Anomalous Diffusion Models Involving Regularized General Fractional Derivatives with Sonin Kernels

Abstract

In this paper, we introduce a general fractional master equation involving regularized general fractional derivatives with Sonin kernels, and we discuss its physical characteristics and mathematical properties. First, we show that this master equation can be embedded into the framework of continuous time random walks, and we derive an explicit formula for the waiting time probability density function of the continuous time random walk model in form of a convolution series generated by the Sonin kernel associated with the kernel of the regularized general fractional derivative. Next, we derive a fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels from the continuous time random walk model in the asymptotical sense of long times and large distances. Another important result presented in this paper is a concise formula for the mean squared displacement of the particles governed by this fractional diffusion equation. Finally, we discuss several mathematical aspects of the fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels, including the non-negativity of its fundamental solution and the validity of an appropriately formulated maximum principle for its solutions on the bounded domains.

1. Introduction

Today, mathematical models involving fractional integrals, derivatives, and fractional differential equations are actively employed in various fields such as physics [1,2,3], financial economics [4,5], engineering [6,7,8], linear viscoelasticity [9,10], and bioengineering and medicine [11], to mention only a few of many relevant research areas. One of the most discussed and well investigated case studies of Fractional Calculus (FC) applications is the modeling of anomalous diffusion or anomalous transport processes that can be roughly characterized as those that do not follow the Gaussian statistics on long times, see, e.g., [12,13,14] and the references therein. In particular, the mean squared displacement (MSD) of diffusing particles is not a linear function of time, unlike in the case of conventional diffusion. For anomalous diffusion, this linear dependence is replaced by other relationships, such as power law functions of time, or the MSD is not finite.

The most popular mathematical descriptions of anomalous diffusion processes are the Continuous Time Random Walk (CTRW) models at the micro-level and fractional differential equations involving different types of fractional derivatives at the macro-level. The CTRW models were first introduced in [15] for the description of some transport processes which could be intepreted as a result of multiparticle motion. Assuming the particles are independent, their behavior can be described by the waiting time and jump length probability density functions (PDFs) of individual particles. A strong connection between the CTRW model and fractional differential equations was first established in [16], where a fractional master equation involving the Caputo fractional derivative was embedded into the framework of the general CTRW model. This equation was derived for a special waiting time PDF given in terms of the Mittag-Leffler function depending on a parameter (order of the fractional derivative in the fractional diffusion equation). It is worth mentioning that for this waiting time PDF, the fractional master equation was shown to be rigorously equivalent to the corresponding CTRW model and not just asymptotically as in the subsequent publications, see, e.g., [13,14]. In [17], a closed-form formula for solutions to the fractional master equation involving the Caputo fractional derivative was derived in terms of a special case of the Fox H-function, and its properties were investigated. In particular, the MSD of the diffusing particles governed by this fractional master equation was shown to be proportional to a power law function of time with the exponent being equal to the order of the fractional derivative.

Another significant result on the fractional diffusion equation with the Caputo fractional derivative with order between zero and one and the spatial Laplace operator was presented in [18], independently of the CTRW models and prior to publications [16,17]. In [18], the fundamental solution to this equation was derived in terms of the Fox H-function, and its properties were analyzed. In particular, it was shown that the fundamental solution to the fractional diffusion equation is non-negative and normalized and thus that it can be interpreted as a probability density function evolving in time.

The results presented in [16,17,18] demonstrated that the fractional diffusion equation involving the Caputo fractional derivative can serve as a model for a class of anomalous diffusion processes where the MSD of the diffusing particles is proportional to a power law. However, in many applications, deviations of the MSD from the power law with a fixed exponent are observed [19,20,21]. Thus, one needs other, more general fractional master equations that would lead to an extension of the class of functions that describe the MSD of the diffusing particles governed by these equations. Two important classes of such models suggested so far are the fractional differential equations with distributed order derivatives [22,23] and the fractional differential equations with variable order fractional derivatives [24,25]. However, in both cases, no direct connection to the CTRW model has been established until now.

For fractional differential equations with distributed order derivatives, the fundamental solution can be interpreted in some important cases as a PDF [26]. Moreover, the asymptotics of the MSD of diffusing particles governed by this equation has been derived in terms of power-logarithmic functions [22,23]. These properties of solutions to the fractional differential equations with distributed order derivatives suggested their usage as models of the so-called ultra-slow diffusion. Regarding fractional differential equations with variable order fractional derivatives, their mathematical properties and potential physical interpretation as models for anomalous diffusion are still under investigation.

Thus, fractional differential equations with distributed order derivatives and fractional differential equations with variable order derivatives do not provide a satisfactory solution to the problem mentioned above, i.e., a construction of a framework for anomalous diffusion models that would directly follow from the CTRW model, and would lead to some general classes of expressions for the MSD of the diffusing particles governed by these equations.

In this paper, we introduce and investigate a fractional master equation involving regularized general fractional derivatives (GFDs) with Sonin kernels. GFDs with various classes of Sonin kernels have been discussed in several publications, including [27,28,29]. These derivatives encompass most time-fractional derivatives as special cases. Due to the diversity of Sonin kernels [30,31], fractional differential equations with GFDs have became an important tool in applied mathematics. In many recent publications (PDFs, [32,33,34]), the GFDs with Sonin kernels and fractional differential equations with these derivatives were employed for modeling various physical processes and systems.

A key result presented in this paper is derivation of the fractional master equation involving the regularized GFDs with Sonin kernels from the general CTRW model. We show that the waiting time PDF of the CTRW model can be expressed in terms of a convolution series generated by the Sonin kernel associated with the kernel of the regularized GFD. Therefore, this master equation can be explicitly embedded within the CTRW model framework. Moreover, we show that the CTRW model is connected to the fractional diffusion equation involving regularized GFDs with Sonin kernels in the asymptotical sense of long times and large distances.

Another significant finding is a compact formula for the MSD of diffusing particles governed by the fractional diffusion equation involving regularized GFDs, expressed in terms of the Sonin kernels associated with the kernels of the regularized GFDs. Considering various specific cases of Sonin kernels and general formulas for constructing Sonin kernels [30,31], this equation can be adapted to available measurement data describing the MSD of diffusing particles within a certain diffusion process framework. Thus, our approach provides substantially more flexibility needed for the modeling of different kinds of anomalous diffusion processes compared to the fractional diffusion equations introduced so far.

In the last part of the paper, we present some relevant mathematical properties of solutions to the fractional diffusion equation involving regularized GFDs with Sonin kernels. Specifically, we discuss conditions on Sonin kernels that ensure normalization and non-negativity of its fundamental solution, as well as the validity of the maximum principle for solutions on bounded domains.

The rest of the paper is organized as follows. In Section 2, we discuss definitions and basic properties of the GFDs and the corresponding general fractional integrals (GFIs) with Sonin kernels. Section 3 is devoted to the derivation of the general fractional master equation involving regularized GFDs with Sonin kernels from the CTRW model. In doing so, the waiting time PDF of the CTRW model is expressed in terms of a convolution series generated by the Sonin kernel associated with the kernel of the GFD from the fractional master equation. Section 4 focuses on the fractional diffusion equation involving regularized GFDs with Sonin kernels, derived from the CTRW model in the asymptotical sense of long times and large distances. The main result of this section is an explicit formula for the MSD of the diffusing particles governed by this equation, expressed in terms of the Sonin kernel associated with the kernel of the regularized GFD. Finally, we discuss relevant mathematical properties of solutions to the Cauchy problem and to the initial-boundary-value problem formulated for the fractional diffusion equation with regularized GFDs with Sonin kernels.

