Non-Local Kinetics: Revisiting and Updates Emphasizing Fractional Calculus Applications

Abstract

Non-local kinetic problems spanning a wide area of problems where fractional calculus is applicable have been analyzed. Classical fractional kinetics based on the Continuum Time Random Walk diffusion model with the absence of stationary states, real-world problems from pharmacokinetics, and modern material processing have been reviewed. Fractional allometry has been considered a potential area of application. The main focus in the analysis has been paid to the memory functions in the convolution formulation, crossing from the classical power law to versions of the Mittag-Leffler function. The main idea is to revisit the non-local kinetic problems with an update updating on new issues relevant to new trends in fractional calculus.

1. Introduction

Kinetic equations that explain diffusion, chemical reactions, growth processes, and other time-dependent variables with fixed or variable production and destruction rates can be found in a broad range of scientific fields. Typically, these are local kinetic equations where the quantity $N\left(t\right)$ that changes with time balances the rates of creation and destruction to reference [1].

$$\frac{dN}{dt} = -dN\left(t,\tau \right) - pN\left(t,\tau \right),N\left(t,\tau \right) = N\left(t - \tau \right),\tau > 0$$

(1)

It is word noting that in reality, both the destruction $dN\left(t,\tau \right)$ and the production rates $pN\left(t,\tau \right)$ are functionals and depend on the history over a relaxation time $\tau$ that directly indicates a non-locality in time to be encountered in such processes. This can be presented in a simplified form of (1) relating $dN/dt$ to the past values of $N\left(t\right)$ through the functional-differential equation [2] between the cause and the reaction (the effect) due to external physical field action.

$$\mathsf{\Delta }\left(N\right) = {\int }_{-\infty }^{t}K(t - \tau )F\left(\tau \right)d\tau ,$$

(2)

where $\mathsf{\Delta }\left(N\right) = ⟨N\left(t\right)⟩ - {⟨N⟩}_0$, while the kernel $K\left(t\right)$ defines the corresponding correlation function (memory); F(t) is the external physical force amplitude.

Non-local relationship, for instance, between the autocorrelation function of the second order $M_1\left(t\right)$ with correlation functions of high orders (in the Mori–Zwanzig memory formalism) can be expressed as [2,3,4,5,6,7]

$$\frac{dM\left(t\right)}{dt} = {\lambda }_1{M}_1\left(t\right) + {\mathrm{\Omega }}_1{\int }_0^{t}K(t - \tau )M_1\left(t\right)$$

(3)

as a background in statistical physics and particularity related to non-local (fractional kinetics) [6,7]. Here, ${\lambda }_1$ and ${\mathrm{\Omega }}_1$ are characteristic transport parameters. For ${\mathrm{\Omega }}_1 = 0$, for instance, we get a local kinetic equation of first order, while for ${\lambda }_1 = 0$ (3) models evolution processes without stationary states. The construction (3) resembles the fading memory approach explored in the sequel with cases involving non-singular memories (see Section 6).

Similarly, the velocity $v\left(t\right)$ of the tagged particle of mass m in fluid follows the generalized Langevin equation with memory kernel $K\left(\tau \right)$

$$m\frac{dv\left(t\right)}{d} = -\underset{0}{\overset{t}{\int }}K\left(\tau \right)v\left(t - \tau \right)d\tau + F\left(t\right)$$

(4)

in accordance with the Zwanzig–Mori formalism [8,9,10]. The Langevin equation is discussed in Section 2.2 and Section 6.2.

The non-locality naturally comes from the realistic viewpoint that no dynamic process can operate at infinite speeds, and as a result, there exist temporal delays between the causes (reasons) and the results (reactions). This is an instance of the causality principle where the response (or output) cannot come before the cause. In the general case, the non-local counterpart of (1) is

$$\frac{dN}{dt} = -cN\left(t,\tau \right) = -c\underset{0}{\overset{t}{\int }}R\left(t - \tau \right)N\left(\tau \right)d\tau = -cR\left(\tau \right) \ast N\left(\tau \right).$$

(5)

Here $R\left(t - \tau \right)$ is the memory kernel, correlating the current version of N to its past values over the time interval $\tau$; its closest values have stronger contributions than the past (oldest) ones.

The integrals in (2)–(5) are convolution integrals, and the general requirement in these non-local functional relationships, expressing the causality principle, is that $R\left(t\right)$ should be a causal function: $R\left(t\right)$ should be monotonically decaying and vanishing for $t\to \infty$, while for $t < 0$ it should be negligible (i.e., equal to zero because the time t is the processing time (i.e., the time of process duration), but not a time instant [11]. There exist many functions following these fundamental rules, and the decision relies on the actual relaxation mechanisms of the time-evolution processes being modeled. If the initial value of $N\left(t\right)$ is $N_0$ at $t = 0$ when the process starts, then it should be taken into account, and more explicitly, this could be done if the kinetic equations are presented in integral forms (see Section 3).

For instance, kinetic equations are frequently local or memoryless in chemical kinetics, as is given in textbooks. In the sequel, we will review different non-local kinetic equations using different kinds of memory. The fundamental goal of this kind of “academic voyage” is to provide a common point of view on non-local kinetic processes where non-locality may be described by various non-local operators. This is a difficult task since there are different kinetic processes to be modeled as non-local evolutions. For example, when the memories are of the power-law type, then the picture is almost coherent. However, nowadays, there are new operators with non-singular memory kernels, and it would be better to see how this new stream in fractional calculus could be implemented in mathematical models of kinetic processes.

1.1. Aim

The non-local dynamics in various situations with excessive fractional calculus application is the main topic of this paper. It compiles examples from statistical physics and practical concerns with a variety of approaches to solutions, allowing us to see common problems and unresolved issues when fractional calculus is applied broadly to kinetic problems. The main goal, the reference quotation approach, and the paper organization are explained next.

The main goal

This study’s major objective is to organize a common view of non-local kinetic models with a focus on applications of fractional calculus. Most of this article’s parts describe the dominant findings on non-local kinetics, which are based on applications of statistical physics and irreversible thermodynamics. However, there are several straightforward situations arising from actual (physical or technical) issues where the use of fractional calculus is advantageous over the local (in time) kinetic models we are familiar with from textbooks.

It is worth noting that power-law kinetics and fractional operators based on it constitute the foundation for most cases in the literature and our study (fractional operators with singular kernels) too. However, as we shall show, memories based on the non-singular Mittag-Leffler function can simulate non-locality. To show that there are new trends in non-local modeling, we attempt to connect the results from diffusion models with non-singular fractional operators, such as Caputo–Fabrizio and Atangana–Baleanu derivatives, and models based on the Langevin equation with a Mittag-Leffler noise. Although there are not many extensive reference quotations, as will be detailed below, we hope that this research will be a thorough source of knowledge.

Remark: On the references quotation style. Last but not least, before starting the non-local kinetic analysis, we have to say something about the style of this article and the reference quotations. This text primarily refers to some significant contributions to the field of non-local kinetics of various physical performances as opposed to other articles where there are repetitions of the main results that are already well-known and where efforts are stressed on the development of various mathematical techniques, which are poorly motivated and have solutions that are not sufficiently analyzed. This manner of presenting and analyzing the issue at hand can be thought of as an approach in which the major goal is to portray a red line across many issues that share a common basic philosophy in the context of non-local kinetics, even though they have diverse physical origins. This is the author’s point of view implemented in what follows, and to some extent, it might be acceptable or not, but it allows us to see how some issues could be interpreted and collated within a common framework.

1.2. Further Text Organization

The text is organized into two main parts: Fractional kinetics with operators with singular kernels (i.e., Riemann–Liouville and Caputo derivatives)-(Section 2, Section 3, Section 4 and Section 5), and Fractional kinetics with operators with non-singular kernels—(Section 6). From our point of view, this allows a better exposition of the results developed and facilitates the comparative analysis.

In detail, the article is organized as follows: Section 2 considers the well-established kinetic formalism based on the Continuous Time Random Walk (CTRW) mechanism following the style of Zaslavky [12] as well-developed from a methodological point of view. This allows, after that, easy to compare different results. This large section spans: Kinetics of normal and anomalous transport (Section 2.1), Gaussian processes (Section 2.1.1), Levy processes (Section 2.1.2), fractional kinetic equations (Section 2.1.5–Section 2.1.6), fractional kinetic through Langevin equation (Section 2.2), ultraslow diffusion (Section 2.4), fractional kinetic formalism (Section 3), fractional kinetics of dissolution and sorption (Section 4) coming from pharmacodynamics (Section 4.1), pharmacokinetics through compartment models (Section 4.2), the kinetics of decomposition of nanoparticles (Section 4.3), and fractional allometry (Section 5).

Fractional kinetics with operators with non-singular kernels (Section 6) is presented from three points of view: Fractional kinetics exhibited by a diffusion equation with various memory kernels (Section 6.1), fractional kinetic via the Langevin equation with a Mittag-Leffler noise (Section 6.2) and approximate solution of a fractional diffusion equation with Atangana–Baleanu fractional operator (Section 6.3).

2. The Background of Fractional Kinetics

In what follows, representing the main idea of fractional kinetics development briefly, we credit the basic study of Zaslavsky [12] used here as a background allowing us to demonstrate the various branches of the problem. We presume that the Zaslavsky explanation is more systematic, which enables us to provide a concise and comprehensive background of fractional (non-local) kinetics. It is important to remember that this approach is only relevant to power-law memories related to the CTRW transport mechanism, which existed for years before new operators with non-singular memories appeared (see references [13,14] discussed in (Section 6)). In general, the application of fractional calculus for the modeling of non-local kinetics begins probably with the works of Mandelbrot [15], about the initial steps in the strong formulation of fractional kinetics are available in [12,16].

2.1. Non-Local (Fractional) Kinetics of Normal and Anomalous Transport

The dynamics of a given system allows splitting into two different time scales [12,16]: short time scale $\mathsf{\Delta }t$ and long time scale ${\tau }_{col}$. The short time scale $\mathsf{\Delta }t$ is, in general, related to the process of particle collisions while ${\tau }_{col}$ is associated with the time interval between adjacent collisions. The term “collision” should be considered in a broad sense, such as a change in energy, momentum, mass, concentration, or other process variables of interest [12,16].

For example, the Fokker–Plank–Kolmogorov (KPK) equation is a kinetic equation relating the probability density $W\left(x,t;x^{\prime },t^{\prime }\right)$ of observing a particle to be the position x at the time t, if it was at the position $x^{\prime }$ at the time $t^{\prime } \le t$ [12,16]. Then, considering a Markov-type process, a chain equation can be written as

$$W\left(x_3,t_3,x_1,t_1\right) = \int d{x}_{2}W\left(x_3,t_3,x_2,t_2\right)W\left(x_2,t_2,x_1,t_1\right).$$

(6)

The typical assumption for W is [12]: $W\left(x,t;x^{\prime },t^{\prime }\right) = W\left(x,x^{\prime };t - t^{\prime }\right)$. Then, the evolution of $W\left(x,x^{\prime };t - t^{\prime }\right)$ over an infinitesimal time interval $\mathsf{\Delta }t = t^{\prime } - t$, applying the Taylor expansion

$$W\left(x,x_0;t + \mathsf{\Delta }t\right) = W\left(x,x_0;t\right) + \frac{\partial W\left(x,x_0;t\right)}{\partial t}\mathsf{\Delta }t + ...$$

(7)

which is valid upon the limit $\mathsf{\Delta }t\to 0$, such that

$${lim}_{\mathsf{\Delta }t\to 0}\left[\frac{W\left(x,x_0;t + \mathsf{\Delta }t\right) - W\left(x,x_0;t\right)}{\mathsf{\Delta }t}\right] = \frac{\partial W\left(x,x_0;t\right)}{\partial t}.$$

(8)

The limit (8) yields specific physical constraints commented further in this article. The notation $P\left(x,t\right) = W\left(x,x_0;t\right)$ neglects the initial position (coordinate) $x_0$ and allows (8) to be converted to

$$\frac{\partial P\left(x,t\right)}{\partial t} = {lim}_{\mathsf{\Delta }t\to 0}\frac{1}{\mathsf{\Delta }t}\left\{\int dyW\left(x,y;\mathsf{\Delta }t\right)P\left(y,t\right) - P\left(x,t\right)\right\}.$$

(9)

The important moment here is the use of two distribution functions $P\left(x,t\right)$ and $W\left(x,y;\mathsf{\Delta }t\right)$ instead $W\left(x,p;x^{\prime },p^{\prime }\right)$. The former distribution is targeted to the time interval where $t\to \infty$ (see the condition $t \gg {\tau }_{col} \gg \mathsf{\Delta }t$ mentioned above). However, opposing $P\left(x,t\right)$, the distribution $W\left(x,y;\mathsf{\Delta }t\right)$ is related to short-time transition, i.e., for $\mathsf{\Delta }t\to 0$ (for $\mathsf{\Delta }t = 0$ there is no transition (relaxation) process if the velocity is finite), and this means that $W_{\mathsf{\Delta }t\to 0}\left(x,y;\mathsf{\Delta }t\right) = \delta \left(x - y\right)$, where $\delta$ is the Dirac delta function. Expansion over the $\delta$ function [12,16] is

$$W\left(x,y;\mathsf{\Delta }t\right) = \delta \left(x - y\right) + A\left(y;\mathsf{\Delta }t\right){\delta }^{\prime }\left(x - y\right) + \frac{1}{2}B\left(y;\mathsf{\Delta }t\right){\delta }^{″}\left(x - y\right)$$

