The formal constructions of the operators nowadays referred to as the general fractional integrals (GFIs) and the general fractional derivatives (GFDs) were suggested for the first time by Sonin in [27]. In this paper, Sonin extended Abel’s method for solving the Abel integral equation presented in [1,2] to a class of the integral equations with the so-called Sonin kernels. He recognized that the basic ingredient of Abel’s method for solving the Abel integral equation (in the modern notations) is nothing else than a simple formula for the power law kernels and suggested its generalisation in the form where * stands for the Laplace convolution. Nowadays the condition (3) is referred to as the Sonin condition and the functions that satisfy this conditions are called the Sonin kernels.

The simplest pair of the Sonin kernels are the power law kernels $h_{\alpha }$ and $h_{1-\alpha }$ that was known already to Abel, see the relation (2). In [27], Sonin introduced an important class of the Sonin kernels that can be represented in the following form: where the functions ${\kappa }_1={\kappa }_1\left(t\right)$ and $k_1=k_1\left(t\right)$ are analytical on $\mathbb{R}$ and their coefficients satisfy the following infinite system of linear equations with a triangular matrix:

In particular, he derived the famous pair of the Sonin kernels in terms of the Bessel function $J_{\nu }$ and the modified Bessel function $I_{\nu }$.

Following Abel, Sonin formally solved the integral equation with the Sonin kernel $\kappa$ and represented its solution in the form where the kernel k is the Sonin kernel associated to the kernel $\kappa$ through the Sonin condition (3). In particular, for the Abel integral equation (1), the solution takes the well-known form:

However, Sonin did not interpreted his results in terms of the Fractional Calculus. This was done by Kochubei much later in his paper [13], where among other things he introduced and investigated the regularized GFD or the GFD of the Caputo type in the form with the Sonin kernel k from a special class $\mathcal{K}$ of kernels described in terms of their Laplace transforms (see [13] for details). Kochubei also considered the GFD of the Riemann-Liouville type

Evidently, the regularized GFD (7) and the GFD (8) are connected each to other by the relation

On the space of absolutely continuous functions, another useful representation of the regularized GFD is valid in the following form:

For the Sonin kernels k from the class $\mathcal{K}$, Kochubei showed existence of the associated completely monotonic Sonin kernels $\kappa$ and introduced the corresponding GFI as follows:

It is worth mentioning that the constructions of the GFI and GFDs with the Sonin kernels were first formally introduced by Sonin in [27]. However, their theory essentially depends on the kernels or on the classes of kernels and on the spaces of functions where these operators are studied. Say, even in the case of the power law Sonin kernels $h_{\alpha }$ and $h_{1-\alpha }$ that generate the Riemann-Liouville fractional integral and the Riemann-Liouville and Caputo fractional derivatives, the properties of these Fractional Calculus (FC) operators are very different on different spaces of functions. In [13], Kochubei introduced and investigated a very important special case of the GFI and GFDs with the Sonin kernels from the class $\mathcal{K}$. However, one of the conditions posed on the kernels from $\mathcal{K}$ (the Laplace transform of the kernel k has to be a Stieltjes function) is very restrictive. As a consequence, not all of the known Sonin kernels belong to this class.

In the subsequent publications devoted to the GFI and GFDs, essentially larger classes of the Sonin kernels compared to those suggested in [27] and [13] were introduced and investigated. In [15,16], the GFI and GFDs with the Sonin kernels from the class ${\mathcal{S}}_{-1}$ as well as fractional differential equations with these GFDs were studied. The Sonine kernels from ${\mathcal{S}}_{-1}$ are the functions continuous on ${\mathbb{R}}_{+}$ that have an integrable singularity of the power law type at the origin (see [15,16] for details). The class ${\mathcal{S}}_{-1}$ is a very general one and contains both the kernels (4), (5) introduced by Sonin and the Kochubei class $\mathcal{K}$ of the Sonin kernels.

In [17], the Sonin condition (3) was generalized and the GFI and GFDs of arbitrary order were introduced (the operators (7), (8), and (11) with the Sonin kernels have a "generalized order" less than one). In [18], the so-called 1st level GFD was suggested. This derivative is constructed as a composition of two GFIs with different kernels and the first order derivative and contains both the GFD (8) and the regularized GFD (10) as its particular cases. In the recent publications by Tarasov, the multi-kernel approach for definition of the GFI and GFDs, the general vector calculus based on the GFI and GFDs, and the Riesz form of the general FC operators in the multi-dimensional space were suggested, see [28,29,35], respectively. We also refer to the paper [5], where the left- and right-hand sided GFIs and GFDs on a finite interval were introduced and investigated. For an overview of the recent publications devoted to the fractional differential equations, both ordinary and partial, with the GFDs we refer to the recent survey paper [19].

