hamiltonian systems, Journal of Physics A, 38, 5929–5943.
V.E. Tarasov, 2005c, Fractional hydrodynamic equations for fractal media, Annals of Physics, 318, 286–307.
V.E. Tarasov, 2005d, Electromagnetic field of fractal distribution of charged particles, Physics of Plasmas, 12, 082106.
V.E. Tarasov, 2005e, Multipole moments of fractal distribution of charges, Modern Physics Letters B, 19, 1107–1118.
V.E. Tarasov, 2006a, Magnetohydrodynamics of fractal media, Physics of Plasmas, 13, 052107.
V.E. Tarasov, 2006b. Electromagnetic fields on fractals, Modern Physics Letters A, 21, 1587–1600.
V.E. Tarasov, 2006c, Fractional statistical mechanics, Chaos, 16, 033108.
V.E. Tarasov, 2006d, Continuous limit of discrete systems with long-range interaction, Journal of Physics A, 39, 14895–14910.
V.E. Tarasov, 2006e, Map of discrete system into continuous, Journal of Mathematical Physics, 47, 092901.
V.E. Tarasov, 2007, Liouville and Bogoliubov equations with fractional derivatives, Modern Physics Letters B, 21, 237–248.
V.E. Tarasov, 2008, Fractional vector calculus and fractional Maxwell’s equations, Annals of Physics, 323, 2756–2778.
V.E. Tarasov, G.M. Zaslavsky, 2006a, Fractional dynamics of coupled oscillators with long-range interaction, Chaos, 16, 023110.
V.E. Tarasov, G.M. Zaslavsky, 2006b, Fractional dynamics of systems with longrange interaction, Communications in Nonlinear Science and Numerical Simulation, 11, 885–898.
E.I. Veliev, N. Engheta, 2004, Fractional curl operator in reflection problems, 10th International Conference on Mathematical Methods in Electromagnetic Theory, Sept. 14–17, Ukraine, IEEE, 228–230.
A.A. Vlasov, 1966, Statistical Distribution Functions, Nauka, Moscow. In Russian.
A.A. Vlasov, 1978, Nonlocal Statistical Mechanics, Nauka, Moscow. In Russian.
S.W. Wheatcraft, M.M. Meerschaert, 2008, Fractional conservation of mass, Advances in Water Resources, 31, 1377–1381.
Chen Yong, Yan Zhen-ya, Zhang Hong-qing, Applications of fractional exterior differential in three-dimensional space, Applied Mathematics and Mechanics, 24, 256–260.
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Tarasov, V.E. (2010). Fractional Vector Calculus. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_11
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DOI: https://doi.org/10.1007/978-3-642-14003-7_11
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