Fractional Vector Calculus and Fractional Maxwell's Equations

By Vasily E. Tarasov. 2008

320 citations recorded by the source corpus.

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s 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. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/20832bdbf7c66eac4e404f655eecb5049cd4d4e9

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Citations in the corpus

Fractional calculus of variations for a combined Caputo derivative (2011, background): “, [2, 6, 11, 20, 23, 24, 26, 36, 42] and many others in this journal “Fract.”
On a new class of fractional partial differential equations (2014, background): “Indeed, the literature on fractional derivatives in one dimension is quite extensive4, while in several dimensions there have been other possible de nitions due to Meerschaert, Mortensen and Wheatcraft [22] or Tarasov [32].”
A fractional calculus of variations for multiple integrals with application to vibrating string (2010, background): “We mention the recent paper [42], where some fractional versions of the classical results of Green, Stokes and Gauss are obtained via Riemann-Liouville and Caputo fractional operators.”
Review of Some Promising Fractional Physical Models (2013, background): “ long history from 1695. The history of fractional vector calculus is not so long. It has little more than ten years and can be reduced to small number of papers (about twenty articles) (see Refs. in [64]). A consistent fractional vector calculus is important for application in the following researc”
Fractional calculus: A survey of useful formulas (2013, unknown): “[21–24] and the references therein.”
TEMPERED FRACTIONAL CALCULUS (2015, background): “They are now used to model a wide variety of problems in physics [24, 35,38,50,51,55,65], finance [23,27,37,42,60,61], biology [2,4,22,26,36], and hydrology [1,7,8,15,17,62].”