It is worth mentioning that the GFI and GFDs introduced in the publications mentioned above have already found there first applications. In the papers [30,31,32,33,34], Tarasov suggested several non-local physical theories based on the GFI and GFDs with the Sonin kernels including the general fractional dynamics, the general non-Markovian quantum dynamics, the general non-local electrodynamics, the non-local classical theory of gravity, and the non-local statistical mechanics. Furthermore, the GFI and GFDs with the Sonin kernels were used in some mathematical models for anomalous diffusion and in the linear viscoelasticity, see, e.g., [6,7,11,21,22,25].

Another active direction of research in modern FC concerns an object that does not possess a counterpart in the world of integer order derivatives, namely, the so-called fractional derivatives of distributed order. This interest has a clear physical background. Recently, a lot of attention in modelling of the anomalous diffusion processes was attracted by the so-called ultra-slow diffusion that is characterized by the logarithmic behaviour of the mean squared displacement of the diffusing particles, see, e.g., [8,9,23,26], and the references therein. One of the most promising approaches for description of such processes is by means of the time-fractional diffusion equations with the fractional derivatives of distributed order. From the mathematical viewpoint, the fractional derivatives of distributed order and the fractional differential equations with these derivatives were studied, e.g., in [4,10,12,14,20,36].

Until now, the definitions of the fractional derivatives of distributed order were based on the conventional fractional derivatives and especially on the Riemann-Liouville and the Caputo derivatives. The main subject of this paper is in introducing a concept of the GFDs of distributed order and in investigation of their basic properties. These operators are a generalization of both the GFDs and the fractional derivatives of distributed order introduced so far. Moreover, in this paper, we also define the corresponding fractional integrals of distributed type and prove the fundamental theorems of FC for these integrals and the GFDs distributed order.

The rest of the paper is organized as follows: In the 2nd Section, a one-parametric class of the Sonin kernels is introduced and the GFDs of distributed order with the kernels from this class are defined and investigated. The 3rd Section is devoted to fractional integral of distributed type and some connections between the GFDs and the fractional integral of distributed type in form of two fundamental theorems of FC. In the final 4th Section, some examples of the GFDs of distributed order and the corresponding fractional integrals of distributed type are presented.

First we remind the readers on the definitions of the Riemann-Liouville and the Caputo fractional derivatives of distributed order on the interval $(a,b),0\le a<b\le 1$, respectively: where the weight function w satisfies the properties $w\in C\left(\right[a,b\left]\right)$, $w\left(\alpha \right)\ge 0$$\forall \alpha \in [a,b]$ and $w\left(\alpha \right)\neg \equiv 0$, the Riemann-Liouville fractional derivative of the order $\alpha ,0<\alpha <1$ is defined by and the Caputo fractional derivative of the order $\alpha ,0<\alpha <1$ is given by

Please note that in some FC publications, the distributed order derivatives (12) and (13) are defined on the interval $(a,b)$ with $0\le a<b\le 2$. However, in this paper, we restrict ourselves to the case $0\le a<b\le 1$.

For a generalization of the distributed order fractional derivatives (12) and (13) to the case of the GFDs of distributed order, we need a class of the Sonin kernels that explicitly depend on a parameter that can be interpreted as a "generalized order" of the corresponding GFDs.

In what follows, we deal with the Sonin kernels $k_{\alpha }\left(t\right)$ and their associated kernels ${\kappa }_{\alpha }\left(t\right)$ that satisfy the following two constraints:

C1) The Sonin condition holds valid for all $\alpha \in (a,b)$, where $0\le a<b\le 1.$

C2) The kernels k and $\kappa$ can be represented as follows: where $k_1,{\kappa }_1\in C([0,+\infty )\times [a,b])$ and $k_1(0,\alpha )\ne 0,{\kappa }_1(0,\alpha )\ne 0,t>0,\alpha \in (a,b)$.

