General Fractional Vector Calculus

TL;DR

This paper develops a comprehensive general fractional vector calculus framework, extending classical vector calculus theorems to non-local fractional operators with general kernels, applicable to complex regions and coordinate systems.

Contribution

It introduces self-consistent definitions of fractional vector operators and proves fundamental theorems for a broad class of non-local fractional calculus.

Findings

Proposed new definitions of fractional gradient, curl, divergence, and integrals.

Proved fractional Green, Stokes, and Gauss theorems for complex regions.

Extended fractional calculus to orthogonal curvilinear coordinates.

Abstract

A generalization of fractional vector calculus 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 in the Luchko approach. This paper proposed the following: (1) 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. (2) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (3) The general fractional gradient, Green, Stokes and Gauss 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. Therefore we can state that we proposed a calculus, which is a general fractional vector calculus. 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 (Leibniz) rule, chain rule, and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.