How Many Fractional Derivatives Are There?

Funding: This work was partially funded by National Funds through the FCT-Foundation for Science and Technology within the scope of the CTS Research Unit—Center of Technology and Systems/UNINOVA/FC /NOVA, under the reference UIDB/00066/2020, and also by FCT through IDMEC, under LAETA, project UID/EMS/50022/2020. Publisher Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland.In this paper, we introduce a unified fractional derivative, defined by two parameters (order and asymmetry). From this, all the interesting derivatives can be obtained. We study the one-sided derivatives and show that most known derivatives are particular cases. We consider also some myths of Fractional Calculus and false fractional derivatives. The results are expected to contribute to limit the appearance of derivatives that differ from existing ones just because they are defined on distinct domains, and to prevent the ambiguous use of the concept of fractional derivative.publishersversionpublishe

On the Numerical Computation of the Mittag-Leffler Function

The Mittag-Leffler function (MLF) plays an important role in many applications of fractional calculus, establishing a connection between exponential and power law behaviors that characterize integer and fractional order phenomena, respectively. Nevertheless, the numerical computation of the MLF poses problems both of accuracy and convergence. In this paper, we study the calculation of the 2-parameter MLF by using polynomial computation and integral formulas. For the particular cases having Laplace transform (LT) the method relies on the inversion of the LT using the fast Fourier transform. Experiments with two other available methods compare also the computational time and accuracy. The 3-parameter MLF and its calculation are also considered.publishe

Dark fermentative hydrogen production: from concepts to a sustainable production

ABSTRACT: The use of renewable sources and environmentally friendly processes is considered a priority for the construction of a sustainable energy future. The harmful impact of fossil fuels and the fact that we are reaching a disrupting point regarding environmental damage require the rapid implementation of new energy systems and a substantial increase in the use of alternative, unconventional energy sources. Hydrogen (H2) is considered one of the most promising sources as a clean energy vector, because of its high energy density (120 MJ/kg) and carbon-free combustion (Argun and Kargi, 2011). Hydrogen is the simplest and most abundant element on earth; however, it barely exists in nature in its molecular state. Instead, it is almost always found as part of other compounds from which it should be separated, either by thermochemical processes or through biological conversion.info:eu-repo/semantics/publishedVersio

Fractional Operators, Dirichlet Averages, and Splines

Fractional differential and integral operators, Dirichlet averages, and splines of complex order are three seemingly distinct mathematical subject areas addressing different questions and employing different methodologies. It is the purpose of this paper to show that there are deep and interesting relationships between these three areas. First a brief introduction to fractional differential and integral operators defined on Lizorkin spaces is presented and some of their main properties exhibited. This particular approach has the advantage that several definitions of fractional derivatives and integrals coincide. We then introduce Dirichlet averages and extend their definition to an infinite-dimensional setting that is needed to exhibit the relationships to splines of complex order. Finally, we focus on splines of complex order and, in particular, on cardinal B-splines of complex order. The fundamental connections to fractional derivatives and integrals as well as Dirichlet averages are presented

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator. Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions

Global existence of solutions for a fractional Caputo nonlocal thermistor problem

We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions