Fractional Vector Calculus and Fractional Maxwell's Equations

TL;DR

This paper reviews the development of fractional vector calculus, introduces new formulations of fundamental theorems, and applies these to derive fractional Maxwell's equations and wave equations.

Contribution

It presents a consistent formulation of fractional vector calculus and extends classical theorems to fractional order, enabling the derivation of fractional Maxwell's equations.

Findings

Formulated fractional Green's, Stokes', and Gauss's theorems.

Developed fractional generalizations of exterior calculus.

Derived fractional nonlocal Maxwell's and wave equations.

Abstract

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.