General Fractional Vector Calculus

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 theo...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autor principal:	Vasily E. Tarasov
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	vector calculus fractional calculus fractional dynamics Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/e6e776f548514c558f5b7011d441cbb9
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:e6e776f548514c558f5b7011d441cbb9
record_format	dspace
spelling	oai:doaj.org-article:e6e776f548514c558f5b7011d441cbb92021-11-11T18:20:45ZGeneral Fractional Vector Calculus10.3390/math92128162227-7390https://doaj.org/article/e6e776f548514c558f5b7011d441cbb92021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2816https://doaj.org/toc/2227-7390A 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.Vasily E. TarasovMDPI AGarticlevector calculusfractional calculusfractional dynamicsMathematicsQA1-939ENMathematics, Vol 9, Iss 2816, p 2816 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	vector calculus fractional calculus fractional dynamics Mathematics QA1-939
spellingShingle	vector calculus fractional calculus fractional dynamics Mathematics QA1-939 Vasily E. Tarasov General Fractional Vector Calculus
description	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.
format	article
author	Vasily E. Tarasov
author_facet	Vasily E. Tarasov
author_sort	Vasily E. Tarasov
title	General Fractional Vector Calculus
title_short	General Fractional Vector Calculus
title_full	General Fractional Vector Calculus
title_fullStr	General Fractional Vector Calculus
title_full_unstemmed	General Fractional Vector Calculus
title_sort	general fractional vector calculus
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/e6e776f548514c558f5b7011d441cbb9
work_keys_str_mv	AT vasilyetarasov generalfractionalvectorcalculus
_version_	1718431916575686656

General Fractional Vector Calculus

Ejemplares similares