(10)

with two normalizing conditions imposed on the transfer probability $W\left(x,y;\mathsf{\Delta }t\right)$:

$$\int W\left(x,y;\mathsf{\Delta }t\right)dx = 1$$

(11)

$$\int W\left(x,y;\mathsf{\Delta }t\right)dy = 1$$

(12)

The coefficients $A\left(y;\mathsf{\Delta }t\right)$ and $B\left(y;\mathsf{\Delta }t\right)$ in (10) are meaning of moments of $W\left(x,y;\mathsf{\Delta }t\right)$, namely

$$A\left(y;t\right) = \int dx\left(y - x\right)W\left(x,y;\mathsf{\Delta }t\right) \equiv ⟨⟨\mathsf{\Delta }y⟩⟩$$

(13)

$$B\left(y;t\right) = \int dx{\left(y - x\right)}^{2}W\left(x,y;\mathsf{\Delta }t\right) \equiv ⟨⟨{\left(\mathsf{\Delta }y\right)}^2⟩⟩$$

(14)

The integration of the expansion (10) with respect to x has no informative sense [12], while the integration concerning y (bearing in mind (12)) yields

$$A\left(y;\mathsf{\Delta }t\right) = \frac{1}{2}\frac{\partial B\left(y;\mathsf{\Delta }t\right)}{\partial y}$$

(15)

That implies that we get the Landau relationship of microscopic reversibility [12]

$$⟨⟨\mathsf{\Delta }y⟩⟩ = \frac{1}{2}\frac{\partial }{\partial y}⟨⟨{\left(\mathsf{\Delta }y\right)}^2⟩⟩$$

(16)

Further, using the expansion (10) and reversibility normalization (12), and as the final step, the so-called Kolmogorov conditions, we may establish the following limits [12,16]:

$${lim}_{\mathsf{\Delta }t\to 0}\frac{1}{\mathsf{\Delta }t}⟨⟨\mathsf{\Delta }t⟩⟩ \equiv \mathbf{A}\left(x\right)$$

(17)

$${lim}_{\mathsf{\Delta }t\to 0}\frac{1}{\mathsf{\Delta }t}⟨⟨{\left(\mathsf{\Delta }t\right)}^2⟩⟩ \equiv \mathbf{B}\left(x\right)$$

(18)

$${lim}_{\mathsf{\Delta }t\to 0}\frac{1}{\mathsf{\Delta }t}⟨⟨{\left(\mathsf{\Delta }t\right)}^m⟩⟩ = 0,m > 2$$

(19)

Due to the Kolmogorov irreversibility conditions, the above results allow for getting the Kolmogorov equation [12]

$$\frac{\partial P\left(x,t\right)}{\partial t} = -\frac{\partial \mathbf{A}P\left(x,t\right)}{\partial x} + \frac{1}{2}\frac{{\partial }^2\mathbf{B}P\left(x,t\right)}{\partial x^2}$$

(20)

This equation is of a diffusion type and describes an irreversible process (transition). With (16) we may obtain the final form of the diffusion equation

$$\frac{\partial P\left(x,t\right)}{\partial t} = \frac{1}{2}\frac{\partial }{\partial x}\left\{\frac{\partial }{\partial x}\mathbf{D}P\left(x,t\right)\right\},\mathbf{D} = \mathbf{B} = {lim}_{\mathsf{\Delta }t\to 0}\frac{⟨⟨{\left(\mathsf{\Delta }x\right)}^2⟩⟩}{\mathsf{\Delta }t}$$

(21)

Thus, defining the diffusion coefficient $\mathbf{D}$ (from (21)) it is obvious why the dimension of $\mathbf{D}$ is $\left[L^2{T}^{-1}\right]$, that is in the SI system $\left[m^2{s}^{-1}\right]$). This definition still does not rely on how the mean squared displacement ${\left(\mathsf{\Delta }x\right)}^2$ scales to the time. However, all results obtained so far and the definition (21) show that $\mathbf{D}$ is related to Gaussian diffusion, i.e., ${\left(\mathsf{\Delta }x\right)}^2\propto t$. However, we may see that $\mathbf{D}$ can be interpreted as a local rate of ${\left(\mathsf{\Delta }x\right)}^2$, that is $\frac{d{\left(\mathsf{\Delta }x\right)}^2}{dt} = \mathbf{D}$, and if $\mathbf{D} = const.$, we have a normal diffusion. However, if $\frac{d{\left(\mathsf{\Delta }x\right)}^2}{dt} \equiv t^{\alpha - 1}$, or in other words, we have a time-dependent diffusion coefficient $\mathbf{D} = \mathbf{D}\left(t\right)$, then with ${\left(\mathsf{\Delta }x\right)}^2 \equiv t^{\alpha }$ for $\alpha < 1$ we get a subdiffusion transport, while for $\alpha > 1$ there is a superdiffusive process. Deviations from the normal scaling ${\left(\mathsf{\Delta }x\right)}^2\propto t$ will also be discussed in Section 2.1.1, but the comments above simply explain why local in time diffusion equations having time-dependent diffusion coefficients (local in time formulation) can model anomalous (commonly subdiffusion) transport, and the result is the same as that obtained by solving time-fractional diffusion equations.

The formulation (21) has a divergent form and is related to the condition of conservation of the number of particles (the continuity equation) with a flux J [12,16,17]

$$\frac{\partial P\left(x,t\right)}{\partial t} = \frac{\partial J}{\partial t},J = \frac{1}{2}\mathbf{D}\frac{\partial P\left(x,t\right)}{\partial x}$$

(22)

The condition (19), as well as the (17)–(19) impose an additional relationship

$$\mathbf{A}\left(x\right) = \frac{1}{2}\frac{\partial }{\partial x}\mathbf{B}\left(x\right) = \frac{1}{2}\frac{\partial }{\partial x}\mathbf{D}\left(x\right)$$

(23)

defining $\mathbf{A}\left(x\right)$ as a convective component of the particle flux. If $\mathbf{D} = const.$, then this component of the flux is zero.

2.1.1. Normal Transport: Gaussian Process

For the sake of simplicity, with $\mathbf{D} = const.$ and $x\in \left(-\infty ,\infty \right)$ we get the Gaussian distribution [12,16,17].

$$P\left(x,t\right) = \frac{1}{\sqrt{2\pi \mathbf{D}t}}exp\left(-\frac{x^2}{2\mathbf{D}t}\right).$$

(24)

The odd moments of this distribution are zero, but the second moment is proportional to the time, i.e., $⟨x^2⟩ = \mathbf{D}t$ or $⟨x⟩ \equiv t$. Then, the higher moments are [12]: $⟨x^{2m}⟩ = {\mathbf{D}}_m{t}^m$ with $m = 1,2,...$.

Moreover, there are two important issues [12]: (1) All moments of $P\left(x,t\right)$ are finite because the distribution decays exponentially for $t\to \infty$, and (2) $P\left(x,t\right)$ is invariant to the symmetry group (renormalization)

$$\widehat{R}:x^{\prime } = \lambda x,t^{\prime } = {\lambda }^{2}t.$$

(25)

In general, the time evolution of the moment $⟨x^m⟩$ is termed transport [12] (of particle, momentum, mass, or energy). For $⟨x^2⟩ = \mathbf{D}t$ and $⟨x^{2m}⟩ = {\mathbf{D}}_m{t}^m$ we have normal transport. In addition, the Gaussian packet [12]

$$P\left(x,t\right) = \frac{1}{\sqrt{2\pi \mathbf{D}t}}exp\left(-\frac{{\left(x - ct\right)}^2}{2\mathbf{D}t}\right)$$

(26)

involving the traveling wave symmetry, $x - ct$ also is a solution to the equation

$$\frac{\partial P\left(x,t\right)}{\partial t} + c\frac{\partial P\left(x,t\right)}{\partial x} = \frac{1}{2}\frac{{\partial }^{2}P\left(x,t\right)}{\partial x^2}.$$

(27)

where $⟨x⟩ = ct$ and $⟨{\left(x - ⟨x⟩\right)}^2⟩ = \mathbf{D}t$, thus, resembling the outcomes of the distribution (24).

2.1.2. Anomalous Transport: Levy Process

The Levy distribution is related to distributions with infinite second moments [12] and appears in various processes deviating from the normal (Gaussian) distribution. In such a case the normalized distribution $P_L\left(x\right)$, that is $\underset{-\infty }{\overset{\infty }{\int }}{P}_L\left(x\right)dx = 1$, the characteristic function is

$$P_L\left(q\right) = \underset{-\infty }{\overset{\infty }{\int }}{e}^{iqx}{P}_L\left(x\right)dx$$

(28)

If there are two different random variables $x_1$ and $x_2$, then their linear combination $c_3{x}_3 = c_1{x}_1 + c_2{x}_2$ will be considered stable if all variables (i.e., $x_1,x_2,x_3$) are distributed by the same function $P_L\left(x\right)$. The Gaussian distribution (24) is a stable distribution and its second moment $⟨x^2⟩$ is finite.

Now, let us consider the equation [12,18]

$$P\left(x_3\right)d{x}_3 = P\left(x_1\right)P\left(x_2\right)\delta \left(x_3 - \frac{c_1}{c_3}{x}_1 - \frac{c_2}{c_3}{x}_2\right)d{x}_{1}d{x}_2,c_3{x}_3 = c_1{x}_1 + c_2{x}_2$$

(29)

From the definition of $P_L\left(q\right)$ it follows

$$P\left(c_{3}q\right) = P\left(c_{1}q\right)P\left(c_{2}q\right)\Rightarrow lnP\left(c_{3}q\right) = lnP\left(c_{1}q\right) + lnP\left(c_{2}q\right)$$

(30)

With a solution involving an arbitrary parameter $\alpha$, namely

$$ln{P}_{\infty }\left(c_{3}q\right) = {\left(c_{3}q\right)}^{\alpha } = c_{3}exp\left(-i\alpha \left(signq\right)\right){\left|q\right|}^{\alpha },{\left(\frac{c_1}{c_3}\right)}^{\alpha } + {\left(\frac{c_2}{c_3}\right)}^{\alpha } = 1$$

(31)

the Levy distribution with a Levy index $\alpha$ is a distribution having a characteristic function

$$P_{\infty }\left(q\right) = exp\left(-c_3{\left|q\right|}^{\alpha }\right)$$

(32)

The Gaussian distribution is a special case when $\alpha = 2$, while $\alpha = 1$ is leading to the Cauchy distribution $P_1\left(x\right) = \frac{c_3}{\pi }\frac{1}{x^2 + c_3}$.

Asymptotically, for large $\left|x\right|$ we get [12,18]

$$P_{\infty }\left(x\right) \equiv \frac{1}{\pi }\alpha c_3\mathrm{\Gamma }\left(\alpha \right)sin\frac{\pi \alpha }{2}\frac{1}{{\left|x\right|}^{\alpha + 1}}$$

(33)

The existence of the Levy process requires $0 < \alpha \le 2$ thus, assuring the positiveness of (28)

2.1.3. Simplified Approach to Fractional Transport Kinetic Equation

Further, assuming a transitional probability $W\left(x,x_0;t + \mathsf{\Delta }t\right)$, where $\mathsf{\Delta }t$ is infinitesimal, the following expansion is possible

$${lim}_{\mathsf{\Delta }t\to 0}\frac{1}{{\left|\mathsf{\Delta }t\right|}^{\beta }}\left\{W\left(x,x_0;t + \mathsf{\Delta }t\right) - W\left(x,x_0;t\right)\right\} = \frac{{\partial }^{\beta }}{\partial t^{\beta }}W\left(x,x_0;t\right) = \frac{{\partial }^{\beta }}{\partial t^{\beta }}P\left(x,t\right)$$

(34)

Here the notation used for the development of (9) is applied, and a general approach arrives at a fractional in-time derivative of $P\left(x,t\right)$ (precisely a Riemann–Liouville derivative, see the definition in Appendix A.1.2).