The class of the Sonin kernels that satisfy the conditions C1) and C2) will be denoted by ${}_{*}{S}_{\alpha }$. Evidently, any associated kernel ${\kappa }_{\alpha }$ to a kernel $k_{\alpha }\in {}_{*}{S}_{\alpha }$ also belongs to the class ${}_{*}{S}_{\alpha }$ and $k_{\alpha }$ is its associated kernel.

Please note that the kernels from the class ${}_{*}{S}_{\alpha }$ are functions of two variables: $t(t\in {\mathbb{R}}_{+})$ and $\alpha (\alpha \in (a,b\left)\right)$. The simplest and very important example of this kind are the Sonin kernels $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right),\alpha \in (0,1)$ and ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right),\alpha \in (0,1)$ of the Riemann-Liouville fractional derivative and the Riemann-Liouville fractional integral, respectively.

In general, the Sonin kernels from the class ${}_{*}{S}_{\alpha }$ and their associated kernels are the power law functions $h_{1-\alpha }\left(t\right)$ and $h_{\alpha }\left(t\right)$ that are disturbed by (multiplied with) some continuous functions (compare to the kernels (4) and (5) introduced by Sonin). These continuous functions can depend on $\alpha$ and/or other parameters or not. Most of the known Sonin kernels (see, e.g., [15,16]) belong to the class ${}_{*}{S}_{\alpha }$ and thus our theory will cover many known particular cases including those presented in the examples from our paper, see the 4th Section. However, as mentioned in [24], the Sonin kernels can also possess other kinds of singularities at the origin, say, the ones of the power-logarithmic type. Such kernels are not covered by our theory.

It is also worth mentioning that for any fixed $\alpha \in (a,b)$, the Sonin kernels from the class ${}_{*}{S}_{\alpha }$ are functions of the t-variable that belong to the space $C_{-1}(0,+\infty )$ defined as follows:

The space $C_{-1}(0,+\infty )$ was employed in several publications devoted to the GFIs and GFDs with the Sonin kernels from the class ${\mathcal{S}}_{-1}$ (see, e.g., [15,16]):

For a fixed $\alpha \in (a,b)$, the Sonin kernels from the class ${}_{*}{S}_{\alpha }$ introduced above evidently belong to the class ${\mathcal{S}}_{-1}$. Thus, we can use the results derived in these publications for analysis of the GFDs and GFIs with the kernels from the class ${}_{*}{S}_{\alpha }$.

In what follows, we also employ another useful subspace of the space $C_{-1}(0,+\infty )$ defined by

For a kernel $k_{\alpha }\in {}_{*}{S}_{\alpha }$, the GFD (8) and the regularized GFD (10) are defined for all values of $\alpha$ from the interval $(a,b)$ as follows:

Now we proceed with the definitions of the corresponding general fractional derivatives of distributed order.

Now we derive a useful representation of the regularized GDOFD (24) that by definition is the following iterated integral

The representation (25) ensures that the corresponding double integral is absolutely integrable. Thus, by using Fubini’s theorem we can interchange the order of integration in the iterated integral and get the following formula: where and * denotes the Laplace convolution.

Applying the same procedure to the GDOFD (23), we arrive at the analogous representation where the kernel $k^w$ is defined as in (28).

As in the case of the GDOFD (24), under some additional conditions (see the next section), the GDOFD (23) can be interpreted as a GFD of the Riemann-Liouville type with the kernel $k^w\in {\mathcal{S}}_{-1}$.

Now we discuss an important relation between the GDOFD (23) and the regularized GDOFD (24). To derive it, we employ the relation that holds true for any $u\in C_{-1}^1(0,\infty )$ and for any kernel from the class ${\mathcal{S}}_{-1}$, see [15,16]. As already mentioned, for a fixed $\alpha \in (a,b)$, the kernels from the class ${}_{*}{S}_{\alpha }$ belong to the class ${\mathcal{S}}_{-1}$. Using the formula (30) we thus arrive at the relation

Some examples of the GDOFDs introduced above are provided in Section 4.

