Related papers: General Fractional Vector Calculus
H\"older functions represent mathematical models of nonlinear physical phenomena. This work investigates the general conditions of existence of fractional velocity as a localized generalization of ordinary derivative with regard to the…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case…
In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional…
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on…
The theory of fractional calculus has developed in a number of directions over the years, including: the formulation of multiple different definitions of fractional differintegration; the extension of various properties of standard calculus…
Local fractional derivative and integrals are revealed as one of useful tools to deal with everywhere continuous but nowhere differentiable functions in fractal areas ranging from fundamental science to engineering. In this paper, a…
We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak…
Gauss' and Stokes' theorems are fundamental results in vector calculus and important tools in physics and engineering. When students are asked to describe the meaning of Gauss' divergence theorem, they often use statements like this: "The…
The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In…
Fractional diffusion equations imply non-Gaussian distributions that generalise the standard diffusive process. Recent advances in fractional calculus lead to a class of new fractional operators defined by non-singular memory kernels,…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…
Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…
In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in…
Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of…