Results 1 to 10 of about 364 (182)

Fractional Derivatives and the Fundamental Theorem of Fractional Calculus [PDF]

Fractional Calculus and Applied Analysis, 2020
In this paper, we address the one-parameter families of the fractional integrals and derivatives defined on a finite interval. First we remind the reader of the known fact that under some reasonable conditions, there exists precisely one unique family of the fractional integrals, namely, the well-known Riemann-Liouville fractional integrals.
Yuri Luchko, Luchko Yuri
exaly   +4 more sources

General Fractional Vector Calculus

Mathematics, 2021
A generalization of fractional vector calculus (FVC) 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.
Vasily E. Tarasov
doaj   +3 more sources

General Fractional Calculus in Multi-Dimensional Space: Riesz Form

Mathematics, 2023
An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949.
Vasily E. Tarasov
doaj   +3 more sources

The General Fractional Integrals and Derivatives on a Finite Interval

Mathematics, 2023
The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis.
Mohammed Al-Refai, Yuri Luchko
doaj   +3 more sources

General Fractional Integrals and Derivatives with the Sonine Kernels

Mathematics, 2021
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.
Yuri Luchko
doaj   +3 more sources

Scale-Invariant General Fractional Calculus: Mellin Convolution Operators

Fractal and Fractional, 2023
General fractional calculus (GFC) of operators that is defined through the Mellin convolution instead of Laplace convolution is proposed. This calculus of Mellin convolution operators can be considered as an analogue of the Luchko GFC for the Laplace ...
Vasily E. Tarasov
doaj   +3 more sources

General Fractional Calculus: Multi-Kernel Approach

Mathematics, 2021
For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one ...
Vasily E. Tarasov
doaj   +3 more sources

Operational Calculus for the General Fractional Derivatives of Arbitrary Order

Mathematics, 2022
In this paper, we deal with the general fractional integrals and the general fractional derivatives of arbitrary order with the kernels from a class of functions that have an integrable singularity of power function type at the origin.
Maryam Al-Kandari, Latif A-M. Hanna, Yuri Luchko   +2 more
doaj   +3 more sources

Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Mathematics, 2021
In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero.
Yuri Luchko
doaj   +3 more sources

Operational Calculus for the 1st-Level General Fractional Derivatives and Its Applications

Mathematics
The 1st-level General Fractional Derivatives (GFDs) combine in one definition the GFDs of the Riemann–Liouville type and the regularized GFDs (or the GFDs of the Caputo type) that have been recently introduced and actively studied in the fractional ...
Maryam Alkandari, Yuri Luchko
doaj   +3 more sources