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General fractional calculus: Multi-kernel approach

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 based either on the power of one Sonine kernel, or the convolution of one Sonine 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.


Introduction
The theory of integro-differential operators and equations is important tool to describe systems and processes with non-locality in space and time. Among such operators, an important role is played by the integrals and derivatives of non-integer orders [1,2,3,4,5], [6,7]. These operators are called fractional derivatives (FD) and fractional integrals (FI). For these operators, generalizations of first and second fundamental theorems of standard calculus are satisfied. This is one of the main reasons to state that these operators form a calculus, which is called fractional calculus.
Equations with derivatives of non-integer orders with respect to time and space are important tools to describe non-locality in time and space in physics [8,9], biology [10], and economics [11,12], for example. To describe various physical, biological, economic phenomena with nonlocality, it is important to have a wide range of operators that allow us to describe various types of nonlocality [13]. Nonlocality is determined by the form of the kernel of the operator, which are fractional integrals (FI) and fractional derivatives (FD) of non-integer orders. Therefore it is important to have a fractional calculus that allows us to describe non-locality in a general form.
An important turning point in formulation of general fractional calculus (GFC) was the results obtained by Anatoly N. Kochubei in his work [14] (see also [15]) in 2011. In this work, the general fractional integral (GFI) and general fractional derivatives (GFD) of the Riemann-Liouville and Caputo type are defined, for which the general fundamental theorems are proved (see Theorem 1 in [14]). In addition, the relaxation and diffusion equations [14,16], and then the growth equation [17], which are contain GFD, are solved. In fact, the term "general fractional calculus" (GFC) was introduced in article [14]. This approach to GFC is based on the concept of Sonine pairs of mutually associated kernels proposed in the work [18] (see also [19]). The integral equations of the first kind with Sonine kernels, and the GFI and GFD of the Liouville and Marchaud type are described in [20,21]. After Kochubei article [14], works on general fractional calculus and some of its applications began to be published (see [22,23,24,25,26] and references therein).
The next revolutionary step in the construction of the general fractional calculus was proposed in the works of Yuri Luchko in 2021 [27,28,29]. In artiles [27,28] a general fractional integrals and derivatives of arbitrary order have been proposed. The general fundamental theorems of GFD are proved for the GFI and GFDs of Riemann-Liouville and Caputo in [27,28]. Operational calculus for equations with general fractional derivatives with the Sonine kernels is proposed in [29].
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 ( ( ), ( )) ∈ with > 1 for the GFI and GFD kernels from the Sonine pairs of kernels ( ( ), ( )) ∈ −1 = L 1 , [27,28]. These two possible approaches to define kernels of GDI and GFD, which are proposed in [27,28], can be briefly described as follows.
2) In article [28], the kernels ( ), ( ) of GFI and GFD, which belong to the Luchko set , are considered in the form In works [27,28], the proposed approaches are based either on the powers of one Sonine kernel [27], or the convolution of one Sonine kernel with the kernels of the integer-order integrals [28]. In applications of fractional calculus and GFC, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels.
Let us give some examples of possible expansions of operator kernels of the general fractional calculus. In the beginning we proposed to define the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pairs, by using ( ), ( ) ∈ −1 (0, ∞) in Definition 5. Then we can consider, for example, the following pairs of kernels from the Luchko set . I) As an example of a generalization of the first approach, we can use the kernels that are the Laplace convolution of different types of kernels. For example, one of the ways to define the kernels ( ( ), ( )) ∈ ℒ with > 1 is to remove the restrictions ( ) = ( ) and ( ) = ( ) for all = 1, … , , which are used in (1). II) Another example is removing the restriction on using only one pair of Sonine kernel in (3). For example, we can consider the Luchko pairs ( ( ), ( )) ∈ L with > 1 in the form instead of (3), where ( ), ( ) ∈ −1,0 (0, ∞), and ( * )( ) = {1} for all = 1, … , . (7) III) As a more general example of the pair of kernels from , we can consider the Laplace convolutions of kernel pairs ( ( ), ( )) from the Luchko sets L such that for all = 1, … . These possible approaches of extensions are based on the statement: the triple ℛ −1 = ( −1 (0, ∞), * , +), is a commutative ring without divisors of zero [30,27], 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, thus, nonlocality, are important for describing systems and processes with nonlocality in space and time.
In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed as an extension of the Luchko approaches, which is described in [27,28]. Extension is achieved by using different types (subsets) of Sonine and Luchko pairs of kernels in definitions general fractional integrals and derivatives of multi-kernel form. 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 sets of kernel pairs are subsets of the Luchko set of kernel pairs. The first and second fundamental theorems for the proposed general fractional derivatives and integrals of multi-kernel form are proved in this paper.

