General Fractional Integrals and Derivatives with the Sonine Kernels

TL;DR

This paper develops a comprehensive theory of general fractional integrals and derivatives using Sonine kernels, including new classes with singularities, fundamental theorems, and properties of iterated operators.

Contribution

It introduces a new class of Sonine kernels with integrable singularities and proves fundamental theorems for fractional calculus based on these kernels.

Findings

Established fundamental theorems for fractional derivatives with Sonine kernels.

Constructed and analyzed properties of n-fold fractional integrals and derivatives.

Identified special classes of Sonine kernels with power-type singularities.

Abstract

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 and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the $n$-fold general fractional integrals and derivatives and study their properties.