Furthermore, an expansion for $W\left(x,x_0;\mathsf{\Delta }t\right)$, resembling (10), but different in its sense, with appropriate fractal dimensions $\alpha$ and ${\alpha }_1$, is possible [12]

$$\begin{array}{c}\hfill W\left(x,y;\mathsf{\Delta }t\right) = \delta \left(x - y\right) + A\left(y;\mathsf{\Delta }t\right){\delta }^{\left(\alpha \right)}\left(x - y\right) + B\left(y;\mathsf{\Delta }t\right){\delta }^{\left({\alpha }_1\right)}\left(x - y\right),\\ \hfill 0 < \alpha < {\alpha }_1\le 2\end{array}$$

(35)

The coefficient $B\left(y;\mathsf{\Delta }t\right)$ can be developed from the moment of W, that is [12]

$$⟨⟨{\left|\mathsf{\Delta }x\right|}^{{\alpha }_1}⟩⟩ \equiv {\int dx\left|x - y\right|}^{{\alpha }_1}W\left(x,y;\mathsf{\Delta }t\right) = \mathrm{\Gamma }\left({\alpha }_1 + 1\right)B\left(y;\mathsf{\Delta }t\right)$$

(36)

The integration of (35) with respect to y yields

$$\frac{{\partial }^{\alpha }\mathbf{A}\left(x\right)}{\partial {\left(-x\right)}^{\alpha }} + \frac{{\partial }^{{\alpha }_1}\mathbf{B}\left(x\right)}{\partial {\left(-x\right)}^{{\alpha }_1}} = 0$$

(37)

$$\begin{array}{c}\hfill \mathbf{A}\left(x\right) = {lim}_{\mathsf{\Delta }t\to 0}\frac{A\left(x,\mathsf{\Delta }t\right)}{{\left(\mathsf{\Delta }t\right)}^{\beta }},\mathbf{B}\left(x\right) = {lim}_{\mathsf{\Delta }t\to 0}\frac{B\left(y,\mathsf{\Delta }t\right)}{{\left(\mathsf{\Delta }t\right)}^{\beta }} = \\ \hfill = \frac{1}{\mathrm{\Gamma }\left(1 + {\alpha }_1\right)}{lim}_{\mathsf{\Delta }t\to 0}\frac{⟨⟨{\left|\mathsf{\Delta }x\right|}^{{\alpha }_1}⟩⟩}{{\left(\mathsf{\Delta }t\right)}^{\beta }}\end{array}$$

(38)

If it is assumed that ${\alpha }_1 = \alpha + 1$, then (37) can be simplified as [12]

$$\frac{{\partial }^{\alpha }}{\partial {\left(-x\right)}^{\alpha }}\left[\mathbf{A}\left(x\right) + \frac{\partial \mathbf{B}\left(x\right)}{\partial x}\right] = 0$$

(39)

$$\mathbf{A}\left(x\right) = \frac{1}{\mathrm{\Gamma }\left(1 + \alpha \right)}{lim}_{\mathsf{\Delta }t\to 0}\frac{⟨⟨{\left|\mathsf{\Delta }x\right|}^{\alpha }⟩⟩}{{\left(\mathsf{\Delta }t\right)}^{\beta }},\mathbf{B}\left(x\right) = \frac{1}{\mathrm{\Gamma }\left(2 + \alpha \right)}{lim}_{\mathsf{\Delta }t\to 0}\frac{⟨⟨{\left|\mathsf{\Delta }x\right|}^{\alpha + 1}⟩⟩}{{\left(\mathsf{\Delta }t\right)}^{\beta }}$$

(40)

Further, based on the expansions (35) and (38) we get

$$\frac{{\partial }^{\beta }P\left(x,t\right)}{\partial t^{\beta }} = \frac{{\partial }^{\alpha }}{\partial {\left(-x\right)}^{\alpha }}\left\{\mathbf{A}\left(x\right)P\left(x,t\right)\right\} + \frac{{\partial }^{{\alpha }_1}}{\partial {\left(-x\right)}^{{\alpha }_1}}\left\{\mathbf{B}\left(x\right)P\left(x,t\right)\right\}$$

(41)

For ${\alpha }_1 = \alpha + 1$ the simplification results in [12]

$$\frac{{\partial }^{\beta }P\left(x,t\right)}{\partial t^{\beta }} = \frac{{\partial }^{\alpha }}{\partial {\left(-x\right)}^{\alpha }}\left\{\mathbf{B}\left(x\right)\frac{\partial P\left(x,t\right)}{\partial x}\right\}$$

(42)

For $\alpha = 1$ we get the normal diffusion equation with $\mathbf{B}\left(x\right) = \frac{1}{2}\mathbf{D}$. In cases when $\mathbf{B}\left(x\right)$ can be neglected the simplified version of (41) is [12]

$$\frac{{\partial }^{\beta }P\left(x,t\right) = }{\partial t^{\beta }}\frac{{\partial }^{\alpha }}{\partial {\left|x\right|}^{\alpha }}\left\{\mathbf{A}P\left(x,t\right)\right\}$$

(43)

For $\beta = 1$ and $\alpha = 2$ we get the normal diffusion model and the dimension of A is $\left[m^2/s\right]$ while in (43) it should be $\left[m^{\alpha }/s^{\beta }\right]$. For $0 < \beta < 1$ and $\alpha = 2$ we get the so-called time-fractional heat wave equation

$$\frac{{\partial }^{\beta }P\left(x,t\right)}{\partial t^{\beta }} = \frac{{\partial }^2}{\partial x^2}\left\{\mathbf{A}P\left(x,t\right)\right\}$$

(44)

known as the equation of fractional Brownian motion [12,16,17] and the dimension of A is $\left[m^2/s^{\beta }\right]$.

For $\beta = 1$ and $1 < \alpha < 2$ we get

$$\frac{\partial P\left(x,t\right)}{\partial t} = \frac{{\partial }^{\alpha }}{\partial {\left|x\right|}^{\alpha }}\left\{\mathbf{A}P\left(x,t\right)\right\}$$

(45)

and the model is relevant to the Levy process. In this case the dimension of A is $\left[m^{\alpha }/s\right]$.

The parameters $\beta$, $\alpha$ and ${\alpha }_1$ are termed critical parameters [12]. We can see further that when time and space fractional operators are applied, we will call them fractional orders.

2.1.4. Fractional Kinetic Equation of Anomalous Transport: Some Comments

From a physical and mathematical standpoint, every one of the fractional kinetic equations thus far presented is an individual issue that is pertinent to specific instances. It is important to know and define the space-time interval regarding the coordinate (or momentum) commonly noted by the variable $x_{min} < x < x_{max}$. As to the time window, it is defined by some $t_{min}$ and $t_{max}$. Thus, we have time and space windows [12], with finite moments of $P\left(x,t\right)$, which are integrable (the infinite intervals (windows) yield infinite moments of the distributions).

Further, the positiveness of the distribution should be assured, that is, $P\left(x,t\right) \ge 0$ over the window of consideration. For example, $0 < \alpha \le 2$ and $0 < \beta < 1$ correspond to the Levy process [12,16] when $\mathbf{A}$ is considered constant.

Moreover, last but not least, we have to focus on the missing stationary states of the anomalous transport mechanism (commonly mentioned as anomalous diffusion). Why a stationary state cannot be achieved is very clearly explained by the physical theory behind the CTRW (see Section 2.1.6).

2.1.5. Fractional Transport Kinetic Equation: A Series Solution and Outcomes

Let us consider a fractional kinetic equation with source term expressed as [12,16]

$$\frac{{\partial }^{\beta }P\left(x,t\right)}{\partial t^{\alpha }} = \frac{{\partial }^{\alpha }P\left(x,t\right)}{\partial {\left|x\right|}^{\alpha }} + S\left(x,t\right)$$

(46)

that also can be interpreted as a fractional diffusion equation with a source term.

The time series solution (infinite series) [12] is

$$P\left(x,t\right) = \frac{1}{\pi \left|y\right|}\frac{1}{t^{\mu /2}}\sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{{\left|y\right|}^{k\alpha }}\frac{\mathrm{\Gamma }\left(k\alpha + 1\right)}{\mathrm{\Gamma }\left(k\beta + 1\right)}cos\left[\frac{\pi }{2}k\alpha + 1\right],y = x/\sqrt{t^{\mu }}$$

(47)

Here, $\mu = 2\beta /\alpha$ is a transport exponent related to the evolution of the moments when there is a self-similarity of the solution, and $⟨\left|x\right|⟩\sim t^{\beta /2} = t^{\mu /2}$; for example, $y = x/\sqrt{t^{\mu }}$ with $\mu = 1$ coincides the Boltzmann similarity variable known from the normal diffusion process.

For ${\left|x\right|}^2 \gg t^{\mu }$ asymptotically, we get from (47)

$$P\left(x,t\right)\sim \frac{1}{\pi \left|y\right|}\frac{1}{t^{\alpha + 1}}\frac{\mathrm{\Gamma }\left(\alpha + 1\right)}{\mathrm{\Gamma }\left(\beta + 1\right)}sin\frac{\pi \alpha }{2}.$$

(48)

This is a Levy distribution $P_{\alpha }\left(x,t\right)$ with $\beta = 1$. Moreover, it follows from (48) that $⟨{\left|x\right|}^{\gamma }⟩ < \infty$ for $0 < \gamma < \alpha \le 2$.

2.1.6. Fractional Transport Kinetic Equation: Probabilistic Approach

The probabilistic approach addresses the mechanism of the Continuous Time Random Walk (CTRW) mechanism conceived by Montroll and Weiss [19]. Upon the assumed mechanism of CTRW, we have two types of probabilities: probability density $P_j\left(x\right)$ to find the particle at position x after performing j steps, and probability $P\left(x,t\right)$ to find the particle at the position x at the time t. The CTRW mechanism assumes independent changes at each step of particle motion and uniformity of the probability distributions in the space-time windows between consequent steps. Hence, with transitional probability $W\left(x\right)$, we may write [12]

$$P_{j + 1} = \underset{-\infty }{\overset{\infty }{\int }}W\left(x - x^{\prime }\right)P_j\left(x^{\prime }\right)d{x}^{\prime }$$

(49)

$$P\left(x,t\right) = \underset{0}{\overset{t}{\int }}\varphi \left(t - \tau \right)Q\left(x,t\right)d\tau$$

(50)

We can see that $P\left(x,t\right)$ includes a memory mechanism through the probability $\varphi \left(t\right)$ relevant to the particle to be at the position x at the time t, while $Q\left(x,\tau \right)$ is a probability to find the particle the position x at the time t instantly after the jump performance. The memory function, i.e., the probability $\varphi \left(t\right)$ is related to the probability $\psi \left(t\right)$ relevant to a time interval t between two consequent steps of the jumping particle, i.e., $\varphi \left(t\right) = \underset{t}{\overset{\infty }{\int }}d{t}^{\prime }\psi \left(t^{\prime }\right)$. If ${\psi }_j\left(t\right)$ is the probability density to take the $j^{th}$ jump at time instant t, we have

$${\psi }_{j + 1}\left(t\right) = \underset{0}{\overset{t}{\int }}\psi \left(t - \tau \right){\psi }_j\left(\tau \right)d\tau ,{\psi }_1\left(t\right) \equiv \psi \left(t\right)$$

(51)

We can see a memory mechanism in the construction of (51) as a convolution integral. As a consequence, the following equation has been solved [12]

$$Q\left(x,t\right) = \sum _{j = 1}^{\infty }{\psi }_j\left(t\right)P\left(x\right)$$

(52)

It establishes a relationship between discrete steps of random motion walk (jumping steps) and the waiting times at different steps. The solution of (52) with respect to $P(x,t)$ [12,19] is

$$P\left(x,t\right) = \frac{1}{2\pi i}\underset{-i\infty }{\overset{c + \infty }{\int }}d\frac{1}{u}\left[1 - \psi \left(u\right)\right]e^{ut}\left\{\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}dq\frac{e^{-iqx}}{1 - \psi \left(u\right)W\left(q\right)}\right\}$$

(53)

where $\left[\frac{1 - \psi \left(u\right)}{u}\right]$ is an alternative form of $\varphi \left(t\right) = \underset{t}{\overset{\infty }{\int }}\psi \left(t^{\prime }\right)d{t}^{\prime }$.

Hence, the final result (53) relates two probabilities: $W\left(x\right)$ about the jump (step) length and $\psi$ about the time interval between consequent steps, i.e., the waiting time between two consequent random steps.

The CTRW model can be generalized [12,20,21] through the master equation as

$$\frac{\partial P\left(x,t\right)}{\partial t} = \underset{0}{\overset{t}{\int }}\varphi \left(t - \tau \right)\left\{\sum _{x^{\prime }}^{ }W\left(x - x^{\prime }\right)P\left(x^{\prime },t\right) - P\left(x,\tau \right)\right\}d\tau$$

(54)

with appropriate transition probability $W\left(x\right)$ and a memory kernel $\varphi \left(t - \tau \right)$ (termed also as a delay probability function).

Following the results commented above it is possible to assume $\varphi \left(u\right)\sim u^{\beta }$ and $\varphi \left(t\right)\sim t^{-\left(1 + \beta \right)}$. In this context, a generalized fractional (of master equation type) equation of Kolmogorov–Feller construction resembling (54) has been proposed by Saichev and Zaslavsky [16] as

$$\frac{{\partial }^{\beta }P\left(x,t\right)}{\partial t^{\beta }} = \underset{-\infty }{\overset{\infty }{\int }}W\left(t\right)\left[P\left(x - y,t\right) - P\left(x,t\right)\right]dy$$

(55)

Regarding the idea to use the master equation as a starting point for the emergence of fractional kinetics, we refer to [22] where a systematic and easy-to-understand analysis was developed (based on time scale distribution related to the fundamental solution of the space-time fractional diffusion equation).

2.2. Non-Local (Fractional) Kinetics Via the Langevin Equation

The fractal Brownian motion (see (44)) where the mean squared displacement is proportional to a power-law of the time with non-integer exponent, that is $⟨{\left(\mathsf{\Delta }x\right)}^2⟩\propto t^{\alpha }$, $\alpha \ne 1$, in many cases this can be considered as a performance of self-similarity of the diffusion process [23,24]. This type of motion is interesting not only as a mathematical challenge but from the side of physical interpretations of solutions developed. It is well-known that such type of motion can be considered as the Ornstein–Uhlenbeck process described by the Langevin stochastic differential equation concerning the velocity v of the Brownian particle

$$\frac{dv}{dt} = -\gamma v + F\left(t\right)$$

(56)

Here, the motion is damped by the liquid (medium) friction coefficient $\gamma$ (with dimensions $\left[1/s\right]$) and forces by a random source $F\left(t\right)$ (with dimensions $\left[m/s^2\right]$).