To introduce the general fractional integrals of distributed type (GFIDs), in what follows, we impose some additional conditions on the kernels $k_{\alpha }\left(t\right)$ from the class ${}_{*}{S}_{\alpha }$:

C3) The Laplace transform exists for all $\rho >0$,

C4) The Laplace transform $K_{\alpha }\left(\rho \right)$ satisfies the following standard conditions:

The class of the kernels from ${}_{*}{S}_{\alpha }$ that satisfy the conditions C3) and C4) will be denoted by ${}_{*}{\widehat{S}}_{\alpha }$. In what follows, we always consider the operators with the kernels from the class ${}_{*}{\widehat{S}}_{\alpha }$, i.e., the kernels that satisfy the conditions C1)-C4).

For a definition of the GFIDs, we need some auxiliary results. First, we mention an evident relation between the Laplace transform $K^w\left(\rho \right)$ of the kernel $k_w$ given by (28) and the Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$. Moreover, for $k_{\alpha }\in {}_{*}{\widehat{S}}_{\alpha }$, the Laplace transform $K^w\left(\rho \right)$ satisfy the conditions (32) and (33).

The last formula, the Laplace convolution theorem, and the known Laplace transform formula for the first order derivative result in the following useful representation of the Laplace transform for the regularized GDOFD (24):

Now let us consider the following initial-value problem for the fractional differential equation with the regularized GDOFD (24):

Assuming existence of the Laplace transform of the function g for all $\rho >0$, we apply the Laplace transform to the above equation and using the formula (35), we get the relation

The solution u to the problem (36) can be formally represented as follows:

The convolution theorem for the Laplace transform leads then to the representation of the solution u in the time-domain, where the kernel ${\psi }_w$ is defined in terms of the inverse Laplace transform:

The function ${\psi }_w$ will play the role of a kernel of the GFID. In the following theorem, we provide some important characteristics of this function.

Motivated by the form (37) of the solution to the fractional differential equation (36) and by Theorem 3.1, we now proceed with defining the GFIDs.

As shown in Theorem 3.1, the kernel ${\psi }_w$ of the GFID (39) is from the class ${\mathcal{S}}_{-1}$ of the Sonin kernels. Thus, the operator (39) is a special case of the GFIs with the kernels from ${\mathcal{S}}_{-1}$ and we can employ the results already derived for the GFIs on the space $C_{-1}(0,\infty )$ (see, e.g., [15,16,17] and subsequent publications). In particular, the following properties are worth mentioning:

According to Theorem 3.1, the kernel $k^w$ of the GDOFD (23) and of the regularized GDOFD (24) is an associated Sonin kernel to the kernel ${\psi }_w$ of the GFID (39). Thus, we can apply the first and the second fundamental theorems for the GFDs and the GFIs with the Sonin kernels from the class ${\mathcal{S}}_{-1}$ derived in [15] and arrive at the following important results:

For the proofs of the fundamental theorems for the GFDs and the GFIs with the Sonin kernels from the class ${\mathcal{S}}_{-1}$ we refer the interested readers to [15].

In this section, we discuss three particular examples of the Sonin kernels from the class ${}_{*}{\widehat{S}}_{\alpha }$ and the corresponding general fractional operators of distributed order.

1st Example: We start with the power law kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ of the Riemann-Liouville and the Caputo fractional derivatives with the associated kernel ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)$ of the Riemann-Liouville fractional integral.

In this case, the GDOFDs introduced in this paper are nothing else than the Riemann-Liouville and the Caputo fractional derivatives of distributed order on the interval $(0,1)$ defined as in (12) and (13), respectively. As mentioned in Introduction, the distributed order fractional derivatives of the Riemann-Liouville and the Caputo types are well-studied (see, e.g., [4,10,12,14,20], and [36]) and have many applications. In this example, we look at these operators from the viewpoint of our general theory.

For the kernels $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ and ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)$, we set $k_1(t,\alpha )={\kappa }_1(t,\alpha )=1$ and $a=0,b=1$ and show now that they belong to the class ${}_{*}{\widehat{S}}_{\alpha }$ of the Sonin kernels introduced in Section 2. Indeed, they evidently satisfy the conditions C1) and C2).

Moreover, the Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ does exist for $\rho >0$ and can be written down in explicit form:

For the function $K_{\alpha }\left(\rho \right)={\rho }^{\alpha -1}$, the conditions (32) and (33) are evidently satisfied and thus the kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ belongs to the class ${}_{*}{\widehat{S}}_{\alpha }$ of the Sonin kernels.

This means that all of the results that were presented in the previous sections including the properties of the corresponding GFID and the fundamental theorems of FC for the GDOFDs and the GFID hold true. However, we found instructive to do some independent calculations and derivations and to establish some explicit formulas that are not possible in the general case.