2. Definitions and Basic Properties of the GFDs and GFIs

The general fractional master equation we are dealing with in this paper involves the GFDs with Sonin kernels. This derivative in different forms and with Sonin kernels from different classes was treated in a series of recent publications, including [27,28,29]. In this paper, we employ the general fractional integral (GFI), the GFD, and the regularized GFD that are defined as follows, respectively:

$$\left({\mathbb{I}}_{\left(\kappa \right)}f\right)\left(t\right):=(\kappa \ast f)\left(t\right)={\int }_0^t\kappa (t-\tau )f\left(\tau \right)d\tau ,t>0,$$

(1)

$$\left({\mathbb{D}}_{\left(k\right)}f\right)\left(t\right):={\displaystyle \frac{d}{dt}}(k\ast f)\left(t\right)={\displaystyle \frac{d}{dt}}\left({\mathbb{I}}_{\left(k\right)}f\right)\left(t\right),t>0,$$

(2)

$${(}_{\ast }{\mathbb{D}}_{\left(k\right)}f)\left(t\right):=\left({\mathbb{D}}_{\left(k\right)}f\right)\left(t\right)-f\left(0\right)k\left(t\right),t>0,$$

(3)

where the operation ∗ stands for the Laplace convolution and the kernels $\kappa$ and k satisfy the condition

$$(\kappa \ast k)\left(t\right)\equiv 1,t>0.$$

(4)

Condition (4) was first introduced by Sonin in [35] and is referred to as the Sonin condition. The functions that satisfy the Sonin condition are called Sonin kernels. For a given Sonin kernel $\kappa$, the kernel k is referred to as its associated Sonin kernel.

The power law functions

$$\kappa \left(t\right)=h_{\alpha }\left(t\right),k\left(t\right)=h_{1-\alpha }\left(t\right),0<\alpha <1\mathrm{with}{h}_{\alpha }\left(t\right):={\displaystyle \frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)}},\alpha >0$$

(5)

constitute the most known and probably most important pair of Sonin kernels. Originally, these kernels were introduced by Abel in [36,37] for the analytical treatment of the tautochrone problem.

The GFI (1), the GFD (2), and the regularized GFD (3) with the power law Sonin kernels (5) are reduced to the Riemann-Liouville fractional integral (6) and to the Riemann-Liouville and Caputo fractional derivatives (7) and (8) of the order $\alpha$ ($0<\alpha <1$), respectively:

$$\left(I_{0+}^{\alpha }f\right)\left(t\right):=(h_{\alpha }\ast f)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau ,t>0,$$

(6)

$$\left(D_{RL}^{\alpha }f\right)\left(t\right):={\displaystyle \frac{d}{dt}}\left(I_{0+}^{1-\alpha }f\right)\left(t\right)={\displaystyle \frac{d}{dt}}(h_{1-\alpha }\ast f)\left(t\right),t>0,$$

(7)

$$\left(D_C^{\alpha }f\right)\left(t\right):=\left(D_{RL}^{\alpha }f\right)\left(t\right)-f\left(0\right)h_{1-\alpha }\left(t\right).$$

(8)

The Formula (6) defines the Riemann-Liouville fractional integral for any $\alpha >0$ or more generally for any $\alpha \in \mathbb{C}$ with $\Re \left(\alpha \right)>0$ and the operator $I_{0+}^0$ is interpreted as the identity. As to the Caputo fractional derivative, in most FC publications it is defined by the expression

$$\left(D_C^{\alpha }f\right)\left(t\right):=(h_{1-\alpha }\ast f^{\prime })\left(t\right)=\left(I_{0+}^{1-\alpha }{f}^{\prime }\right)\left(t\right),$$

(9)

that follows from (8) for any absolutely continuous function f.

Another important pair of Sonin kernels that will be used in this paper is given by the formula [38,39,40]

$$\kappa \left(t\right)=h_{1-\alpha +\beta }\left(t\right)+h_{1-\alpha }\left(t\right),0<\beta <\alpha <1,$$

(10)

$$k\left(t\right)=t^{\alpha -1}{E}_{\beta ,\alpha }(-t^{\beta }),$$

(11)

where $E_{\beta ,\alpha }$ stands for the two-parameters Mittag-Leffler function that is defined by the following convergent series:

$$E_{\beta ,\alpha }\left(z\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\Gamma (\beta k+\alpha )}},\beta >0,\alpha ,z\in \mathbb{C}.$$

(12)

More examples of Sonin kernels can be found in [30,31,38].

It is worth mentioning that the properties of the GFIs and the regularized GFDs essentially depend on the classes of functions that their kernels belong to. In the rest of this section, we focus on the GFIs and the regularized GFDs with Sonin kernels $\kappa ,k$ that belong to the space $C_{-1}(0,+\infty )$ defined as follows:

$$C_{-1}(0,+\infty ):=\{f:f\left(t\right)=t^p{f}_1\left(t\right),t>0,p>-1,f_1\in C[0,+\infty )\}.$$

(13)

In [38,39] and subsequent publications, the class of Sonin kernels from the space $C_{-1}(0,+\infty )$ was referred to as ${\mathcal{L}}_1$. It is a very broad class that includes most of the known Sonin kernels. Still, it is worth mentioning that there exist some Sonin kernels with power-logarithmic asymptotics at the origin that do not belong to this class; see [41] for examples of such kernels.

In further discussions, we need some properties of the GFIs, the GFDs, and the regularized GFDs. As shown in [38,39], the GFIs and the GFDs with Sonin kernels that belong to the class ${\mathcal{L}}_1$ build a kind of calculus. In particular, the following analogies of the first and second fundamental theorems of calculus hold valid for the GFI, the GFD, and the regularized GFD:

(1)

The GFD (2) is a left inverse operator to the GFI (1) on the space $C_{-1}(0,+\infty )$ defined as in (13):

$$\left({\mathbb{D}}_{\left(k\right)}{\mathbb{I}}_{\left(\kappa \right)}f\right)\left(t\right)=f\left(t\right),f\in C_{-1}(0,+\infty ),t>0,$$

(14)

and the regularized GFD (3) is a left inverse operator to the GFI (1) on the space ${\mathbb{I}}_{\left(k\right)}\left(C_{-1}(0,+\infty )\right)$:

$${(}_{\ast }{\mathbb{D}}_{\left(k\right)}{\mathbb{I}}_{\left(\kappa \right)}f)\left(t\right)=f\left(t\right),f\in {\mathbb{I}}_{\left(k\right)}\left(C_{-1}(0,+\infty )\right),t>0,$$

(15)

where ${\mathbb{I}}_{\left(k\right)}\left(C_{-1}(0,+\infty )\right):=\{f:f\left(t\right)=\left({\mathbb{I}}_{\left(k\right)}\varphi \right)\left(t\right),\varphi \in C_{-1}(0,+\infty )\}$.

(2)

The relations

$$\left({\mathbb{I}}_{\left(\kappa \right)}{\mathbb{D}}_{\left(k\right)}f\right)\left(t\right)=f\left(t\right)-\left({\mathbb{I}}_{\left(k\right)}f\right)\left(0\right)\kappa \left(t\right),t>0,f\in C_{-1,\left(k\right)}^1(0,+\infty ),$$

(16)

and

$$\left({\mathbb{I}}_{\left(\kappa \right)}{}_{\ast }\mathbb{D}_{\left(k\right)}f\right)\left(t\right)=f\left(t\right)-f\left(0\right),t>0,f\in C_{-1}^1(0,+\infty ),$$

(17)

hold valid, where

$$C_{-1,\left(k\right)}^1(0,+\infty )=\{f\in C_{-1}(0,+\infty ):\left({\mathbb{D}}_{\left(k\right)}f\right)\in C_{-1}(0,+\infty )\},$$

(18)

and

$$C_{-1}^1(0,+\infty ):=\{f\in C_{-1}(0,+\infty ):f^{\prime }\in C_{-1}(0,+\infty )\}.$$

(19)

For the Riemann-Liouville fractional integral with the kernel $\kappa \left(t\right)=h_{\alpha }\left(t\right)$ and the Riemann-Liouville and Caputo fractional derivatives with the kernel $k\left(t\right)=h_{1-\alpha }\left(t\right)$, $0<\alpha <1$, the relations (16) and (17) take the following well-known form:

$$\left(I_{0+}^{\alpha }{D}_{0+}^{\alpha }f\right)\left(t\right)=f\left(t\right)-\left(I_{0+}^{1-\alpha }f\right)\left(0\right)h_{\alpha }\left(t\right),t>0,$$

(20)

and

$$\left(I_{0+}^{\alpha }{}_{\ast }D_{0+}^{\alpha }f\right)\left(t\right)=f\left(t\right)-f\left(0\right),t>0.$$

(21)

3. The General Fractional Master Equation Involving Regularized GFDs

In this section, we apply the method suggested in [16] to demonstrate that the general fractional master equation involving regularized GFDs with Sonin kernels can be embedded into the framework of the CTRW model. More precisely, we show that this general fractional master equation is equivalent to the CTRW model with a certain waiting time PDF that can be expressed in terms of a convolution series generated by the Sonin kernel associated with the kernel of the regularized GFD.

3.1. The CTRW Model

In this subsection, we briefly present the main equations of the CTRW model. This model is based on the idea that the displacement $\mathbf{x}\in {\mathbb{R}}^n$ of a random walker by each single jump and the waiting times between two successive jumps are governed by certain PDFs that we denote by $\lambda \left(\mathbf{x}\right)$ and $\psi \left(t\right)$, respectively. In what follows, we assume that there is no correlation between displacements and waiting times, that at time $t=0$ the random walker is located at position $\mathbf{x}$ = $\mathbf{0}$, and that the locations $\mathbf{x}$ of the random walker are either some discrete points from ${\mathbb{R}}^n$ or the whole ${\mathbb{R}}^n$. In the last case, the sums in the Formulas (22) and (26) have to be interpreted as the integrals over ${\mathbb{R}}^n$.

Denoting by $u(\mathbf{x},t)$ the unknown probability to find the random walker at the location $\mathbf{x}$ at time t, the law of total probability leads to the following integral equation [42]:

$$u(\mathbf{x},t)={\int }_0^t\psi (t-t^{\prime })\sum _{{\mathbf{x}}^{\prime }}\lambda (\mathbf{x}-{\mathbf{x}}^{\prime })u({\mathbf{x}}^{\prime },t^{\prime })d{t}^{\prime }+\delta \left(\mathbf{x}\right)\Psi \left(t\right),$$

(22)

where $\Psi =\Psi \left(t\right)$ stands for the survival probability at the initial position that is defined by the formula

$$\Psi \left(t\right)=1-{\int }_0^t\psi \left(t^{\prime }\right)d{t}^{\prime }.$$

(23)

Applying the Fourier and the Laplace transforms to the relation (22), we obtain the equation

$$\widehat{\tilde{u}}(\zeta ,s)=\tilde{\psi }\left(s\right)\widehat{\lambda }\left(\zeta \right)\widehat{\tilde{u}}(\zeta ,s)+\tilde{\Psi }\left(s\right),$$

(24)

that can be solved for the Fourier-Laplace transform $\widehat{\tilde{u}}(\zeta ,s)$ of solution $u(\mathbf{x},t)$ to the integral Equation (22), and we arrive at the formula

$$\widehat{\tilde{u}}(\zeta ,s)={\displaystyle \frac{\tilde{\Psi }\left(s\right)}{1-\tilde{\psi }\left(s\right)\widehat{\lambda }\left(\zeta \right)}}.$$

(25)

We remind the readers that the Fourier and Laplace transforms are defined by the following formulas, respectively:

$$\widehat{f}\left(\zeta \right)=\mathcal{F}\{f\left(\mathbf{x}\right);\zeta \}=\sum _{\mathbf{x}}{e}^{i\zeta \mathbf{x}}f\left(\mathbf{x}\right),\zeta \in {\mathbb{R}}^n,$$

(26)

$$\tilde{f}\left(s\right)=\mathcal{L}\{f\left(t\right);s\}={\int }_0^{\infty }{e}^{-st}f\left(t\right)dt,\Re \left(s\right)>C_f.$$

(27)

The Formula (23) in the Laplace domain takes the form

$$\tilde{\Psi }\left(s\right)={\displaystyle \frac{1}{s}}-{\displaystyle \frac{1}{s}}\tilde{\psi }\left(s\right).$$

(28)

Substituting this relation into the right-hand side of the expression (25), we obtain the following solution formula to the integral Equation (22) in the Fourier-Laplace domain:

$$\widehat{\tilde{u}}(\zeta ,s)={\displaystyle \frac{1-\tilde{\psi }\left(s\right)}{s}}{\displaystyle \frac{1}{1-\tilde{\psi }\left(s\right)\widehat{\lambda }\left(\zeta \right)}}.$$

(29)

3.2. The General Fractional Master Equation

In this subsection, we introduce a general fractional master equation involving the regularized fractional GFD with the Sonin kernel k in the form

$$\left({}_{\ast }\mathbb{D}_{\left(k\right)}u(\mathbf{x},·)\right)\left(t\right)=\sum _{{\mathbf{x}}^{\prime }}\omega (\mathbf{x}-{\mathbf{x}}^{\prime })u({\mathbf{x}}^{\prime },t),$$

(30)

equipped with the initial condition

$$u(\mathbf{x},0)=\delta \left(\mathbf{x}\right),$$

(31)

where the fractional transition rates $\omega \left(\mathbf{x}\right)$ satisfy the relation ${\sum }_{{\mathbf{x}}^{\prime }}\omega \left({\mathbf{x}}^{\prime }\right)=0$ and the domain of the variable ${\mathbf{x}}^{\prime }$ is either a discrete subset of ${\mathbb{R}}^n$ or the whole ${\mathbb{R}}^n$. In the last case, the sums in the Equations (30) and (32) have to be interpreted as integrals over ${\mathbb{R}}^n$.

Due to the Formula (17), Equations (30) and (31) can be transformed to the integral equation

$$u(\mathbf{x},t)=\left({\mathbb{I}}_{\left(\kappa \right)}\sum _{{\mathbf{x}}^{\prime }}\omega (\mathbf{x}-{\mathbf{x}}^{\prime })u({\mathbf{x}}^{\prime },·)\right)\left(t\right)+\delta \left(\mathbf{x}\right),$$

(32)

where ${\mathbb{I}}_{\left(\kappa \right)}$ is the GFI with the Sonin kernel $\kappa$ associated with the Sonin kernel k of the regularized GFD in Equation (30).