A generalization of (56) can be obtained if the time-derivative is replaced by a fractional one (Riemann–Liouville, but it is better by Caputo type)

$$\frac{{\partial }^{\alpha }v}{\partial t^{\alpha }} = -\gamma v + F\left(t\right)$$

(57)

Applying the fractional integral operator (Riemann–Liouville) (see Appendix A) for basic definitions related to fractional operators) to (57) we get [23]

$$v - v_0 = I_t^{\alpha }\left(-\gamma v + F\left(t\right)\right)$$

(58)

The solution in terms of the two-parameter Mittag-Leffler function $E_{\alpha ,\beta }$ [23] is

$$v = v_0{E}_{\alpha ,1}\left(-\gamma t^{\alpha }\right) + \underset{0}{\overset{t}{\int }}F\left(s\right){\left(t-s\right)}^{\alpha - 1}{E}_{\alpha ,\alpha }\left[ - \gamma {\left(t - s\right)}^{\alpha }\right]ds$$

(59)

This allows the mean squared displacement of particles to be defined as

$$⟨\left(\mathsf{\Delta }{x}^2\right)⟩ = \underset{0}{\overset{t}{\int }}\underset{0}{\overset{t}{\int }}⟨v\left(t_1\right)v\left(t_2\right)⟩d{t}_{1}d{t}_2$$

(60)

Then, from (59) we have

$$⟨\left(\mathsf{\Delta }{x}^2\right)⟩ = v_0^2{\left[t{E}_{\alpha ,1}\left(-\gamma t^{\alpha }\right)\right]}^2 + \frac{q}{{\gamma }^2}\left\{t - 2t{E}_{\alpha ,2}\left(-\gamma t^{\alpha }\right) + \underset{0}{\overset{t}{\int }}{E}_{\alpha ,1}\left(-\gamma t^{\alpha }\right)dt\right\}$$

(61)

For long times we have asymptotics $E_{\alpha ,1}\left(-\gamma t^{\alpha }\right)\propto E_{\alpha ,2}\left(-\gamma t^{\alpha }\right)\propto {\left(\gamma t\right)}^{-\alpha }$. This allows us to see that the integral in the second term of (61) is converging for $t\to \infty$ when $1/2 < \alpha < 1$, and, therefore, we get asymptotics of the mean squared displacement $⟨\left(\mathsf{\Delta }{x}^2\right)⟩ = \frac{q}{{\gamma }^2}t$ as in the integer-order Langevin Equation (56).

However, the meaning of particle velocity v (which to the original Langevin equation is defined as a local variable) can be reconsidered and suggesting it as a non-local variable$v\left(t\right)$ [23]. In such a case, the distance traveled is defined as

$$x = \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{v\left(s\right)}{{\left(t - s\right)}^{1 - \alpha }}ds$$

(62)

Physically, this means that the fractal paths of the curve describing the particle motion are averaged and that the velocity at the time t depends on its recent values, over the interval $t - s$, rather than on the oldest ones. Under the assumptions of such a non-local construction, the mean square displacement is [23]

$$\begin{array}{cc}\hfill & ⟨\left(\mathsf{\Delta }{x}^2\right)⟩ = I_{t_1}^{\alpha }{I}_{t_2}^{\alpha }⟨v\left(t_1\right)v\left(t_2\right)⟩ = v_0^2\frac{1}{{\gamma }^2}{\left[E_{\alpha ,1}\left(-\gamma t^{\alpha }\right)\right]}^2 + \hfill \\ \hfill & 2\frac{q}{{\gamma }^2}{t}^{2\alpha - 1}\sum _{k,l = 1}^{\infty }\frac{{\left(-\gamma \right)}^{k + 1}}{\mathrm{\Gamma }\left(\alpha + \alpha k\right)\mathrm{\Gamma }\left(\alpha l\right)}\frac{\mathrm{\Gamma }\left(\alpha k + \alpha l + \alpha - 1\right)}{\mathrm{\Gamma }\left(\alpha k + \alpha l + 2\alpha \right)}{t}^{\alpha \left(k + l\right)}\hfill \end{array}$$

(63)

For long times the first term of the extended version in (63) approaches $v_0^2/{\gamma }^2$ while the sum in the second term converges.

Assuming that $1/2 < \alpha < 1$ we get

$$⟨\left(\mathsf{\Delta }{x}^2\right)⟩\propto N\frac{q}{{\gamma }^2}{t}^{2\alpha - 1},1 < N < \frac{2}{\mathrm{\Gamma }\left(\alpha \right)}$$

(64)

This is a result matching the yield of the solution corresponding to the fractional Brownian motion.

Now, extending the non-locality over the random force (source) $F\left(t\right)$, Equation (58) can be expressed as [23]

$$v = v_0 - \gamma \underset{0}{\overset{t}{\int }}v\left(s\right)ds + \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{F\left(s\right)}{{\left(t - s\right)}^{\alpha }}ds$$

(65)

The random source $F\left(t\right)$ is not $\delta$ correlated but can be expressed as a fractional derivative (Riemann–Liouville in the original notation) of white noise [23]

$$F\left(t\right) = D_t^{1 - \alpha }g\left(t\right),⟨g\left(t_1\right)g\left(t_2\right)⟩ = q\delta \left(t_1 - t_2\right)$$

(66)

The analytical solution of (65) is [23]

$$v = v_0{e}^{-\gamma t} + \underset{0}{\overset{t}{\int }}F\left(s\right){\left(t - s\right)}^{\alpha - 1}{E}_{\alpha ,1}\left[-\gamma \left(t - s\right)\right]ds$$

(67)

Then, for the mean squared displacement, when $t\to \infty$, we have [23]

$$⟨\left(\mathsf{\Delta }{x}^2\right)⟩\propto \frac{1}{\left(2\alpha - 1\right){\left(\mathrm{\Gamma }\left(\alpha \right)\right)}^2}\frac{q}{{\gamma }^2}{t}^{2\alpha - 1}$$

(68)

It is also reasonable to consider the case when the non-locality is concentrated only in the friction coefficient since we may suggest that the reaction of the medium to the particle motion is not instant (that is, this reaction is not local) and the source $F\left(t\right)$ is Gaussian white noise. In such a case, we may write [23]

$$v = v_0 - \gamma \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{v\left(s\right)}{{\left(t - s\right)}^{\alpha }}ds + \underset{0}{\overset{t}{\int }}v\left(s\right)ds$$

(69)

Then, the dissipative force $f_{diss}$ is proportional to the fractional derivative of the velocity $v\left(t\right)$, i.e., $f_{diss} = D_t^{1 - \alpha }v\left(t\right)$ and the solution is [23]

$$v = v_0{E}_{\alpha ,1}\left(-\gamma t^{\alpha }\right) + \underset{0}{\overset{t}{\int }}F\left(s\right)E_{\alpha ,\alpha }\left(-\gamma {\left(t - s\right)}^{\alpha }\right)ds$$

(70)

and

$$⟨\left(\mathsf{\Delta }{x}^2\right)⟩\propto M\frac{q}{{\gamma }^2}{t}^{3 - 2\alpha },1 < M < \frac{1}{\mathrm{\Gamma }\left(1 + \alpha \right)}$$

(71)

At the end of this section, we refer to the works of Bazzani et al. [25] (and the references therein), where linear and non-linear versions of the Langevin equation with memory effects expressed by Caputo fractional derivatives are studied. We also refer to the deep studies of Kupferman [6,7] where the Kac–Zwanzig bath model is the starting point for developing the fractional kinetic equation: see Section 4 of [7] on the solution of the generalized Langevin equation (see also Section 6.2 for generalized Langevin equations). In addition, deep studies on anomalous diffusion based on the generalized Langevin equations are available in [26,27,28,29,30,31,32].

The anomalous processes modeled by the non-local Langevin equation, regardless of which term of it the non-locality is assigned, have no stationary states, as was already mentioned regarding the kinetics of the transport modeled by the CTRW mechanism, for example. This conclusion must be made because it follows naturally from the solutions developed. Although we have different asymptotics, the lack of stationarity is a fundamental property.

2.3. Subdiffusion and Superdiffusion

Nowadays anomalous diffusion is commonly associated with the power-law with anomalous exponent $\alpha \ne 1$ and a generalized diffusion coefficient $K_{\alpha }$ with a dimension $\left[m^2/s^{\alpha }\right]$ [33,34]

$$⟨x^2\left(t\right)⟩ \cong K_{\alpha }{t}^{\alpha }$$

(72)

depending on the exponent $\alpha$ we have: subdiffusion for $0 < \alpha < 1$, normal diffusion when $\alpha = 1$, and superdiffusion for $\alpha > 1$.

The subdiffusive and superdiffusive transports can be characterized based on different waiting times (73) and jump lengths variance (74), which are finite or infinite, within the framework of the generalized CTRW transport mechanism that is

$$⟨\tau ⟩ = \underset{0}{\overset{\infty }{\int }}\tau \psi \left(\tau \right)d\tau$$

(73)

$$⟨\mathsf{\Delta }{x}^2⟩ = \underset{-\infty }{\overset{\infty }{\int }}\left(\mathsf{\Delta }{x}^2\right)\lambda \left(\mathsf{\Delta }x\right)d\left(\mathsf{\Delta }x\right).$$

(74)

With diverging moments of $⟨\tau ⟩$ and $⟨\mathsf{\Delta }{x}^2⟩$ the stochastic process is related to the Levy statistics [34,35] since following it sums of independent and identically distributed random variables with diverging moments are stable distributions.

Finite $⟨\tau ⟩$ and $⟨\mathsf{\Delta }{x}^2⟩$ can be observed in regular Brownian motion with $\alpha = 1$ where the diffusion coefficient is $K_{\alpha = 1} = ⟨\mathsf{\Delta }{x}^2⟩/\left(2⟨\tau ⟩\right)$ (see earlier Equations (22) and (23)) in the limiting sense of random walk [34].

2.3.1. Subdiffusion

With jump length sufficiently narrow distributed such that $⟨\delta x^2⟩$ is finite [34], the probability density function corresponds to Gaussian motion ($\delta$ is the Dirac delta function) or to jumps on a lattice of spacing b such that $\lambda \left(\mathsf{\Delta }x\right) = \left[\delta \left(\mathsf{\Delta }x - a\right) + \delta \left(\mathsf{\Delta }x + a\right)\right]/2$. Simultaneously, since there are no successive jumps with equal time steps supposed to appear with waiting times $\tau$ distributed asymptotically as a power law [34] for $\tau \to \infty$, with a scaling factor (characteristic time scale of the process) ${\tau }_0$

$$\psi \left(\tau \right) \simeq {\tau }_0^{\alpha }{\tau }^{-\left(1 + \alpha \right)},0 < \alpha < 1$$

(75)

Following CTRW theory, the mean squared displacement with power-law probability density function $P\left(x,t\right)$ of the waiting times (75) following (72) is subdiffusive and is governed by a fractional diffusion equation with a diffusion coefficient $K_{\alpha } = ⟨\mathsf{\Delta }{x}^2⟩/2{\tau }_0^{\alpha }$ (see also Equation (44)) in terms of the Riemann–Liouville fractional integral $D_t^{1 - \alpha }$

$$\frac{\partial P\left(x,t\right)}{\partial t} = K_{\alpha }{D}_t^{1 - \alpha }\frac{{\partial }^{2}P\left(x,t\right)}{\partial x^2}$$

(76)

2.3.2. Superdiffusion

If the waiting times are correlated in a way that at each step, they are modified by small increments (positive or negative), that is ${\tau }_i = {\tau }_{i - 1} + \mathsf{\Delta }{\tau }_i$, then this could result in a short waiting time or in a long waiting time. In the context of CTRW, the current waiting time can be presented as a sum of increments [34]: ${\tau }_i = \left|\mathsf{\Delta }{\tau }_0 + \mathsf{\Delta }{\tau }_2 + ...\mathsf{\Delta }{\tau }_i\right|$ (the absolute value means that the waiting times should be positive). Therefore, the increments $\mathsf{\Delta }{\tau }_i$ have to follow an assumed probability distribution.

A symmetric Levy stable law is considered, for instance, with a power-law mean squared displacement (72) and anomalous exponent $\alpha$ defined as [34]

$${\alpha }_{sup} = \frac{\gamma }{1 + \gamma },0 \le \gamma \le 2$$

(77)

with $\alpha \left(\gamma = 0\right) = 0$ and $\alpha \left(\gamma = 2\right) = 2/3$.

At the limit, $\gamma = 2$ we get a stretched exponential $P\left(x,t\right) \simeq exp\left(-c{t}^{1/2}\right)$, but for $1 < \alpha < 2$ the power-law $P\left(x,t\right) \simeq t^{-\alpha }$ takes place [34]. The average waiting time in the superdiffusive transport process must have an increasing function and diverges towards the end of many steps [34].

