General Fractional Vector Calculus

Abstract

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

Trends, directions for further research, and some open problems of fractional calculus

“…For the first time, a consistent mathematical formulation of fractional vector calculus was proposed in work [135], [112, pp. 241-264], which includes fractional generalizations of the differential operations (gradient, divergence, curl), the integral operations (flux, circulation), and the relationship of these operators in the form the generalized Gauss, Stokes, and Green theorems (see also [136] and references therein).…”

“…However, for fractional operators with distributed lag, which are proposed in [154] and used in [109,, the fundamental theorems of FC have not been proved in general. Note that p.d.f of the gamma distribution is the Sonine kernel, and we can use the general FC [38][39][40][41]136,155] in this case and some other distributions on the positive semi-axis. This is an important direction of future research in FC and its applications in economics and physics.…”

Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

“…The solution method employed in [22] is the technique of the convolution series that are a far-reaching generalization of the power series with both integer and fractional exponents. Finally, we note the papers [23][24][25], where the theory presented in [18][19][20][21][22] was applied for the formulation of a general fractional dynamics, a general non-Markovian quantum dynamics, and a general fractional vector calculus.…”

The 1st Level General Fractional Derivatives and some of their Properties

“…The Cauchy problems for the fractional differential equations were considered in [15] in the case of the Kochubei GFDs and in [16,18] in the case of the Sonine GFDs. We also mention the papers [26,27,28], where the theory presented in [13,14,15,16,17,18,25] was applied for formulation of a general fractional dynamics, a general non-Markovian quantum dynamics, and a general fractional vector calculus, respectively.…”