The 1st Level General Fractional Derivatives and some of their Properties

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 order derivative and an integral operator of convolution type. Depending on succession of these operators, the general fractional derivatives of the Riemann-Liouville and of the Caputo types were defined and studied. The main objective of this paper is a construction of the 1st level general fractional derivatives that comprise both the general fractional derivative of the Riemann-Liouville type and the general fractional derivative of the Caputo type. We also provide some of their properties including the 1st and the 2nd fundamental theorems of Fractional Calculus for these derivatives and the suitably defined general fractional integrals.Comment: 17 pages. arXiv admin note: text overlap with arXiv:2202.0472

On complete monotonicity of solution to the fractional relaxation equation with the nnth level fractional derivative

In this paper, we first deduce the explicit formulas for the projector of the $n$th level fractional derivative and for its Laplace transform. Then the fractional relaxation equation with the $n$th level fractional derivative is discussed. It turns out that under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag-Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the 2nd level derivative.Comment: 15 pages, 1 figur

General fractional integrals and derivatives with the Sonine kernels

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. For the general fractional integrals and derivatives with the Sonine kernels, the two fundamental theorems of Fractional Calculus are proved. Then we construct the $n$-fold general fractional integrals and derivatives and study their properties.Comment: 17 page