General Fractional Calculus: Multi-Kernel Approach

Vasily E Tarasov

doi:10.3390/math9131501

Abstract

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 Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

Highlights

The theory of integro-differential operators and equations is an important tool to describe systems and processes with non-locality in space and time
Nonlocality is determined by the form of the kernel of the operator, which are fractional integrals (FI) and fractional derivatives (FD)
In articles [27,28], two possible approaches to construct general fractional integrals and derivatives of arbitrary order, which satisfy general fundamental theorems of general fractional calculus (GFC), are proposed. These approaches are based on building the Luchko pairs ( M (t), N (t)) ∈ Ln with n > 1 for the GFI and general fractional derivatives (GFD) kernels from the Sonin pairs of kernels (μ(t), ν(t)) ∈

Summary

Introduction

The theory of integro-differential operators and equations is an important tool to describe systems and processes with non-locality in space and time. In articles [27,28], two possible approaches to construct general fractional integrals and derivatives of arbitrary order, which satisfy general fundamental theorems of GFC, are proposed These approaches are based on building the Luchko pairs ( M (t), N (t)) ∈ Ln with n > 1 for the GFI and GFD kernels from the Sonin pairs of kernels (μ(t), ν(t)) ∈. R−1 = (C−1 (0, ∞), ∗, +), is a commutative ring without divisors of zero [27,30], where the multiplication ∗ is the Laplace convolution and + the standard addition of functions These examples and other possible approaches to expanding the variety of types of kernels of operators of general fractional calculus and, nonlocality, are important for describing systems and processes with nonlocality in space and time. Sonin [31] is used as “Sonin”, and not in the French translation “Sonine” that is used in some papers

Luchko Set of Kernel Pairs and Its Subsets

General Fractional Integral and Derivatives

Fundamental Theorems of General Fractional Calculus

Conclusions