Applying the Fourier and the Laplace transforms to the Formula (32) leads to the equation

$$\widehat{\tilde{u}}(\zeta ,s)=\tilde{\kappa }\left(s\right)\widehat{\omega }\left(\zeta \right)\widehat{\tilde{u}}(\zeta ,s)+{\displaystyle \frac{1}{s}},$$

(33)

that can be solved for the Fourier-Laplace transform $\widehat{\tilde{u}}(\zeta ,s)$ of the unknown function $u(\mathbf{x},t)$ as follows:

$$\widehat{\tilde{u}}(\zeta ,s)={\displaystyle \frac{1}{s}}{\displaystyle \frac{1}{1-\tilde{\kappa }\left(s\right)\widehat{\omega }\left(\zeta \right)}}.$$

(34)

3.3. The General Fractional Master Equation as a CTRW Model

For embedding of the general fractional master Equation (30) along with the initial condition (31) into the framework of the CTRW model provided by the Equations (22) and (23), we suppose that their solutions and thus their Fourier-Laplace transforms given by the Formulas (29) and (34), respectively, are identically equal, i.e., the following relation holds valid:

$${\displaystyle \frac{1-\tilde{\psi }\left(s\right)}{s}}{\displaystyle \frac{1}{1-\tilde{\psi }\left(s\right)\widehat{\lambda }\left(\zeta \right)}}={\displaystyle \frac{1}{s}}{\displaystyle \frac{1}{1-\tilde{\kappa }\left(s\right)\widehat{\omega }\left(\zeta \right)}}.$$

(35)

By elementary transformations, the variables $\zeta$ and s in the relation (35) can be separated and we arrive at the formula

$${\displaystyle \frac{\widehat{\lambda }\left(\zeta \right)-1}{\widehat{\omega }\left(\zeta \right)}}={\displaystyle \frac{\tilde{\kappa }\left(s\right)(1-\tilde{\psi }\left(s\right))}{\tilde{\psi }\left(s\right)}}.$$

(36)

Because the variables $\zeta$ and s are independent from each other, the functions on the left- and the right-hand sides of the Equation (36) have to be equal to the same constant and thus we get two important relations:

$${\displaystyle \frac{\widehat{\lambda }\left(\zeta \right)-1}{\widehat{\omega }\left(\zeta \right)}}=C,C\in \mathbb{R},C\ne 0,$$

(37)

$${\displaystyle \frac{\tilde{\kappa }\left(s\right)(1-\tilde{\psi }\left(s\right))}{\tilde{\psi }\left(s\right)}}=C,C\in \mathbb{R},C\ne 0.$$

(38)

In what follows, without loss of generality, we set $C=1$ in the Formulas (37) and (38).

Solving the Equation (38) with $C=1$ for $\tilde{\psi }$, we arrive at the following relation between the Laplace transforms of the waiting time PDF $\psi$ from the CTRW model and the Sonin kernel $\kappa$ associated with the kernel k of the regularized GFD from the general fractional master Equation (30):

$$\tilde{\psi }\left(s\right)={\displaystyle \frac{\tilde{\kappa }\left(s\right)}{1+\tilde{\kappa }\left(s\right)}}.$$

(39)

Now we assume that $\tilde{\kappa }\left(s\right)\to 0$ as $s\to +\infty$ (see the condition K3 for Sonin kernels from the class $\mathcal{K}$ presented in Section 4.3) and obtain the series representation of the Laplace transform of the waiting time PDF $\psi$ in the form

$$\tilde{\psi }\left(s\right)=\tilde{\kappa }\left(s\right)\sum _{n=0}^{\infty }{(-\tilde{\kappa }\left(s\right))}^n=\sum _{n=0}^{\infty }{(-1)}^n{\left(\tilde{\kappa }\left(s\right)\right)}^{n+1},$$

(40)

that holds valid for $\Re \left(s\right)>C\left(\kappa \right)$, where $C\left(\kappa \right)$ is a constant which depends on the kernel $\kappa$.

Applying the inverse Laplace transform to the Equation (40), we arrive at the formula

$$\psi \left(t\right)=\sum _{n=0}^{\infty }{(-1)}^n{\kappa }^{<n+1>}\left(t\right),$$

(41)

where the notation $f^{<n>}$ stands for the convolution power defined as follows:

$$f^{<n>}\left(t\right):=\left\{\begin{array}{cc}f\left(t\right),\hfill & n=1,\hfill \\ \underset{n\mathrm{times}}{\underbrace{(f\ast \dots \ast f)}}\left(t\right),\hfill & n=2,3,\dots .\hfill \end{array}\right.$$

(42)

The series as on the right-hand side of the Formula (41) was introduced for the first time in [39] (see also [43] for more details) and was there called the convolution series. As has been shown in [39,43], for any kernel $\kappa \in C_{-1}(0,+\infty )$, this convolution series is convergent for all $t>0$ and defines a function that belongs to the space $C_{-1}(0,+\infty )$.

In the FC literature, the convolution series of the type as in the right-hand side of the Formula (41) is denoted by $l_{\kappa ,\mu }$ and defined as follows:

$$l_{\kappa ,\mu }\left(t\right):=\sum _{n=0}^{+\infty }{\mu }^n{\kappa }^{<n+1>}\left(t\right).$$

(43)

Using this notation, we can represent the Formula (41) in the form

$$\psi \left(t\right)=l_{\kappa ,-1}\left(t\right),$$

(44)

where the convolution series $l_{\kappa ,-1}\left(t\right)$ is defined as in (43).

In the case of the kernel $\kappa \left(t\right)=h_{\alpha }\left(t\right),0<\alpha <1$ of the Riemann-Liouville fractional integral, the formula ${\kappa }^{<n+1>}\left(t\right)=h_{\alpha }^{<n+1>}\left(t\right)=h_{(n+1)\alpha }\left(t\right)$ is valid (see, e.g., [39,43]) and the convolution series (43) takes the form

$$l_{\kappa ,\mu }\left(t\right)=\sum _{j=0}^{+\infty }{\mu }^j{h}_{(j+1)\alpha }\left(t\right)=t^{\alpha -1}\sum _{j=0}^{+\infty }{\displaystyle \frac{{\mu }^j{t}^{j\alpha }}{\Gamma (j\alpha +\alpha )}}=t^{\alpha -1}{E}_{\alpha ,\alpha }\left(\mu t^{\alpha }\right),$$

(45)

where the two-parameters Mittag-Leffler function $E_{\alpha ,\alpha }$ is defined as in the Equation (12).

