Related papers: On fractional calculus with general analytic kerne…
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…
We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new…
The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).
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…
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 paper deals with a fractional derivative introduced by means of the Fourier transform. The explicit form of the kernel of general derivative operator acting on the functions analytic on a curve in complex plane is deduced and the…
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…
(Draft 3) A generalized differential operator on the real line is defined by means of a limiting process. These generalized derivatives include, as a special case, the classical derivative and current studies of fractional differential…
The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which…
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…
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…
The goal of this communication is to propose a generalized notion of the "traditional derivative". This generalization includes the fractional derivatives such as the Riemann-Liouville, Gruenwald-Letnikov, Weyl, Riesz, Caputo, Marchaud…
The following material was created with the idea of being used for an introductory fractional calculus course. A recapitulation of the history of fractional calculus is presented, as well as the different attempts at fractional derivatives…
Most of the known Fourier transforms associated with the equations of mathematical physics have a trivial kernel, and an inversion formula as well as the Parseval equality are fulfilled. In other words, the system of the eigenfunctions…
Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the…
In this paper, we first discuss the convolution series that are generated by the Sonine kernels from a class of functions continuous on the real positive semi-axis that have an integrable singularity of power function type at the point…
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 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…
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…
The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation.The fractional Euler-Lagrange equations were obtained and two examples were studied.