For the power law kernels, the formula (34) takes the form

Because $w\in C[0,1]$ and ${\rho }^{\alpha -1}$ is integrabale and of one sign for $0<\alpha <1$, applying the mean value theorem for the last integral yields the relation for some ${\alpha }_0$ ($0<{\alpha }_0<1)$. We also mention that $w\left({\alpha }_0\right)\ne 0$ because of the evident inequality $K^w\left(\rho \right)>0$ for any $\rho >0$.

Now we get the representation

Because the function at the right-hand of the last formula has finitely many singular point and tends to 0 as $\rho \to \infty$, its inverse Laplace transform is well-defined and can be represented in explicit form ([3], p. 1027): where $E_1$ is the exponential integral $E_1\left[t\right]:={\int }_1^{\infty }\frac{e^{-tx}}{x}dx$.

2nd Example: In this example, we consider the kernels with $\mu \ge 0$. It is well-known (see, e.g. [15]) that for any $\alpha \in (0,1)$ the function $k_{\alpha }\left(t\right)$ is a Sonin kernel and ${\kappa }_{\alpha }\left(t\right)$ is its associated Sonin kernel. Furthermore, direct calculations show that these kernels are from the class ${}_{*}{S}_{\alpha }$ of the Sonin kernels with $a=b=1$.

The Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$ can be explicitly evaluated:

For this function, the condition C4) is also satisfied and thus the kernels $k_{\alpha }$ and ${\kappa }_{\alpha }$ belong to the class ${}_{*}{\widehat{S}}_{\alpha }$ of the Sonin kernels.

Then we proceed with the formula (34) that takes the form

Because ${(\rho +\mu )}^{\alpha }\in C[0,1]$ and the weight function w is integrabale and of one sign for $0<\alpha <1,$ applying the mean value theorem for the last integral yields the relation for some ${\alpha }_0(0<{\alpha }_0<1)$ and with $W={\int }_0^{1}w\left(\alpha \right)d\alpha .$

Then we get the formula

By definition, the kernel function ${\psi }_w\left(t\right)$ of the GFID is the inverse Laplace transform of the last expression. Thus we arrive at the representation

As we see, the kernel ${\psi }_w$ has the from of the kernel ${\kappa }_{\alpha }$ with a certain ${\alpha }_0\in (0,1)$ that depend on $\mu$ and the weight function w. Thus, it is well-defined and belongs to the class ${\mathcal{S}}_{-1}$ of the Sonin kernels as stated in Theorem 3.1.

3rd Example: In the last example, we consider the Sonin kernels (see, e.g. [15])

First we represent the kernel $k_{\alpha }$ in the form

It is easy to verify that $k_{\alpha }\left(t\right)\in {}_{*}{S}_{\alpha }$ provided that $0\le a<b<1.$ Similarly, ${\kappa }_{\alpha }\left(t\right)\in {}_{*}{S}_{\alpha },$ provided that $0<a<b\le 1.$ Thus, $k_{\alpha },{\kappa }_{\alpha }\in {}_{*}{S}_{\alpha }$, provided that $0<a<b<1$ (in the previous examples, we had the case $a=0$ and $b=1$). In the further derivations, we assume that this condition holds valid.

The Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$ can be explicitly evaluated:

For $0<\alpha ,\gamma <1$, this function satisfies the condition C4) and thus the kernels $k_{\alpha }$ and ${\kappa }_{\alpha }$ belong to the class ${}_{*}{\widehat{S}}_{\alpha }$ of the Sonin kernels.

The formula (34) takes now the form

Because ${\rho }^{\alpha }\in C[a,b]$ and w is integrabale and of one sign for $a<\alpha <b,$ applying the mean value theorem for the last integral yields the relation for some ${\alpha }_0(a<{\alpha }_0<b)$ and with $W={\int }_a^{b}w\left(\alpha \right)d\alpha .$

From the last formula, we get

Applying the inverse Laplace transform to the right-hand side of the last formula, we arrive at the following representation for the kernel ${\psi }_w$ of the corresponding GFID:

This function is well-defined for $t>0$, belongs to the space $C_{-1}(0,\infty )$ and is a Sonin kernel from the class ${\mathcal{S}}_{-1}$ as predicted by Theorem 3.1.