For the Sonin kernel $\kappa \left(t\right)=h_{1-\alpha +\beta }\left(t\right)+h_{1-\alpha }\left(t\right),0<\beta <\alpha <1$ (see Formula (10)), the convolution series (43) takes the following form [39]:

$$l_{\kappa ,\mu }\left(t\right)={\displaystyle \frac{1}{\mu t}}\sum _{j=0}^{+\infty }\sum _{l_1+l_2=j}{\displaystyle \frac{j!}{l_1!l_2!}}{\displaystyle \frac{{\left(\mu t^{1-\alpha +\beta }\right)}^{l_1}{\left(\mu t^{1-\alpha }\right)}^{l_2}}{\Gamma (l_1(1-\alpha +\beta )+l_2(1-\alpha ))}}=$$

$${\displaystyle \frac{1}{\mu t}}{E}_{(1-\alpha ,1-\alpha +\beta ),0}(\mu t^{1-\alpha },\mu t^{1-\alpha +\beta }),$$

where $E_{(1-\alpha ,1-\alpha +\beta ),0}$ is a particular case of the multinomial Mittag-Leffler function defined by the expression [44]:

$$E_{({\alpha }_1,\dots ,{\alpha }_m),\gamma }(z_1,\dots ,z_m):=\sum _{j=0}^{+\infty }\sum _{l_1+\cdots +l_m=j}{\displaystyle \frac{j!}{l_1!\times ·\times l_m!}}{\displaystyle \frac{{\prod }_{i=1}^m{z}_i^{l_i}}{\Gamma (\gamma +{\sum }_{i=1}^m{\alpha }_i{l}_i)}}.$$

(46)

It is also worth mentioning that the Sonin condition (4) in the Laplace domain takes the form

$$\tilde{\kappa }\left(s\right)·\tilde{k}\left(s\right)={\displaystyle \frac{1}{s}},\Re \left(s\right)>0,$$

(47)

that leads to another representation of the relation (39):

$$\tilde{\psi }\left(s\right)={\displaystyle \frac{1}{1+s\tilde{k}\left(s\right)}},$$

(48)

where k is the Sonin kernel of the GFD from the fractional master Equation (30).

Another important feature of the convolution series $l_{\kappa ,\mu }\left(t\right)$ defined by (43) is that this function is the main component of the solution formulas for linear fractional differential equations with GFDs and constant coefficients [39,45]. In particular, it is the unique solution to the following initial-value problem for the fractional differential equation involving the GFD with the kernel k [45]:

$$\left\{\begin{array}{c}\left({\mathbb{D}}_{\left(k\right)}y\right)\left(t\right)=\mu y\left(t\right),t>0,\hfill \\ \left({\mathbb{I}}_{\left(k\right)}y\right)\left(0\right)=1.\hfill \end{array}\right.$$

(49)

Taking into account the representation (44) of the waiting time PDF $\psi$ in terms of the convolution series $l_{\kappa ,-1}\left(x\right)$, the function $\psi$ can be also interpreted as the eigenfunction of the GFD ${\mathbb{D}}_{\left(k\right)}$ to the eigenvalue $\mu =-1$ that satisfies the non-local initial condition in terms of the GFI with the Sonin kernel k:

$$\left\{\begin{array}{c}\left({\mathbb{D}}_{\left(k\right)}\psi \right)\left(t\right)=-\psi \left(t\right),t>0,\hfill \\ \left({\mathbb{I}}_{\left(k\right)}\psi \right)\left(0\right)=1.\hfill \end{array}\right.$$

(50)

As an example, let us discuss the case of the power law Sonin kernels $\kappa \left(t\right)=h_{\alpha }\left(t\right)$ and $k\left(t\right)=h_{1-\alpha }\left(t\right)$, $0<\alpha <1$ that was analyzed for the first time in [16]. As mentioned in Section 2, the GFI with the kernel $\kappa \left(t\right)=h_{\alpha }\left(t\right)$ is the Riemann-Liouville fractional integral and the regularized GFD with the kernel $k\left(t\right)=h_{1-\alpha }\left(t\right)$ is the Caputo fractional derivative. Thus, the fractional master Equation (30) takes the form

$$\left({}_{\ast }D_{0+}^{\alpha }u(\mathbf{x},·)\right)\left(t\right)=\sum _{{\mathbf{x}}^{\prime }}\omega (\mathbf{x}-{\mathbf{x}}^{\prime })u({\mathbf{x}}^{\prime },t),0<\alpha <1.$$

(51)

Equipped with the initial condition

$$u(\mathbf{x},0)=\delta \left(\mathbf{x}\right),$$

(52)

this equation is equivalent to the integral equation with the Riemann-Liouville fractional integral in the form

$$u(\mathbf{x},t)=\left({\mathbb{I}}_{0+}^{\alpha }\sum _{{\mathbf{x}}^{\prime }}\omega (\mathbf{x}-{\mathbf{x}}^{\prime })u({\mathbf{x}}^{\prime },·)\right)\left(t\right)+\delta \left(\mathbf{x}\right),0<\alpha <1.$$

(53)

Due to the well-known formula

$$\mathcal{L}\{h_{\alpha }(·);s\}={\tilde{h}}_{\alpha }\left(s\right)=s^{-\alpha },\alpha >0,\Re \left(s\right)>0,$$

(54)

our derivations demonstrate that the fractional master equation in its integral form (53) is a particular case of the CTRW model with the Laplace transform of the waiting time PDF given by the formula (see the Equation (48))

$$\tilde{\psi }\left(s\right)={\displaystyle \frac{1}{1+s{\tilde{h}}_{1-\alpha }\left(s\right)}}={\displaystyle \frac{1}{1+s^{\alpha }}},0<\alpha <1,\Re \left(s\right)>0.$$

(55)

The inverse Laplace transform of the right-hand side of the Formula (55) is well-known (see, e.g., [16]) and we arrive at the following representation of the waiting time PDF $\psi$ of the CTRW model in terms of the two-parameters Mittag-Leffler function defined as in (12):

$$\psi \left(t\right)=t^{\alpha -1}{E}_{\alpha ,\alpha }\left(-t^{\alpha }\right),0<\alpha <1.$$

(56)

Of course, the relation (56) immediately follows also from the general Formula (44) and the Formula (45) for the convolution series $l_{\kappa ,\mu }\left(t\right)$ generated by the Sonin kernel $\kappa \left(t\right)=h_{\alpha }\left(t\right),0<\alpha <1$.

It is also worth mentioning that the function on the right-hand side of Formula (56) can be interpreted as a PDF for all values of $\alpha \in (0,1]$ (see, e.g., [46]) and that this function is the eigenfunction of the Riemann-Liouville fractional derivative to the eigenvalue $\mu =-1$ that satisfies the non-local initial condition $\left(I_{0+}^{\alpha }\psi \right)\left(0\right)=1$ (see the Formula (50)).

Finally, we remark that the Formulas (37) and (38) provide a two-sided relation between the displacement PDF $\lambda$ and waiting time PDF $\psi$ of the CTRW model (22) and the fractional transition rates $\omega$ and the Sonin kernel k of the regularized GFD from the general fractional master Equation (30). Using this connection, the general fractional master equation can be interpreted as a CTRW model and vice versa, the CTRW model can be represented in the form of a general fractional master equation.

4. The Fractional Diffusion Equation with Regularized GFDs

In this section, we derive the fractional diffusion equation involving regularized GFDs with Sonin kernels from the CTRW model in the asymptotical sense of long times and large distances. Then we establish a formula for the MSD of a random walker (diffusing particle) governed by this equation in terms of the Sonin kernel associated with the kernel of the regularized GFD and demonstrate this formula on several known and new examples. We also discuss some relevant mathematical properties of solutions to the Cauchy and initial-boundary value problems for the fractional diffusion equation involving regularized GFDs with Sonin kernels.