2.4. Slow and Ultraslow Diffusions

The mean square displacement of the Brownian particle motion raises linearly in time $⟨x^2\left(t\right)⟩ \simeq t$ and refers to normal diffusion, while the power-law $⟨x^2\left(t\right)⟩ \simeq t^{\alpha }$, $\alpha \ne 1$, is observed in the so-called anomalous diffusion, as several times discussed above. Now, we address physical problems where the time evolution of $⟨x^2\left(t\right)⟩$ is much slower and scales to power of the logarithm in time such as [36,37,38]

$$⟨x^2\left(t\right)⟩ \simeq {\left(logt\right)}^{\beta },\beta > 1$$

(78)

These are the cases of slow and ultraslow diffusions (also called logarithmic diffusion) [39,40,41,42,43] or strong anomaly [44] with $\beta = 4$ (when the particle motion is performed in a quenched random force field). Similar behavior has been observed in dense colloids [39], concrete relaxation [41], crack propagation in metals [34], polymer physics [45] and in other processes with complex materials [46]. Some of the systems with ultraslow performances exhibit probability density functions of the displacement as a function $\chi = x/{\left(logt\right)}^{\beta /2}$ and for large $\left|\chi \right|$ (at the wings) the decay is exponential $f\left(\chi \right) \simeq exp\left( - A\left|\chi \right|\right)$ [42].

2.4.1. Ultraslow Diffusion via Generalized Fokker–Plank Equation

The distributed Fokker–Plank equation concerning probability density function $f_u\left(x,t\right)$ in terms of the Caputo distributed order derivative [47,48] can be expressed as [42] (with a weight function $p\left(\alpha \right)$ as a dimensionless non-negative function such that the normalization condition $\underset{0}{\overset{1}{\int }}p\left(\alpha \right)d\alpha = 1$ is satisfied)

$$\underset{0}{\overset{1}{\int }}d\alpha {\tau }_u^{\alpha - 1}p\left(\alpha \right)\frac{{\partial }^{\alpha }{f}_u\left(x,t\right)}{\partial t^{\alpha }} = L_{FP}{f}_u\left(x,t\right),f_u\left(x,t\right) = \delta \left(t\right)$$

(79)

In (79) $\frac{{\partial }^{\alpha }{f}_u\left(x,t\right)}{\partial t^{\alpha }}$ is the well-known Caputo derivative (see the definitions in Appendix A.1.1), while ${\tau }_u$ is the process characteristic time (a positive constant).

$L_{FP}$ is the Fokker–Plank operator under external potential $U\left(x\right)$ (with m as particle mass and ${\gamma }_u$ as friction coefficient) defined as

$$L_{FP} = \frac{\partial }{\partial x}\frac{U^{\prime }\left(x\right)}{m{\gamma }_u} + D\frac{{\partial }^2}{\partial x^2}$$

(80)

The distributed order Fokker–Plank Equation (79) with $p\left(\alpha \right)$ as a simple power-law

$$p\left(\alpha \right) = \beta {\alpha }^{\beta - 1}$$

(81)

with $\beta > 0$ (as it follows from the normalization condition) have a solution as follows.

It is worth noting that only the case $\beta \to 0$ relates to the overall process behavior. In addition, for $p\left(\alpha \right) = \delta \left(\alpha - 1\right)$ the regular Fokker–Plank equation is recovered from (79). Further, $p\left(\alpha \right) = \delta \left(\alpha - {\alpha }_0\right)$, $0 < {\alpha }_0 < 1$ results is (79). The solution, through the Laplace-Fourier transform, yields asymptotics for small and long times [42]

$$f_u\left(x,t\right) = \frac{\beta }{4D}\frac{1}{\sqrt{tln\left(\frac{t}{{\tau }_u}\right)}}exp\left[-\sqrt{\frac{\beta }{D}}\frac{\left|x\right|}{\sqrt{tln\left(\frac{t}{{\tau }_u}\right)}}\right],\frac{t}{{\tau }_u} \ll 1$$

(82)

$$f_u\left(x,t\right) \approx \frac{1}{\sqrt{4D{\tau }_u}}\sqrt{\frac{\mathrm{\Gamma }\left(\beta + 1\right)}{{\left(ln\frac{t}{{\tau }_u}\right)}^{\beta }}}exp\left[-\sqrt{\frac{\mathrm{\Gamma }\left(\beta + 1\right)}{D{\tau }_u}}\frac{\left|x\right|}{\sqrt{{\left(ln\frac{t}{{\tau }_u}\right)}^{\beta }}}\right],\frac{t}{{\tau }_u} \gg 1$$

(83)

These results allow the mean squared displacement asymptotics to be established [42]

$$⟨\left(x{\left(t\right)}^2\right)⟩\propto \left\{\begin{array}{cc}\hfill & \frac{2D}{\beta }tln\left(\frac{t}{{\tau }_u}\right),\frac{t}{{\tau }_u} \ll 1\hfill \\ \hfill & \frac{2D{\tau }_u}{\mathrm{\Gamma }\left(\beta + 1\right)}{\left[ln\left(\frac{t}{{\tau }_u}\right)\right]}^{\beta },\frac{t}{{\tau }_u} \gg 1\hfill \end{array}\right.$$

(84)

2.4.2. Ultraslow Diffusion via a Structured Derivative Model

The structural derivative model proposed in [40] develops the idea that the general ultraslow diffusion can be presented by the scaling relationship

$$⟨x^2\left(t\right)⟩ \simeq {\left[E_{\alpha }^{-1}\left(t\right)\right]}^{\beta },0 < \alpha < 1$$

(85)

where $E_{\alpha }^{-1}\left(t\right)$ is the inverse of the Mittag-Leffler function $E_{\alpha }\left(t\right)$ (see the definition in Appendix A.1.4).

Recall, that the exponential function $exp\left(t\right)$ is the special case of $E_{\alpha }\left(t\right)$ for $\alpha = 1$, thus, $E_{\alpha }^{-1}\left(t\right)$ is a generalization of the logarithmic function $ln\left(t\right)$. The inverse Mittag-Leffler function $E_{\alpha }^{-1}\left(t\right)$ increases slower than the logarithmic function and recovers the faster scaling relationship (78) for $\alpha = 1$.

Chen et al. [39,40] proposed a diffusion model

$$\frac{\partial p\left(x,t\right)}{\partial t} = { }_0{K}_t^{1 - \alpha }\left[D_{\alpha }\frac{{\partial }^{2}p\left(x,t\right)}{\partial x^2}\right],t > 1,-\infty < x < \infty$$

(86)

with structure fractional operator (derivative) with a pseudo-Riemann–Liouville construction (see the Note below for explanations) [39,40]

$${}_0{K}_t^{1 - \alpha }\left[p\left(x,t\right)\right] = \frac{d}{dt}\underset{0}{\overset{t}{\int }}k\left(t - s\right)p\left(x,s\right)ds$$

(87)

The kernel $k\left(t\right)$ is a structural function constructed on the basis of (78) with some assumptions: from (86) it follows that [39,40]

$$\frac{d}{dt}\left[⟨x^2\left(t\right)⟩\right] = {}_0{K}_t^{1 - \alpha }2D_{\alpha }$$

(88)

and

$${ }_0{K}_t^{1 - \alpha } = \frac{d}{dt}{\left(ln{t}^{\alpha }\right)}^{\beta }$$

(89)

From (89), by the Laplace transform, it follows that the structural function (i.e., the memory kernel) [40] is

$$k\left(t\right) = \beta \frac{\alpha {\left(ln{t}^{\alpha }\right)}^{\beta - 1}}{t}$$

(90)

and that

$$⟨x^2\left(t\right)⟩ \simeq \underset{0}{\overset{t}{\int }}k\left(s\right)ds,0 < \alpha < 1,t > 0$$

(91)

For $\alpha = 1$ we have $⟨x^2\left(t\right)⟩ \simeq k\left(t\right)$.

Thus, the structural function $k\left(t\right)$ for the faster logarithmic law is [39,40]

$$k_{log}\left(t\right) = \beta \frac{{\left(lnt\right)}^{\beta - 1}}{t}$$

(92)

while for the slower inverse Mittag-Leffler law is [39,40]

$$k_E\left(t\right) = \beta {\left[E_{\alpha }^{-1}\left(t\right)\right]}^{\beta - 1}\frac{d}{dt}\left[E_{\alpha }^{-1}\left(t\right)\right]$$

(93)

The diffusion model (86) has no analytical solution [39,40]

Note: The structural derivative in time (denoted as $\frac{S}{st}$) is defined as [40]

$$\frac{Sp\left(x,t\right)}{st} = {lim}_{t_1\to t}\frac{p\left(x,t_1\right) - p\left(x,t\right)}{k\left(t_1\right) - k\left(t\right)}$$

(94)

For $k\left(t\right) = t$, we get the classical (local) derivative, but for $k\left(t\right) = t^{\alpha }$ we have a local fractal derivative.

The definition (94) is local in time because a memory effect through a convolution is not involved. For the ultraslow diffusion, the structural derivative equation model is formulated with $k\left(t\right) = E_{\alpha }^{-1}\left(t\right)$ as [40]

$$\frac{Sp\left(x,t\right)}{st} = {lim}_{t_1\to t}\frac{p\left(x,t_1\right) - p\left(x,t\right)}{E_{\alpha }^{-1}\left(t_1\right) - E_{\alpha }^{-1}\left(t\right)}$$

(95)

3. Fractional Kinetics Formalism

The dynamics of some systems can be presented as the time evolution of a dynamic variable $X\left(t\right)$ as

$$\frac{dX\left(t\right)}{dt} = \mathrm{\Omega }\left(t\right),\mathrm{\Omega }\left(t\right) = \sum _k{\omega }_k$$

(96)

Here $\mathrm{\Omega }\left(t\right)$ is a sum of driving forces depending on the environment where the modeled dynamic process occurs. As in the preceding section, the driving forces could be any stochastic noise, concentration potential, birth or death of particles, etc. In some specific contexts, (96) can be considered a Liouville equation [49] allowing the build-up of a Fokker–Plank construction upon assumed conditions of a thermal bath. We will not extend this specific topic here except to emphasize that the formal approach to constructing fractional kinetic equations considered further in this section has its background.

3.1. Local Kinetic Equation: Well-Known Background

In the beginning, we start with the well-known integer-order (local) kinetic equation of zero-order

$$\frac{d}{dt}N\left(t\right) = -c_0,c_0 > 0,N_0 = N\left(t = 0\right)$$

(97)

with a solution

$$N\left(t\right) = N_0 - c_{0}t$$

(98)

Further, the first-order local kinetic equation

$$\frac{d}{dt}N\left(t\right) = -c_{0}N\left(t\right),c_0 > 0,N_0 = N\left(t = 0\right)$$

(99)

with a solution

$$\underset{0}{\overset{t}{\int }}\frac{dN\left(t\right)}{N\left(t\right)}dt = -\underset{0}{\overset{t}{\int }}{c}_{0}dt\Rightarrow N\left(t\right) = N_{0}exp\left(-c_{0}t\right)$$

(100)

3.2. Non-Local Kinetics with Singular Memories

Let us consider again (99) and integrate it, but presenting it in a general manner, as it was considered in [1,50,51,52]

$$N\left(t\right) = N_0 - c_0{D}_t^{-1}N\left(t\right)$$

(101)

With ${ }_0{D}_t^{-1}$ denoting the standard Riemann integral (the same that (100)) yields the exponential function. Then, the introduction of non-locality has been done only by replacing ${ }_0{D}_t^{-1}$ with ${ }_0{D}_t^{-\alpha }$ where $0 < \alpha < 1$, in (101). The final result is

$$N\left(t\right) = N_{0}f\left(t\right) - c_0{D}_t^{-\alpha }N\left(t\right)$$

(102)

This approach to some extent casts doubts because such replacement of ${ }_0{D}_t^{-1}$ by ${ }_0{D}_t^{-\alpha }$ is possible if the initial form of the kinetic equation is written as

$$\frac{{\partial }^{\alpha }N\left(t\right)}{\partial t^{\alpha }} = -c_{\alpha }\left(t\right)\Rightarrow N\left(t\right) = -c_{\alpha }{I}^{\alpha }N\left(t\right)\Rightarrow N_0{E}_{\alpha }\left(-c_{\alpha }{t}^{\alpha }\right)$$

(103)

where $D_t^{-\alpha }N\left(t\right) = I^{\alpha }N\left(t\right)$ is the Riemann–Liouville fractional integral (see the definition in Appendix A.1.1).

As in the preceding section, the problems developed here envisage only fractional derivatives with singular (power-law) memory kernels used to model the time non-locality. Now, we like to see what the solutions will be for different functions $N\left(t\right) = c_{0}f\left(t\right)$ in the right-hand side of the kinetic equation.

3.3. Zero-Order Kinetics

If we get the time-fractional analog of the zero-order kinetic Equation (97) we have

$$\frac{{\partial }^{\alpha }N\left(t\right)}{\partial t^{\alpha }} = -c_0\Rightarrow -\frac{c_0}{\mathrm{\Gamma }\left(\alpha + 1\right)}{t}^{\alpha }$$

(104)

However, in the integral version (102) the case with $N_0\left(t\right) = N_{0}f\left(t\right) = N_0$, i.e., with $f\left(t\right) = 1$ [50,51] yields a solution through the one-parameter Mittag-Leffler function

$$N\left(t\right) = N_0\sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k{\left(ct\right)}^k}{\mathrm{\Gamma }\left(\alpha k + 1\right)} = N_0{E}_{\alpha }\left(-c_{\alpha }{t}^{\alpha }\right)$$

(105)

Bearing in mind that $E_{\alpha }\left(-c^{\alpha }t = 0\right) = 1$, then (105) is a physically correct solution of (102). However, the principal question is: Why the results (104) and (105) are different? In other words, is the integral form of the kinetic equation (102) physically correct? Mathematically, the solution (105) is correct, but the model formulated by replacing the integral is a violation of physics. Some examples with solutions will be considered next.