Luchko Set of Kernel Pairs and its Subsets
Let us give definitions of some exteansion of the concept of the Luchko pairs of kernels , which is given in [28] and some subset of kernels from . In these definitions, we use the function spaces Then the set of such kernel pairs ( ( ), ( )) will be called the Luchko pairs and it will be denoted as . The kernels ( ), ( ) will be called the Luchko kernels, and the pair ( ( ), ( )) will be called the Luchko pairs. [28, p.7]. In article [28, p.7], the set is defined as a set of the kernels ( ), ∈ −1 (0, ∞) and ( ) ∈ −1,0 (0, ∞) ⊂ −1 (0, ∞) that satisfy condition (10).
Note that as a special case of , , we can consider the kernels from the subset ⊂ . In general, we assume that ( ( ), ( )) ∈ , which may not belong to the subset . In other words, the kernels ( ), ( ), which are used in (19) and can be represented through the Sonine pairs of kernels as (14), are only special case of , .

Remark 4
The subsets , , , , , , and others that are built from the Sonine pairs of kernels cannot cover the entire set of the Luchko pairs . For example, the pair of the kernels ( ) = /2 (2√ ), with − 2 < < − 1, ∈ ℕ, belongs to the Luchko set , [28, p.9], where ( ) and ( ) are are the Bessel and the modified Bessel functions that defined in (25). Proof. The proof of Theorem 5 is similar to the proof of Theorem 3. □ The set of kernel pairs ( ( ), ( )), which belong to and can be represented by expressions that are used in Theorems 4 and 5 will be denoted as ,{ } and ,{ }, , respectively.
Let us give some examples of the kernel pairs from the set 1 (see [27,27] and references therein).

Example 5
The pair of the kernels is well-known in fractional calculus as kernels of the Riemann-Liouville fractional derivatives and integrals [1,4]. This are Sonine pair of kernels, if 0 < 1 < 1.

Example 6
The pair of the kernels
(51) Using these notation, we give the definitions of the general fractional operator for the pairs of kernels from , , with 1 ≤ ≤ , and ≤ ≤ .
The GF operators are defined similarly for subsets , and , of the Luchko set . For the subsets ,{ } and ,{ }, of , the GFI and GFDs are are defined similarly to Definitions 5, 6, 7.

Fundamental Theorems of General Fractional Calculus
The fundamental theorems (FT) of standard calculus for derivatives and integrals of integer order ∈ ℕ are the following. The first FT is written as The second FT is given as (57) Here is the integral of the order ∈ ℕ such that Let us prove FT of the general fractional calculus for multi-kernel approach. Then, for the GF-derivative ( ) of Riemann-Liouville type (7) and the GF-integral (5), the equalities ( ) ( ) ( ) = ( ) (69) holds for ( ) ∈ −1 (0, ∞).
In a more general form, the main idea of this article is as follows: We propose to use different generators (generating set) of pairs of kernels for each Luchko set and powers {1} with = 1, … , − 1 to construct pairs of kernels for Luchko set of the next order . The possibility of this approached is based on the fact that the triple ℛ −1 = ( −1 (0, ∞), * , +), where the multiplication * is the Laplace convolution and + the standard addition of functions, is a commutative ring without divisors of zero [30,27]. Here we mean that a generating set of the Luchko set is a subset of such that almost every kernel of can be expressed as a combination (by using the Laplace convolution) of finitely many kernels of the subset and their associated kernels (their inverses). If the ring ℛ −1 is a ring having a system of generators , then almost every kernel can be represented as a product (the Laplace convolution) of kernels from , and inverse (associated) to them. Note that the number of kernels multiplied by the Laplace convolution can be more than (for example, (ℎ * ℎ )( ) = ℎ + ( )). The mathematical implementation of this idea is a complex and interesting problem that should to be solved. It is not obvious that this idea can be implemented, but research in this direction can lead to interesting results, which will be important for general fractional calculus and its various applications.
The proposed approach to general fractional calculus can be useful for various applications in physics, economics, nonlinear dynamics and for other areas of science. In applications of the general fractional calculus, it is useful to have a wider range of operators, for example, which kernels are defined by using the Laplace convolution of different types of kernels. The importance of the proposed approach to GFC is related with the importance of describing systems and processes with a wider variety of nonlocalities in time and space [8,9,10,11,12], [31,32,33]. The GFC and the proposed multi-kernel approach to GFC can be important to obtaining results concerning of general form of nonlocality, which can be described by general form operator kernels, and not its particular implementations and representations [34]. For example, we can derive general nonlocal maps as exact solutions of nonlinear equations with GFI and GFD at discrete points [34], and a general approach to the construction of non-Markovian quantum theory can be proposed.