4.1. Derivation of the Fractional Diffusion Equation Involving Regularized GFDs

The Formula (37) with $C=1$ provides a relation between the Fourier transforms of the displacement PDF $\lambda$ of the CTRW model and the fractional transition rates $\omega$ of the general fractional master Equation (30) in the form

$$\widehat{\lambda }\left(\zeta \right)=1+\widehat{\omega }\left(\zeta \right).$$

(57)

Let us suppose that the displacement PDF $\lambda$ possesses finite variance ${\sigma }^2$. Without loss of generality, we set ${\sigma }^2=2$ and obtain the asymptotical relation [17]

$$\widehat{\lambda }\left(\zeta \right)\sim 1-{\zeta }^2,\zeta \to 0.$$

(58)

The last formula combined with (57) yields the representation

$$\widehat{\omega }\left(\zeta \right)=\widehat{\lambda }\left(\zeta \right)-1\sim -{\zeta }^2,\zeta \to 0.$$

(59)

On the other hand, the Formula (48) leads to the asymptotical relation

$$\tilde{\psi }\left(s\right)\sim 1-s\tilde{k}\left(s\right),s\to 0,$$

(60)

where k is the Sonin kernel of the regularized GFD from the fractional master Equation (30) that satisfies $s\tilde{k}\left(s\right)\to 0$ as $s\to 0$ (see the condition K3 for the Sonin kernels from the class $\mathcal{K}$ presented in Section 4.3).

Substituting the asymptotical relations (58) and (60) into the right-hand side of the Formula (29) for solution of the CTRW integral Equation (22) in the Fourier-Laplace domain, we obtain the equation

$$\widehat{\tilde{u}}(\zeta ,s)\sim {\displaystyle \frac{s\tilde{k}\left(s\right)}{s}}{\displaystyle \frac{1}{1-(1-{\zeta }^2)(1-s\tilde{k}\left(s\right))}}\sim {\displaystyle \frac{\tilde{k}\left(s\right)}{s\tilde{k}\left(s\right)+{\zeta }^2}},\zeta \to 0,s\to 0.$$

(61)

The Laplace transform formula [27]

$$\mathcal{L}\{\left({}_{\ast }\mathbb{D}_{\left(k\right)}f\right)\left(t\right);s\}=s\tilde{k}\left(s\right)\tilde{f}\left(s\right)-\tilde{k}\left(s\right)f\left(0\right)$$

(62)

for the regularized GFD (3) implicates the representation

$$\widehat{\tilde{u}}(\zeta ,s)={\displaystyle \frac{\tilde{k}\left(s\right)}{s\tilde{k}\left(s\right)+{\zeta }^2}}$$

(63)

for the Fourier-Laplace transform of solution to the Cauchy problem for the fractional diffusion equation with the regularized GFD with the Sonin kernel k in the form

$$\left({}_{\ast }\mathbb{D}_{\left(k\right)}u(\mathbf{x},·)\right)\left(t\right)=\Delta u(\mathbf{x},t),t>0,\mathbf{x}\in {\mathbb{R}}^n,$$

(64)

equipped with the initial condition

$$u(\mathbf{x},0)=\delta \left(\mathbf{x}\right).$$

(65)

Thus, the asymptotical relation (61) means that the Cauchy problem for the fractional diffusion Equation (64) involving regularized GFD can be obtained from the CTRW model (22) with the waiting time PDF in the form (48) and with displacement PDFs with finite variances in the asymptotical sense of long times and large distances.

On the other hand, in the case of displacement PDFs with finite variances, one can derive the fractional diffusion Equation (64) directly from the general fractional master Equation (30) in the asymptotical sense of large distances. Indeed, substituting the asymptotic relation (59) into the Formula (34) for the solution of the general fractional master Equation (30) in the Fourier-Laplace domain and using the Sonin condition in the Laplace domain in the form (47), we obtain the formula

$$\widehat{\tilde{u}}(\zeta ,s)\sim {\displaystyle \frac{1}{s}}{\displaystyle \frac{1}{1+\tilde{\kappa }\left(s\right){\zeta }^2}}={\displaystyle \frac{\tilde{k}\left(s\right)}{s\tilde{k}\left(s\right)+{\zeta }^2}},\zeta \to 0,$$

(66)

and the result follows from comparison of the Formulas (63) and (66).

It is also worth mentioning that the asymptotical relation (66) is valid for any s and not just for $s\to 0$ as the relation (61).

Summarizing the findings of this subsection, the fractional diffusion Equation (64) is closely related to both the CTRW model (22) and to the general fractional master Equation (30), and thus it can serve as a new and very general framework for modeling anomalous diffusion processes. In the rest of this section, we present some additional arguments that confirm this thesis.

4.2. MSD of a Random Walker Governed by the Fractional Diffusion Equation with Regularized GFDs

For a general Sonin kernel $\kappa$, a closed form formula for the inverse Laplace and inverse Fourier transforms of the right-hand side of Formula (63) is not known and probably does not exist at all. However, this formula can be used to calculate the MSD of a random walker governed by the fractional diffusion Equation (64) subject to the initial condition (65).

Indeed, the Laplace transform $\widetilde{〈{\mathbf{x}}^2〉}\left(s\right)$ of the MSD $〈{\mathbf{x}}^2〉\left(t\right)$ satisfies the relation (see, e.g., [17])

$$\widetilde{〈{\mathbf{x}}^2〉}\left(s\right)={\int }_{{\mathbb{R}}^n}{\mathbf{x}}^2\tilde{u}(\mathbf{x},s)d\mathbf{x}=-\left({\nabla }_{\zeta }^2\widehat{\tilde{u}}(\zeta ,s)\right){|}_{\zeta =\mathbf{0}}.$$

(67)

For the function $\widehat{\tilde{u}}(\zeta ,s)$ given by the right-hand side of the Equation (63), routine calculations lead to the formula

$${\nabla }_{\zeta }^2\widehat{\tilde{u}}(\zeta ,s)=-{\displaystyle \frac{2}{s}}\tilde{\kappa }\left(s\right)\left(-4\tilde{\kappa }\left(s\right){(1+{\zeta }^2\tilde{\kappa }\left(s\right))}^{-3}{\zeta }^2+{(1+{\zeta }^2\tilde{\kappa }\left(s\right))}^{-2}\right).$$

(68)

Combining the Formulas (67) and (68), we get the representation

$$\widetilde{〈{\mathbf{x}}^2〉}\left(s\right)={\displaystyle \frac{2}{s}}\tilde{\kappa }\left(s\right),$$

(69)

where $\kappa$ is the Sonin kernel associated with the kernel k of the regularized GFD from the fractional diffusion Equation (64).

The inverse Laplace transform of the right-hand side of the Formula (69) leads to a simple and elegant formula for the MSD of the random walker governed by the fractional diffusion Equation (64) in the form

$$〈{\mathbf{x}}^2〉\left(t\right)=2{\int }_0^t\kappa \left(\tau \right)d\tau =2(1\ast \kappa )\left(t\right)=2(h_1\ast \kappa )\left(t\right).$$

(70)

Now let us consider some known and new examples of the fractional diffusion Equation (64) with different Sonin kernels and calculate the corresponding MSDs of the random walkers governed by this equation.