3.4. Time-Dependent Power-Law Initial Condition

Now, let us consider some examples existing in the literature when the integral equation contains $N_{0}f\left(t\right)$

Power-law decaying $N_0\left(t\right)$

If the initial condition is time-dependent and represents a decaying power-law $N_0\left(t\right) = N_0{t}^{\mu - 1}$ then the integral Equation (102) becomes

$$N\left(t\right) - N_0{t}^{\mu - 1} = -{c^{\alpha }}_0{D}_t^{-\alpha }N\left(t\right)$$

(106)

with a solution is [50,51]

$$N\left(t\right) = N_0\mathrm{\Gamma }\left(\alpha \right)t^{\mu - 1}{E}_{\alpha ,\mu }\left(-c^{\alpha }{t}^{\alpha }\right)$$

(107)

For $\mu = 1$ we get the zero-order case (104) and the solution (105)).

However, let the kinetic equation will be re-written in a differential form

$$\frac{{\partial }^{\alpha }N\left(t\right)}{\partial t^{\alpha }} = N_0{t}^{\beta }$$

(108)

Then, with integration we get

$$N\left(t\right) = \frac{\mathrm{\Gamma }\left(\alpha + 1\right)}{\mathrm{\Gamma }\left(\alpha + \beta + 1\right)}{t}^{\alpha + \beta }$$

(109)

With $\beta = \mu - 1$ (109) yields

$$N\left(t\right) = \frac{\mathrm{\Gamma }\left(\alpha + 1\right)}{\mathrm{\Gamma }\left(\alpha + \mu \right)}{t}^{\alpha + \mu - 1}$$

(110)

The difference between (107) and (110) is obvious, and the origin of this is the starting equations solved. Concerning the integral form of the kinetic Equation (102) which dominates in the literature and especially the case with $N_0\left(t\right) = N_0{t}^{\mu - 1}$, we refer to the correct solution of Tomoski et al. [53]

$$\begin{array}{cc}\hfill & N\left(t\right) = N_0\left[f\left(t\right) + \sum _{n = 1}^{\infty }\frac{{\left(-1\right)}^n{\left(c^{\alpha }\right)}^n}{\mathrm{\Gamma }\left(\alpha n\right)}\left(t^{n\alpha - 1} \ast f\left(t\right)\right)\right] = \hfill \\ \hfill & N_0\left[f\left(t\right) + \sum _{n = 1}^{\infty }{\left(-1\right)}^n{\left(c^{\alpha }\right)}^n{D}_t^{-n\alpha }f\left(t\right)\right]\hfill \end{array}$$

(111)

The second term (the summation) for $t = 0$ is zero, thus, indicating that the initial condition defined by $N\left(t = 0\right) = N_{0}f\left(t\right)$ is correct.

3.5. Time-Dependent (Decaying Tempered Mittag-Leffler Function) Initial Condition

The solution of the integral form of the kinetic equation [52]

$$N\left(t\right) - N_0{t}^{\mu - 1}{E}_{\alpha ,\mu }^{\gamma }\left(-c^{\alpha }{t}^{\mu }\right) = -{c^{\alpha }}_0{D}_t^{-\alpha }N\left(t\right),\alpha > 0,c > 0,\mu > 0$$

(112)

is

$$N\left(t\right) = N_0{t}^{\mu - 2}{E}_{\alpha ,\mu - 1}^{\gamma + 1}\left(-c^{\alpha }{t}^{\alpha }\right)$$

(113)

Here $E_{\alpha ,\mu }^{\gamma }\left(z\right)$ is a generalized Mittag-Leffler function defined as

$$\begin{array}{cc}\hfill & E_{\alpha ,\mu }^{\gamma }\left(z\right) = \sum _{n = 0}^{\infty }\frac{{\left(\gamma \right)}_n{z}^n}{\mathrm{\Gamma }\left(\alpha n + \mu \right)n!},\hfill \\ \hfill & {\left(\gamma \right)}_n = \frac{\mathrm{\Gamma }\left(\gamma + n\right)}{\mathrm{\Gamma }\left(\gamma \right)} = \left\{\begin{array}{c}1,\left(n = 0,\gamma \ne 0\right)\hfill \\ \gamma \left(\gamma + 1\right)...\left(\gamma + n - 1\right),\left(n\in N,\gamma \in C\right)\hfill \end{array}\right.\hfill \end{array}$$

(114)

where ${\left(\gamma \right)}_n$ is Pochammer’s symbol.

Further, with $f\left(t\right) = t^{\mu - 1}{E}_{\alpha ,\mu }\left(-c^{\alpha }{t}^{\mu }\right)$ in (102) we get

$$N\left(t\right) - N_0{t}^{\mu -1}{E}_{\alpha ,\mu }\left( - c^{\alpha }{t}^{\mu }\right) = -{c^{\alpha }}_0{D}_t^{-\alpha }N\left(t\right),\alpha > 0,c > 0,\mu > 0$$

(115)

and the solution is

$$N\left(t\right) = N_0\frac{t^{\mu - 1}}{\alpha }\left[E_{\alpha ,\mu - 1}\left(-c^{\alpha }{t}^{\alpha }\right) - \left(1 - \mu - \alpha \right)E_{\alpha ,\mu - \alpha }\left(-c^{\alpha }{t}^{\alpha }\right)\right]$$

(116)

In the case with $\gamma = 2$ the equation

$$N\left(t\right) - N_0{t}^{\mu - 1}{E}_{\alpha ,\mu }^2\left(-c^{\alpha }{t}^{\mu }\right) = -{c^{\alpha }}_0{D}_t^{-\alpha }N\left(t\right),\alpha > 0,c > 0,\mu > 0$$

(117)

has a solution

$$N\left(t\right) = N_0\frac{t^{\mu - 1}}{\alpha }{E}_{\alpha ,\mu }^3\left( - c^{\alpha }{t}^{\alpha }\right)$$

(118)

3.6. Generalized Fractional Kinetic Differintegral Equation

The following equation has been considered by Saxena et al. [52] and Tomovski et al. [53]

$$a{D}_{0 + }^{\gamma ,\beta }N\left(t\right) - N_{0}f\left(t\right) = b{D}_t^{-\alpha }N\left(t\right),f\left(t\right)\in L\left(0,\infty \right)$$

(119)

under initial condition ${ }_0{D}_t^{\left(1 - \beta \right)\left(1 - \alpha \right)}f\left(0+\right) = c$ (with a, b and c constants).

The initial condition, for instance, to a greater extent, coincides with formulations of the problem solved by (108) and (110). The solution of (119) is [53]

$$N\left(t\right) = \frac{N_0}{a}{\sum _{n = 0}^{\infty }\left(\frac{a}{b}\right)}^n\frac{t^{\gamma + n\left(\alpha + \gamma \right) - 1} \ast f\left(t\right)}{\mathrm{\Gamma }\left(\gamma + n\left(\alpha + \gamma \right)\right)} + c{\sum _{n = 0}^{\infty }\left(\frac{a}{b}\right)}^n\frac{t^{\gamma - \beta \left(1 - \gamma \right) + n\left(\alpha + \gamma \right) - 1}}{\mathrm{\Gamma }\left(\gamma - \beta \left(1 - \gamma \right) + n\left(\alpha + \gamma \right)\right)}$$

(120)

However, a natural question arises: why the non-locality should exist in both sites of (119), and what is the meaning, either physical or mathematical, of this formulation? There is no answer in [53]. We may consider all these brilliant calculations as not physically based because the appearance of non-locality in the integral form of the so-called fractional kinetic equation is not motivated. The fact that the Riemann integral in (101) is mechanically replaced by the fractional Riemann–Liouville integral cannot be accepted as a constitutive equation of fractional kinetics, but as an imitation of the master equation, for example, or it cannot be related to the differential form, from where it originated mechanistically.

4. Fractional Kinetic of Dissolution and Sorption

We now discuss some real cases from everyday life that show the feasibility of fractional calculus. These models are not as complex as the ones that have been examined thus far, but as we all know, sometimes more straightforward designs may convey concepts more clearly than more sophisticated ones.

4.1. Pharmacokinetics: Drug Absorption and Release

This section refers to real applications of the fractional kinetic equation to dynamic modeling of drug absorption and disposition, common problems in pharmacodynamics [54,55]. Let us consider the classical zero-order kinetic equation (expressed here in terms of concentrations).

4.1.1. Basic Kinetic Models

$$\frac{dC}{dt} = k_0\Rightarrow C\left(t\right) = k_{0}t,C(t = 0) = 0$$

(121)

with a linear solution and the constant $k_0$ has a dimension (concentration/time).

Its fractional counterpart

$$\frac{{\partial }^{\alpha }C}{\partial t^{\alpha }} = k_0\Rightarrow C\left(t\right) = \frac{k_0}{\mathrm{\Gamma }\left(\alpha + 1\right)}{t}^{\alpha }$$

(122)

reduces to (121) for $\alpha = 1$, but now the constant $k_0$ has a dimension $\mathit{concentration}/{\left(\mathit{time}\right)}^{\alpha }$

Further, the first-order kinetics

$$\frac{dC}{dt} = k_{1}C\Rightarrow C\left(t\right) = C_{0}exp\left(-k_{1}t\right)$$

(123)

Its fractional counterpart has a solution through the one-parameter Mittag-Leffler function

$$\frac{{\partial }^{\alpha }C\left(t\right)}{\partial t^{\alpha }} = k_{1}C\left(t\right)\Rightarrow C\left(t\right) = C_0\sum _{n = 0}^{\infty }\frac{{\left(-k_{1}t\right)}^n}{\mathrm{\Gamma }\left(\alpha n + 1\right)} = C_0{E}_{\alpha }\left( - k_{1}t\right)$$

(124)

The same kinetic model has been successfully applied to model anomalous luminescence decay process [56].

4.1.2. Drug Release Modeling

For drug release, the fractional form of the Noyes-Whitney equation [54,55] is

$$\frac{{\partial }^{\alpha }\left(C_s - C\right)}{\partial t^{\alpha }} = -k_{dis}\left(C_s - C\right),0 < \alpha < 1$$

(125)

where the driving force is the difference saturation concentration of the solution $C_s$ and its current value; $k_{dis}$ is a mass transfer coefficient.

Equation (125) is equivalent to (122) since it is easy to change variables as $\theta = \left(C_s - C\right)$. Then, the solution is

$$C\left(t\right) = C_s\left[1 - E_{\alpha }\left( - k_{dis}{t}^{\alpha }\right)\right]$$

(126)

Additional information with many examples of fractional calculus applications in pharmacokinetics are available elsewhere [57,58,59].

4.2. Pharmacokinetics:Compartmental Models

4.2.1. Pharmacokinetics: Single Compartment Dynamics

The simplest pharmacokinetic rate equation for drug concentration elimination (in vitro bolus injection) [54,55] is of first-order and its fractional versions (see (124) and (125)), and corresponds to a one-compartment model with a solution in terms of the Mittag-Leffler function

$$\frac{{\partial }^{\alpha }C}{\partial t^{\alpha }} = k_{el}C\left(t\right)\Rightarrow C\left(t\right) = C_0{E}_{\alpha }\left(-k_{el}{t}^{\alpha }\right)$$

(127)

Here $C_0$ is the ratio $\left(dose\right)/\left(apparentvolumeofdistribution\right)$.

It is easy to see that for small times, the solution approaches the stretched exponential function $C\left(t\to 0\right)\propto exp\left(-k_{el}{t}^{\alpha }\right)$ coming from the basic properties of the Mittag-Leffler function; for $\alpha = 1$ we get the integer-order kinetics with the conventional exponential function. In contrast, for long times, $C\left(t\to \infty \right) \equiv t^{-\alpha }$ comes from the asymptotic behavior of the Mittag-Leffler function. Numerical examples are available elsewhere [54,55].

4.2.2. Pharmacokinetics: Multi-Compartment Systems

Now, looking at the multi-compartmental systems, we focus on a simple model (two compartments) where the model-build-up, precisely the fractionalization, is developed physically correctly, and this approach may serve as an instructive example of how this should be done. Hence, the set of equations addressing the kinetic of the system of two compartments [55], formulated in a classical manner (local model), commonly appearing in the textbooks, is

$$\begin{array}{cc}\hfill & \frac{d{A}_1\left(t\right)}{at} = -k_{12}{A}_1\left(t\right) + k_{21}{A}_2\left(t\right) - k_{10}{A}_1\left(t\right) + I_1\left(t\right)\hfill \\ \hfill & \frac{d{A}_2\left(t\right)}{at} = k_{12}{A}_1\left(t\right) - k_{21}{A}_2\left(t\right) - k_{20}{A}_2\left(t\right) + I_2\left(t\right)\hfill \end{array}$$

(128)

Here $A_1\left(t\right)$ and $A_2\left(t\right)$ are masses in the compartments controlled by mass transfer coefficients $k_{i,j}$ ($i = 1,2;j = 1,2$) with dimensions $\left[tim{e}^{-1}\right]$. $I_1\left(t\right)$ and $I_2\left(t\right)$ are the input rates for each compartment, while $k_{10}$ and $k_{20}$ are coefficients related to the elimination from the corresponding compartments.

