Related papers: Fractional Vector Calculus and Fractional Maxwell'…
This paper focuses on the equivalent expression of fractional integrals/derivatives with an infinite series. A universal framework for fractional Taylor series is developed by expanding an analytic function at the initial instant or the…
The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).
Fractional calculus of variation plays an important role to formulate the non-conservative physical problems. In this paper we use semi-inverse method and fractional variational principle to formulate the fractional order generalized…
The fractional integrals and fractional derivatives problem is tackled by using the operator approach. The definition domain E of operators is causal functions.Many properties of fractional integrals are given. Fractional derivatives…
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain…
We generalize the fractional Caputo derivative to the fractional derivative ${{^CD}^{\alpha,\beta}_{\gamma}}$, which is a convex combination of the left Caputo fractional derivative of order $\alpha$ and the right Caputo fractional…
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…
The paper surveys some recent results concerning vector analysis on fractals. We start with a local regular Dirichlet form and use the framework of 1-forms and derivations introduced by Cipriani and Sauvageot to set up some elements of a…
Recently, the research community has been exploring fractional calculus to address problems related to cosmology; in this approach, the gravitational action integral is altered, leading to a modified Friedmann equation, then the resulting…
The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its…
A connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results…
This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper in J.…
We investigate some basic applications of Fractional Calculus (FC) to Newtonian mechanics. After a brief review of FC, we consider a possible generalization of Newton's second law of motion and apply it to the case of a body subject to a…
In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \citet{nigmatullin}, and \citet{zaslavsky} in $\mathbb R^d$ for modeling some physical…
Fractional kinetic theory plays a vital role in describing anomalous diffusion in terms of complex dynamics generating semi-Markovian processes. Recently, the variational principle and associated Levy Ansatz have been proposed in order to…
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…
For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in the works Mathematics 9(6) (2021) 594 and Symmetry 13(5) (2021) 755. In these works, the proposed approaches to formulate this calculus are…
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of…
In fractional calculus there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and…
In the article we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss-Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of…