We start with the case of the power law Sonin kernels $k\left(t\right)=h_{1-\alpha }\left(t\right)$, $\kappa \left(t\right)=h_{\alpha }\left(t\right)$$(0<\alpha <1)$ that has been considered in [16,17]. As already mentioned, the fractional diffusion Equation (64) involving the regularized GFD with the Sonin kernel k takes the form of the fractional diffusion Equation (51) with the Caputo fractional derivative of order $\alpha ,0<\alpha <1$. Applying the relation (70), we obtain the known formula (see, e.g., [17]) for the MSD of the random walker governed by the fractional diffusion Equation (51):

$$〈{\mathbf{x}}^2〉\left(t\right)=2(h_1\ast h_{\alpha })\left(t\right)=2h_{1+\alpha }\left(t\right)={\displaystyle \frac{2t^{\alpha }}{\Gamma (1+\alpha )}}.$$

(71)

The next example is the pair of Sonin kernels $k\left(t\right)=h_{1-\alpha +\beta }\left(t\right)+h_{1-\alpha }\left(t\right)$, $\kappa \left(t\right)=t^{\alpha -1}{E}_{\beta ,\alpha }(-t^{\beta })$$(0<\beta <\alpha <1)$ (see the Formulas (10) and (11)). In this case, the fractional diffusion Equation (64) with the regularized GFD with the Sonin kernel k contains two Caputo fractional derivatives:

$$\left({}_{\ast }D_{0+}^{\alpha }u(\mathbf{x},·)\right)\left(t\right)+\left({}_{\ast }D_{0+}^{\alpha -\beta }u(\mathbf{x},·)\right)\left(t\right)=\Delta u(\mathbf{x},t),t>0,\mathbf{x}\in {\mathbb{R}}^n,0<\beta <\alpha <1.$$

(72)

According to the Formula (70), the MSD of the random walker governed by the Equation (72) takes the form

$$〈{\mathbf{x}}^2〉\left(t\right)=2(h_1\left(\tau \right)\ast {\tau }^{\alpha -1}{E}_{\beta ,\alpha }(-{\tau }^{\beta }))\left(t\right)=2t^{\alpha }{E}_{\beta ,\alpha +1}(-t^{\beta }).$$

(73)

The known asymptotic behavior of the two-parameters Mittag-Leffler function (see, e.g., [46]) leads to the formula

$$〈{\mathbf{x}}^2〉\left(t\right)\sim \left\{\begin{array}{cc}{\displaystyle \frac{2t^{\alpha }}{\Gamma (1+\alpha )}},\hfill & t\to 0+,\hfill \\ {\displaystyle \frac{2t^{\alpha -\beta }}{\Gamma (1+\alpha -\beta )}},\hfill & t\to +\infty .\hfill \end{array}\right.$$

(74)

According to the Formula (74), the behavior of the MSD changes in the course of time between power laws with two different exponents that allows its tuning to anomalous diffusion processes with characteristics that vary in time.

Finally, we discuss the case of Sonin kernels $k\left(t\right)=t^{-\alpha }{E}_{\beta ,1-\alpha }(-t^{\beta })$, $\kappa \left(t\right)=h_{\alpha }\left(t\right)+h_{\alpha +\beta }\left(t\right)$, $(\alpha ,\beta >0,\alpha +\beta <1)$ that are kernels given by the Formulas (10) and (11) in reverse sequence.

For the kernel k, the fractional diffusion Equation (64) takes the form

$$\left({}_{\ast }D_{ML}^{\alpha ,\beta }u(\mathbf{x},·)\right)\left(t\right)=\Delta u(\mathbf{x},t),t>0,\mathbf{x}\in {\mathbb{R}}^n,\alpha ,\beta >0,\alpha +\beta <1,$$

(75)

where ${}_{\ast }D_{ML}^{\alpha ,\beta }$ stands for the regularized GFD with the Sonin kernel in terms of the two-parameters Mittag-Leffler function defined as follows:

$$\left({}_{\ast }D_{ML}^{\alpha ,\beta }f\right)\left(t\right):=({\tau }^{-\alpha }{E}_{\beta ,1-\alpha }(-{\tau }^{\beta })\ast f^{\prime }\left(\tau \right))\left(t\right),\alpha ,\beta >0,\alpha +\beta <1.$$

(76)

The MSD of the random walker governed by Equation (75) is given by the formula

$$〈{\mathbf{x}}^2〉\left(t\right)=2(h_1\ast \kappa )\left(t\right)=2(h_1\ast (h_{\alpha }+h_{\alpha +\beta }))\left(t\right)=$$

$$2(h_{1+\alpha }\left(t\right)+h_{1+\alpha +\beta }\left(t\right))={\displaystyle \frac{2t^{\alpha }}{\Gamma (1+\alpha )}}+{\displaystyle \frac{2t^{\alpha +\beta }}{\Gamma (1+\alpha +\beta )}}.$$

(77)

The asymptotic behavior of the right-hand side of the Formula (77) is similar to one presented in the Formula (74):

$$〈{\mathbf{x}}^2〉\left(t\right)\sim \left\{\begin{array}{cc}{\displaystyle \frac{2t^{\alpha }}{\Gamma (1+\alpha )}},\hfill & t\to 0+,\hfill \\ {\displaystyle \frac{2t^{\alpha +\beta }}{\Gamma (1+\alpha +\beta )}},\hfill & t\to +\infty .\hfill \end{array}\right.$$

(78)

However, the essential difference between the Formulas (74) and (78) is the sign of the parameter $\beta$. As a result, the fractional diffusion Equations (72) and (75) can be employed for modelling of the anomalous diffusion processes with the time-dependent MSD growth rate (the exponent in its power law asymptotics) that becomes both smaller (Equation (72)) and bigger (Equation (75)) in the course of time.

It is also worth mentioning that in the framework of our model, the behavior of the MSD depends on the Sonin kernel $\kappa$ associated with the kernel k of the GFD from the fractional diffusion Equation (64), see the Formula (70). In its turn, the kernel $\kappa$ determines the behavior of the waiting time PDF $\psi$ from the corresponding CTRW model (see the Formula (39)). Thus, the crossover behavior of MSD from short to long times is provoked by variation in the waiting time PDF $\psi$ in the course of time that can be induced by different physical reasons depending on the kind of the anomalous diffusion processes.

4.3. Mathematical Properties of Solutions to the Fractional Diffusion Equation with Regularized GFDs

In this subsection, we discuss some mathematical properties of solutions to the fractional diffusion equation involving regularized GFDs with Sonin kernels that are relevant to its interpretation as a model of the anomalous diffusion processes.

First, following [27], we present an important result regarding the fundamental solution to the fractional diffusion equation involving the regularized GFD with the Sonin kernel k that satisfies the following conditions:

(K1)

The Laplace transform $\tilde{k}\left(s\right)$ of k exists for all real $s>0$.

(K2)

The Laplace transform $\tilde{k}\left(s\right)$ is a Stieltjes function (see [47] for definition and properties of the Stieltjes functions).

(K3)

The Laplace transform $\tilde{k}\left(s\right)$ meets the asymptotical relations $\tilde{k}\left(s\right)\to 0$ and $s\tilde{k}\left(s\right)\to +\infty$ as $s\to +\infty$ and $\tilde{k}\left(s\right)\to +\infty$ and $s\tilde{k}\left(s\right)\to 0$ as $s\to 0$.

The class of Sonin kernels that satisfy the conditions (K1)–(K3) was introduced by Kochubei in [27], see also [48]. We denote this class by $\mathcal{K}$ and refer to it as to the Kochubei class of Sonin kernels.