Commonly, without a deep understanding of how the non-locality (fractionalization) has been introduced, the time –derivatives in (128) are mechanistically replaced by fractional counterparts based on any type of memory kernels. This is a completely wrong approach, and we can see how this should be done correctly. The system (128) can be transformed into a set of integral equations

$$\begin{array}{cc}\hfill & A_1\left(t\right) - A_1\left(0\right) = -k_{12}\underset{0}{\overset{t}{\int }}{A}_1\left(\tau \right) + k_{21}\underset{0}{\overset{t}{\int }}{A}_2\left(\tau \right) - k_{10}\underset{0}{\overset{t}{\int }}{A}_1\left(\tau \right) + \underset{0}{\overset{t}{\int }}{I}_1\left(\tau \right)\hfill \\ \hfill & A_2\left(t\right) - A_2\left(0\right) = k_{12}\underset{0}{\overset{t}{\int }}{A}_1\left(\tau \right) - k_{21}\underset{0}{\overset{t}{\int }}{A}_2\left(\tau \right) - k_{20}\underset{0}{\overset{t}{\int }}{A}_1\left(\tau \right) + \underset{0}{\overset{t}{\int }}{I}_2\left(\tau \right)\hfill \end{array}$$

(129)

Now, the form (128) allows two interpretations:

• The integrals in (129) can be considered as convolution integrals with Dirac delta function as memory kernel; that is, there is no non-locality.

• Since the suggestion in point 1 is unrealistic because any compartment has a finite time to react and there is no instantaneous response, then we may suggest that the integrals in (129) can be assigned specific memory kernels to greater extent approximating the relaxation functions of the compartments.

Hence, Equation (129) can be expressed as

$$\begin{array}{cc}\hfill & A_1\left(t\right) - A_1\left(0\right) = -k_{12}\underset{0}{\overset{t}{\int }}{G}_{12}\left(t,\tau \right)A_1\left(t,\tau \right) + k_{21}\underset{0}{\overset{t}{\int }}{G}_{21}\left(t,\tau \right)A_2\left(\tau \right) - \hfill \\ \hfill & - k_{10}\underset{0}{\overset{t}{\int }}{G}_{10}\left(t,\tau \right)A_1\left(\tau \right) + \underset{0}{\overset{t}{\int }}{I}_1\left(\tau \right)\hfill \\ \hfill & A_2\left(t\right) - A_2\left(0\right) = k_{12}\underset{0}{\overset{t}{\int }}{G}_{12}\left(t,\tau \right)A_1\left(\tau \right) - k_{21}\underset{0}{\overset{t}{\int }}{G}_{21}\left(t,\tau \right)A_2\left(\tau \right) - \hfill \\ \hfill & - k_{20}\underset{0}{\overset{t}{\int }}{G}_{10}\left(t,\tau \right)A_1\left(\tau \right) + \underset{0}{\overset{t}{\int }}{I}_2\left(\tau \right)\hfill \end{array}$$

(130)

The suggestion in the second point by a kernels [55] $G_{ij} = \frac{{\left(t - \tau \right)}^{{\alpha }_{i,j}-1}}{\mathrm{\Gamma }\left({\alpha }_{i,j}\right)}$ transforms the integral in the right-hand side of (130) into Riemann–Liouville integrals. Now, a consequent differentiation (local) concerning time transforms each of these fractional integrals into Riemann–Liouville fractional derivatives of $A_{ij}\left(t\right)$, namely

$$\begin{array}{cc}\hfill & \frac{d{A}_1\left(t\right)}{dt} = -k_{12}{D}_t^{1 - {\alpha }_{12}}{A}_1\left(t\right) + k_{21}{D}_t^{1 - {\alpha }_{21}}{A}_2\left(t\right) - k_{10}{D}_t^{1 - {\alpha }_{10}}{A}_1\left(t\right) + I_1\left(t\right)\hfill \\ \hfill & \frac{d{A}_2\left(t\right)}{dt} = k_{12}{D}_t^{1 - {\alpha }_{12}}{A}_1\left(t\right) - k_{21}{D}_t^{1 - {\alpha }_{21}}{A}_2\left(t\right) - k_{20}{D}_t^{1 - {\alpha }_{10}}{A}_1\left(t\right) + I_2\left(\tau \right)\hfill \end{array}$$

(131)

Therefore, we got a form where the non-localities are only on the right-hand side of the governing equation, thus, following the general principle of the causality of missing instantaneous response to the reason causing the transient transport process (in this case, there is a mass transport)-see the general formulations (1) and (5) at the very beginning.

The initial step of fractionalization, i.e., the transformation from the differential to integral equations and the consequent suggestion of memory kernels, naturally leads to fractional integrals, precisely of the Riemann–Liouville type. Moreover, the differentiation (the transition from (130) to (131)) yields Riemann–Liouville derivatives. They can be replaced by Caputo-derivatives, as was discussed in [55] if the initial conditions are zero. However, if the model (128), only as a thought experiment, in its right-hand side, includes time derivatives $d{A}_{ij}/dt$ the integration (the step (130)) and the introduction of adequate memory kernels different from the power law will yield different fractional kinetic models (we will skip this discussion at this point).

The model (131) can be simplified if we assume that not all integral terms in (130) have memory kernels different from the Dirac delta function. This simply means that at some point (compartment), the instantaneous response is an acceptable approximation if its time scale is negligible concerning the time scale of another point (compartment) where the relaxation (time delay) should be taken into account. Concerning the model (131), there are possibilities for simplifications (see details and explanations in [55]) that yield

$$\begin{array}{cc}\hfill & \frac{d{A}_1\left(t\right)}{dt} = \left[-k_{12} - k_{10}{A}_1\left(t\right)\right]A_1\left(t\right) + {k_{21}}^C{D}_t^{1 - {\alpha }_{21}}{A}_2\left(t\right) + I_1\left(t\right)\hfill \\ \hfill & \frac{d{A}_2\left(t\right)}{dt} = k_{12}{A}_1\left(t\right) - k_{21}{ }^C{D}_t^{1 - {\alpha }_{21}}{A}_2\left(t\right)\hfill \end{array}$$

(132)

In (132), the fractional operators are Caputo derivatives with the assumption that $A_1\left(t = 0\right) = A_2\left(t = 0\right) = 0$. The example discussed in this point is instructive since it clearly shows how fractionalization should be done. This is not a mechanistic step, since a priori information about the relaxations (response function) of the compartments should be known from experiments and what are the suitable memory functions that could approximate them. This is a large discussion that is beyond the scope of this article, but the problem touched on are important for creating adequate fractional models. In addition, we may refer to [60], where a similar two-compartmental model (in terms of Caputo derivatives) of biological systems is analyzed and solved. Fruitful commentaries on the fractionalization of multi-compartmental models are available elsewhere [61].

4.3. Fractional Kinetics of Decomposition

The wide developments and applications of new complex materials based on multi-component structures are challenging trends in modern science and technologies. Diffusive processes in technologies developing such complex materials are under intensive studies [62,63]. In supersaturated solid solutions, where there is a metastable state with limited solubility of one or more components, there is a deviation from the equilibria. As a result, diffusive decompositions with the formations and growth of new phases force the systems into transitions, and therefore, time-dependent processes take places [64,65]. Diffusive decompositions of multi-component solid solutions often perform themselves in a subdiffusive manner [66,67,68].

In the context of the CTRW transport theory, the jump length distribution can be expanded as a Taylor series, and the Green function is a solution of the non-local advection-diffusion equation [66,69]

$$\frac{\partial p\left(x,t\right)}{\partial t} = \underset{0}{\overset{t}{\int }}K\left(t - \tau \right)\left[-\lambda \frac{\partial p\left(x,\tau \right)}{\partial x} + \eta \frac{{\partial }^{2}p\left(x,\tau \right)}{\partial x^2}\right] + p\left(x,0\right)\delta \left(t\right)$$

(133)

With a power-law density function of the waiting times, $\psi \left(t\right) = P\left(\theta .t\right) \simeq A\frac{t^{-\alpha }}{\mathrm{\Gamma }\left(1 - \alpha \right)}$, where A is a scale parameter, we get an asymptotic diffusion equation [16] (in terms of the Riemann–Liouville fractional derivative and with diffusion coefficient $D_{\alpha }$)

$${ }^{RL}{D}_t^{\alpha }p\left(x,t\right) - D_{\alpha }\frac{{\partial }^{2}p\left(x,t\right)}{\partial x^2} = p\left(x,0\right)\frac{t^{-\alpha }}{\mathrm{\Gamma }\left(1 - \alpha \right)}$$

(134)

Fractional Kinetics of Decomposition: An Example

A Stefan problem of an isolated spherical precipitate of radius R in an infinite matrix in the case of subdiffusion-controlled dissolution or growth is formulated as [66]

$$\frac{\partial C}{\partial t} = {D_{\alpha }}^{RL}{D}_t^{1 - \alpha }\left[\frac{{\partial }^{2}C}{\partial r^2} + \frac{2}{r}\frac{\partial C}{\partial r}\right],C\left(R,t\right) = C_l,C\left(\infty ,t\right) = C_m,t > 0$$

(135)

with initial conditions $C\left(r,0\right) = C_m$, $r > R$, where $C_m$ is the concentration in the matrix), and a boundary condition at the moving front

$$\left(C_p - C_l\right)\frac{dR}{dt} = D_{\alpha }{ }^{RL}{D}_t^{1 - \alpha }{\left(\frac{\partial C}{\partial r}\right)}_{r = R}$$

(136)

Under the assumption of slow motion of the interface (the precipitate boundary), that is when $\left|C_m - C_l\right| \ll C_p - C_l$ [70] its possible to assume that $R = const.$ (stationary interface approach). Next, a change of variables as $u\left(r,t\right) = r\left[C\left(r,t\right) - C_m\right]$ (a classical transform from spherical to rectangular geometry) yields one-dimensional fractional diffusion equation [66]

$$\frac{\partial u}{\partial t} = D_{\alpha }\left[{ }^{RL}{D}_t^{1 - \alpha }\frac{{\partial }^{2}u}{\partial r^2}\right]$$

(137)

with boundary conditions: $u\left(r,t\right) = R\left(C_l - C_m\right)$ and $u\left(\infty ,t\right) = 0$, $t > 0$.

With the initial condition $u\left(r,0\right) = 0$ for $r > R$ the model (137) can be expressed as

$${ }^{RL}{D}_t^{1 - \alpha }u = D_{\alpha }\frac{{\partial }^{2}u}{\partial r^2}$$

(138)

With the help of Laplace transform, the solution is

$$C\left(r,t\right) = C_m + \frac{R\left(C_l - C_m\right)}{r}{G}_{ + }^{\alpha /2}\left[t{\left(\frac{r - R}{\sqrt{D_{\alpha }}}\right)}^{-2/\alpha }\right]$$

(139)

where $G_{ + }^{\gamma }\left(t\right)$ is the distribution function of the one-side Levy density $g_{ + }^{\gamma }\left(t\right)$ with characteristic exponent $\gamma$.

For $\alpha = 1$ and $D_{\alpha } = D$ the solution reduces to the integer-order solution [70]

$$C\left(r,t\right) = C_m + \frac{R\left(C_l - C_m\right)}{r}erfc\left(\frac{r - R}{2\sqrt{Dt}}\right)$$

(140)

5. Allometric Fractional Calculus Applications

5.1. Allometry: What Is This?

Allometry is defined as a different measure arising from the scaling of part of bodies (human or animal) with a certain organism or among a group of species [71]. There is no deep theoretical background in the allometric scaling relationships where the common form is between the dependent variable Y observable with a growing rate $\theta$ and X is the measure of the independent variable with a growing rate $\gamma$, then the basic relationship of the allometry, as an ordinary differential equation, is [71,72,73,74,75,76,77,78]

$$\frac{dX}{\gamma X} = \frac{dY}{\gamma Y}$$

(141)

Integrating (141) we directly get

$$Y = a{X}^b,b = \gamma /\theta$$

(142)

and a is an empirical constant (also known as allometric constant), and commonly $b < 1$ The empirical derivation of (142) is based mainly on linear regression fittings of observed data through logarithmic transformation

$$lnY = lna + blnX$$

(143)

The modeling approaches in allometry follow two main directions [71]:

(1) A statistical strategy that uses residual analysis to interpret statistical patterns and discover the reasons for variances in allometric relationships, and

(2) A reductionist approach explaining specific values of allometric relations based on an assumed form of the underlying mechanism of the complex system performance (Reductionism is defined as any of several related philosophical ideas regarding the associations between phenomena which can be described in terms of other simpler or more fundamental phenomena [79]. It could be considered as a philosophical viewpoint that explanations of complex systems could be done as sums of actions of their parts.).

Here, we will avoid deep analysis of allometric relationships (see for this [71]) and will stress the attention on the fact that (142) can be considered as a fractal scaling, allowing to some extent fractional calculus to be applied. Here, we refer to Barenblatt [24] that such scaling relationships cannot appear by chance, but they always reveal self-similarities of the modeled phenomena.

