Related papers: General fractional calculus: Multi-kernel approach
In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some…
Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have…
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…
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…
A generalization of fractional vector calculus 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…
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 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…
General fractional calculus offers an elegant and self-consistent path toward the generalization of fractional calculus to an enhanced class of kernels. Prabhakar's theory can be thought of, to some extent, as an explicit realization of…
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…
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, 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 introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
Most generalized fractional operators rely on prescribed memory kernels, restricting hereditary behavior to predefined forms and limiting flexibility in modeling diverse memory effects. Motivated by these limitations, this paper develops a…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a…
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 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…