Related papers: General fractional integrals and derivatives of ar…

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes…

The problem whether a given pair of functions can be used as the kernels of a generalized fractional derivative and the associated generalized fractional integral is reduced to the problem of existence of a solution to the Sonine equation.…

In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these…

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…

In this article, we introduce a new general definition of fractional derivative and fractional integral, which depends on an unknown kernel. By using these definitions, we obtain the basic properties of fractional integral and fractional…

In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…

Many different types of fractional calculus have been proposed, which can be organised into some general classes of operators. For a unified mathematical theory, results should be proved in the most general possible setting. Two important…

In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new…

Several approaches to the formulation of a fractional theory of calculus of "variable order" have appeared in the literature over the years. Unfortunately, most of these proposals lack a rigorous mathematical framework. We consider an…

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first…

Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann-Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel…

In this paper, we first deal with the general fractional derivatives of arbitrary order defined in the Riemann-Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of…

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce…

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…

This study presents new estimates for fractional derivatives without singular kernels defined by some specific functions. Based on obtained inequalities, we give a useful method to establish the global stability of steady states for…

Definitions of fractional derivative of order $\alpha$ ($0 < \alpha \leq 1$) using non-singular kernels have been recently proposed. In this note we show that these definitions cannot be useful in modelling problems with a initial value…

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…

In the present article, a new method for the evaluation of fractional derivatives of arbitrary real order is proposed. Numerous but inequivalent formulations have been given in the past. Some of them exhibit unsatisfactory properties such…

In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…

A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…