In [27], Kochubei studied the Cauchy problems for ordinary and partial fractional differential equations involving the regularized GFD (3). In particular, he proved that the fundamental solution to the fractional diffusion equation involving the regularized GFD with the Sonin kernel k from the class $\mathcal{K}$ in the form

$$\left({}_{\ast }\mathbb{D}_{\left(k\right)}u(\mathbf{x},·)\right)\left(t\right)=\Delta u(\mathbf{x},t),t>0,x\in {\mathbb{R}}^n,$$

(79)

equipped with the initial condition

$$u(\mathbf{x},0)=u_0\left(\mathbf{x}\right),\mathbf{x}\in {\mathbb{R}}^n,$$

(80)

is locally integrable in t, infinitely differentiable for $\mathbf{x}\ne 0$, and can be interpreted as a spatial PDF evolving in time.

We note that the fundamental solution to the Cauchy problem (79) and (80) is exactly the solution to the fractional diffusion Equation (64) subject to the initial condition (65). Thus, the solution $u(\mathbf{x},t)$ to the fractional diffusion Equation (64) involving the regularized GFD with the Sonin kernel $k\in \mathcal{K}$ is a spatial PDF evolving in time. In its turn, this means that this equation can be interpreted as a model for anomalous diffusion processes. In the framework of this model, Formula (48) provides an explicit relation between the waiting time PDF $\psi$ of the general CTRW model and the Sonin kernel k of the regularized GFD. The MSD of the diffusing particles governed by this equation is given by Formula (70).

Another important aspect of the fractional diffusion equation involving regularized GFDs with Sonin kernels related to its interpretation as a mathematical model for anomalous diffusion processes is the maximum principle for its solutions on the bounded spatial domains. For the first time, this maximum principle was proved in [49].

The results presented in [49] were formulated for the initial-boundary value problems for the fractional diffusion equation involving regularized GFDs with Sonin kernels and a general second-order spatial differential operator. In what follows, we restate the relevant results for the case of the fractional diffusion equation in the form

$$\left({}_{\ast }\mathbb{D}_{\left(k\right)}u(\mathbf{x},·)\right)\left(t\right)=\Delta u(\mathbf{x},t),(x,t)\in \Omega \times (0,T],$$

(81)

subject to the initial condition

$$u(x,t){|}_{t=0}=u_0\left(x\right),x\in \overline{\Omega },$$

(82)

and the boundary condition

$$u(x,t){|}_{(x,t)\in \partial \Omega \times (0,T]}=v(x,t),(x,t)\in \partial \Omega \times (0,T].$$

(83)

In Equations (81)–(83), $\Omega$ is an open and bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\partial \Omega$, $T>0$, and the Sonin kernel k of the regularized GFD from the left-hand side of Equation (81) satisfies the following conditions:

(L1)

$k\in C^1\left({\mathbb{R}}_{+}\right)\cap L_1^{loc}\left({\mathbb{R}}_{+}\right)$,

(L2)

$k\left(\tau \right)>0$ and $k^{\prime }\left(\tau \right)<0$ for $\tau >0$,

(L3)

$k\left(\tau \right)=o\left({\tau }^{-1}\right),\tau \to 0$.

Now we apply Theorem 3.2 from [49] and arrive at the following result:

Let a function $u(x,t),(x,t)\in \overline{\Omega }\times [0,T]$ satisfy the inclusions $u\in C(\overline{\Omega }\times [0,T])$, $u(·,t)\in C^2(\Omega )$ for any $t>0$, and ${\partial }_{t}u(x,·)\in C(0,T]\cap L_1(0,T)$ for any $x\in \Omega$, and the inequality

$$\left({}_{\ast }\mathbb{D}_{\left(k\right)}u(\mathbf{x},·)\right)\left(t\right)-\Delta u(\mathbf{x},t)\le 0,(x,t)\in \Omega \times (0,T].$$

(84)

Then the following maximum principle holds true:

$$\underset{(x,t)\in \overline{\Omega }\times [0,T]}{max}u(x,t)\le max\{\underset{x\in \overline{\Omega }}{max}u(x,0),\underset{(x,t)\in \partial \Omega \times [0,T]}{max}u(x,t),0\}.$$

(85)

For further results regarding the Cauchy and initial-boundary value problems for the fractional diffusion Equation (64) we refer to [27] and to [49], respectively.

5. Discussion and Conclusions

In this paper, for the first time in the FC literature, we introduced a general fractional master equation involving regularized GFDs with Sonin kernels and analyzed its mathematical properties and some of its physical characteristics.

GFDs with various Sonin kernels and more generally with different classes of Sonin kernels are nowadays a hot topic in the FC literature. These derivatives contain most of the time-fractional derivatives as particular cases. Due to diversity of Sonin kernels, the GFDs and the fractional differential equations with GFDs are actively employed for modeling of several physical processes and systems. In particular, such equations with the GFDs with the special Sonin kernels have been already suggested for modeling of anomalous diffusion processes. In this paper, we provided a background for employing the fractional differential equations with the GFDs involving arbitrary Sonin kernels for modeling of anomalous diffusion processes by establishing their connection to the CTRW model.

One of the main results derived in the paper is a close relation between the conventional CTRW model and the general fractional master equation involving regularized GFDs with Sonin kernels. It turns out that this equation can be embedded into the framework of the CTRW model with the waiting time PDFs expressed in terms of the convolution series generated by the Sonin kernels associated with the kernels of the regularized GFDs. In the case of the fractional master equation with the Caputo fractional derivative involving a power law Sonin kernel, this convolution series is reduced to the known waiting time PDF in terms of the two-parameters Mittag-Leffler function.

Another component of the theory presented in the paper is derivation of the fractional diffusion equation involving regularized GFDs with Sonin kernels from the CTRW model in the asymptotical sense of long times and large distances. This connection suggests employing the fractional diffusion equation with regularized GFDs for modeling of the anomalous diffusion processes. In the framework of this model, the MSD of the diffusing particles governed by the fractional diffusion equation involving regularized GFDs was derived in terms of the Sonin kernels associated with the kernels of the regularized GFDs.

Until now, only some particular cases of this anomalous diffusion model with the power law Sonin kernels were discussed in the literature. The power law Sonin kernels in the fractional diffusion equation induce the MSD of the diffusing particles in form of the power law functions with fixed exponents. In the framework of our model, one can employ any Sonin kernels that lead to a variety of possible expressions for the MSD of the diffusing particles governed by the fractional diffusion equation involving regularized GFDs with Sonin kernels. For a given anomalous diffusion process, the MSD of diffusing particles can be measured in the course of time and in some cases its behavior cannot be imitated by a power law with a fixed exponent. In the framework of our model in form of the fractional diffusion equation involving regularized GFDs with Sonin kernels, this behavior can be fitted and simulated more precisely by selecting an appropriate Sonin kernel. As an example, we presented two particular cases of our model with the crossover behavior of MSD from short to long times. Thus, our approach provides substantially more flexibility, which is needed for modeling of different kinds of anomalous diffusion processes, as compared to the fractional diffusion equations introduced so far.

As to the mathematical properties of solutions to the fractional diffusion equation involving regularized GFDs with Sonin kernels, we discussed conditions on Sonin kernels that ensure some important characteristics of any diffusion-type process. These are non-negativity of the fundamental solution to this fractional diffusion equation and validity of the maximum principle for its solutions on the bounded domains.