An interesting point in allometric scaling is the interpretation of time. This is not the chronological time (earlier, concerning the time in the fractional operators, it was termed instant time [11]), but the intrinsic time of the process (the time of duration, this was mentioned in the Introduction) [71].

5.2. Allometry: Towards a Fractional Equation

Let consider that (142) as a result of the following fractional equation

$$\frac{d^{\alpha }Y}{d{X}^{\alpha }} = c\Rightarrow Y = a{X}^{\alpha },a = \frac{c}{\mathrm{\Gamma }\left(\alpha + 1\right)},0 < \alpha < 1$$

(144)

where $\frac{d^{\alpha }Y}{d{X}^{\alpha }}$ could be either Riemann–Liouville or Caputo derivative.

Recall that (144) has the same form as the fractional kinetic equation of zero-order. If $\alpha = 1$, then the results will be $Y = cX$, and we will lose the fractal scaling. Moreover, the independent variable X should be considered continuous (such as the time), and that, to a great extent, is an approximation since the experimental data always are taken at discrete points.

It is worth noting that fractal and fractional modelings are too different issues since in the former there is no memory mechanism, even though the outputs (the final results) have similar forms, but in general the underlying physics are different. Despite this, we will apply fractional kinetic formalism in the analysis of the following example.

5.2.1. A simple Fractional Model of Allometric Scaling Laws: An Example

Zhao et al. [80] interpreted (142) as an opportunity to use a fractional calculus technique to (144) with the Caputo derivative. Experimental data from [81] have been used. With this data set, a particular scaling yields that the allometric relationship should be in the form $Y\left(t\right) \equiv t^b$: the independent variable is the time t, and Y is a mass (see details in [81]).

For better fitting through the model (144), Zhao et al. [80] suggested that the constant c should be replaced by a function of time defined as $f_A\left(t\right) = \mathrm{\Gamma }\left(\alpha + k + 1\right)\left(a_0 + a_{1}t\right)$ and consequently the governing equation is

$${ }^C{D}_t^{\alpha }Y\left(t\right) = f_A\left(t\right),\alpha \in \left(0,1\right),Y\left(0\right) = 0$$

(145)

For $k = 1$, we get $f_A\left(t\right) = \mathrm{\Gamma }\left(\alpha + 2\right)\left(a_0 + a_{1}t\right)$ and the analytical solution is

$$Y\left(t\right) = a_0\left(\alpha + 1\right)t^{\alpha } + a_1{t}^{\alpha + 1}$$

(146)

It was suggested in [80] that $f_A$ can be used in the form

$$f_A\left(t\right) = \mathrm{\Gamma }\left(\alpha + m + 1\right)\sum _{k = 0}^m{a}_k{t}^k$$

(147)

In such a case, the solution of (145) is

$$Y\left(t\right) = \sum _{k = 0}^K\frac{\mathrm{\Gamma }\left(\alpha + m + 1\right)}{\mathrm{\Gamma }\left(\alpha + k + 1\right)}{a}_k{t}^k$$

(148)

where all coefficients $a_1 = a_2 = ... = a_K = 0$ in reality and only $a_0 \ne 0$.

If we look again at (144) and $a_0 = c \ne 0$ we may suggest that $a_0 = c = a\mathrm{\Gamma }\left(\alpha + 1\right)$ then, the solution is

$$\frac{d^{\alpha }Y}{d{X}^{\alpha }} = a\mathrm{\Gamma }\left(\alpha + 1\right)\Rightarrow Y = a{X}^{\alpha }$$

(149)

In such a case, the value of the fractional order $\alpha$ directly comes from the slope in the logarithmic transform (143), and consequently, a can also be justified from the intercept defining $a_0 = a\mathrm{\Gamma }\left(\alpha + 1\right)$. However, this a simplified approach since, as commented in [73], with the conjecture that the exponent b in (142) has a fixed value (in (149) the role of b is played by $\alpha$).

In homogeneous scaling relations, the coefficient a embodies the real physics of the process [82] because statistically, it can be considered as a random variable defined as [73]

$$a^{\prime } = \frac{a}{\epsilon } = \frac{⟨Y⟩}{\epsilon {⟨X⟩}^{\beta }}$$

(150)

With $\beta = b$ held fixed ($\epsilon$ is a scaling constant), there is a single value of the allometric coefficient calculated for each pair $\left(⟨Y⟩,⟨X⟩\right)$ from the data set [73].

5.2.2. Fractional Dynamics in Allometry

West [75] considered the dynamic variable $Z\left(t\right)$ which scales for a positive constant $\lambda$ and satisfies the homogeneous relation

$$Z\left(\lambda t\right) = {\lambda }^{\beta }Z\left(t\right)$$

(151)

The function is concave for $\beta > 1$ and convex when $\beta < 1$. A scaling alone with a non-integer $\beta$ does not guarantee that the function is fractal, but if this function is fractal, then it does scale in this way [75].

In the phase space presentation of the statistical fractals, the phase space variables $\left(z,t\right)$ replace the dynamic variable $Z\left(t\right)$ and the following probability density distribution [75]

$$P\left(z,t\right) = \frac{1}{t^{\mu }}{F}_z\left(\frac{z}{t^{\mu }}\right)$$

(152)

satisfies the scaling relation

$$P\left(z,\lambda t\right) = {\lambda }^{-\mu }P\left(z,t\right)$$

(153)

In the case of the standard diffusion process, the displacement $⟨x⟩$ of a particle from its initial position at the time t with $\mu = 1/2$ and $F_z\left(z{t}^{-\mu }\right)$ is the Gaussian distribution [75]. In general, for complex phenomena, with $\mu \ne 1/2$, $F_z\left(z{t}^{-\mu }\right)$ is not Gaussian, and in the context of this analysis would be an alpha-stable Levy distribution [83].

It is evident the empirical allometric relation $⟨Y⟩ = a{⟨X⟩}^b$ cannot be developed by direct averaging of data since for $b \ne 1$ we have a nonlinear $F\left(X\right) = X^b$ [75].

In the framework of statistical physics, we may apply two major tools [75]: (1) The stochastic Langevin equation with a random force, and (2) The Fokker–Plank equation concerning the phase space evolution of the probability density distribution (pdf). The extension of the evolution of pdf to fractional differential equations has been developed by West [75] by the method of subordination. This approach resulted in the following fractional rate equation

$${ }^{RL}{D}_t^{\alpha }\left[⟨P\left(t\right)⟩\right] - \frac{t^{-\alpha }}{\mathrm{\Gamma }\left(1 - \alpha \right)}P\left(0\right) = ⟨P\left(t\right)⟩$$

(154)

The solution of (154) is in terms of one-parameter Mittag-Leffler function [75]

$$⟨P\left(t\right)⟩ = P\left(0\right)E_{\alpha }\left(-\lambda t^{\alpha }\right)$$

(155)

Further, a fractional phase space equation (time-space equation with a Riesz–Feller fractional derivative ${\partial }_{\left|z\right|}^{\beta }\left(·\right)$) formulated in [75] is

$${ }^{RL}{D}_t^{\alpha }\left[P\left(z,t\right)\right] - \frac{t^{-\alpha }}{\mathrm{\Gamma }\left(1 - \alpha \right)}{P}_0\left(z\right) = K_{\beta }{\partial }_{\left|z\right|}^{\beta }\left[P\left(z,t\right)\right]$$

(156)

For $\alpha = 1$ it reduces to the anomalous diffusion equation

$${ }^{RL}{D}_t^{\alpha }\left[P\left(z,t\right)\right] = K_{\beta }{\partial }_{\left|z\right|}^{\beta }\left[P\left(z,t\right)\right]$$

(157)

The solution of (156) is [75]

$$P\left(z,t\right) = \frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}{e}^{ - ikz}{E}_{\alpha }\left(-K_{\beta }{\left|k\right|}^{\beta }{t}^{\alpha }\right)dk$$

(158)

For $\alpha = 1$, when $E_{\alpha }\left(·\right) = exp\left(·\right)$, this solution is a characteristic function for the alpha-stable Levy distribution in space with $0 < \beta \le 2$.

As an example, let us consider the growth of the total body mass (TBM) of species, modeled by (where the space variable z is the $z = m = TBM$) the equation [75]

$$\frac{\partial P\left(m,t\right)}{\partial t} = \frac{\partial \left[\lambda mP\left(m,t\right)\right]}{\partial m} + K_{\beta }{\partial }_{\left|z\right|}^{\beta }\left[P\left(m,t\right)\right]$$

(159)

The solution is the characteristic function (in the Fourier domain) for a Levy distribution with Levy index$\beta < 2$, namely

$$\tilde{P}\left(k,t\right) = exp\left[-\frac{K_{\beta }}{\lambda \beta }\left(1 - e^{-\lambda \beta t}\right){\left|k\right|}^{\beta }\right]$$

(160)

>For short (early) times) we have $\lambda \beta \gg 1/t$ and the inverse Fourier transform of (160) is

$$P\left(z,t\right) = F{T}^{-1}\left[exp\left(-K_{\beta }{\left|k\right|}^{\beta }t\right);z\right]$$

(161)

The asymptotic form of pdf obtained is the Pareto distribution for average TPM $⟨M_i⟩$ [75]

$$li{m}_{t\to \infty }\left[P\left(⟨M_i⟩\right),t\right]\propto {⟨M_i⟩}^{-\beta }$$

(162)

Similar examples are considered about: urban scaling [75], time series [76], fractional random walk [78].

5.3. Some Comments on Fractional Allometry

To this end, we can see that the literature basis of fractional approaches to allometric scaling is not rich. In any case, we hope that this brief explanation may spark some ideas for more model development. The fundamental constitutive equation of fractional allometry in integral form (142) gives rise to a unique property. As was shown using the model (144), the power-law directly calls the application of fractional operators with power-law memory. This facilitates, to some extent, the modeling since the technology of applications of singular operators to real-world problems is enough rich.

6. Non-Local Dynamics with Non-Singular Memories

This is the second part of the study addressing applications of new operators with non-singular kernels: Caputo–Fabrizio derivative [13] based on exponential (also known as regular) kernel (166) (conceived in 2015), and Atangana–Baleanu derivative [14] with the Mittag-Leffler function (one-parameter) (167) as a memory (conceived in 2016). There are many studies applying these operators, but we do not intend to create a deep analysis where they are applied correctly or wrongly in the models solved. Our goal is to study the non-local dynamics in models, well-known from the fractional kinetic equations discussed in the preceding section, where these new operators are applied and to see what the new effects are and what phenomena (transport processes) the new equations could model.

6.1. Non-Singular Fractional Operators in Diffusion Models

Now, will present some basic results published by Tateishi et al. [84], highly appreciated by the people working on new trends in fractional calculus. In general, we consider a fractional diffusion equation in general form [84], with a constant diffusion coefficient D

$$\frac{\partial \rho \left(x,t\right)}{\partial t} = D{F}_t^{\alpha }\left[\frac{{\partial }^2\rho \left(x,t\right)}{\partial x^2}\right]$$

(163)

where $\rho \left(x,t\right)$ is the probability distribution.

In (163) the fractional operator $F_t^{\alpha }$ is defined in a Riemann–Liouville manner

$$F_t^{\alpha }\rho \left(x,t\right) = \frac{\partial }{\partial t}\underset{0}{\overset{t}{\int }}K\left(t - \tau \right)\rho \left(x,\tau \right)d\tau$$

(164)

compatible with both the singular and non-singular memory kernels under discussion, namely

Power-law kernel

$$K_{PL}\left(t\right) = \frac{t^{\alpha - 1}}{\mathrm{\Gamma }\left(\alpha \right)}$$

(165)

Exponential kernel

$$K_{CF} = bexp\left(-\frac{\alpha }{1 - \alpha }t\right)$$

(166)

Mittag-Leffler function based kernel

$$K_{AB} = b{E}_{\alpha }\left(-\frac{\alpha }{1 - \alpha }{t}^{\alpha }\right)$$

(167)

Waiting Times and Jump Lengths

The main approach is to apply the Fourier–Laplace transform to all kernels.

Power-law kernel:

With this kernel, following all studies commented on earlier and [69], we have for the waiting times $\omega \left(t\right)$ and the jump length $\lambda \left(x\right)$ (in the Laplace domain). The jump length probability distribution $\lambda \left(x\right)$, with an underlying CTRW transport mechanism and a characteristic time ${\tau }_c$, in the Fourier-Laplace domain is

$$\rho \left(k,s\right) = \frac{\phi \left(k\right)}{s + sDK\left(s\right)k^2}$$

(168)

Then

$$\lambda \left(k\right) = 1 - k^{2}D{\tau }_c\Rightarrow \lambda \left(x\right)\propto exp\left[-{\left(\frac{x}{2D{\tau }_c}\right)}^2\right]$$

(169)

and has an asymptotic Gaussian behavior indicating that the jump length is finite irrespective of the type of the kernel $K\left(t\right)$.

As to the waiting time distributions, with $K\left(t\right) = \delta \left(t\right)$, we get the normal diffusion (ND) regime and exponential waiting times

$${\omega }_{ND}\left(t\right) = \frac{1}{{\tau }_c}exp\left(-\frac{t}{{\tau }_c}\right)$$

(170)

For the fractional Riemann–Liouville operator, the waiting time distributions are

$${\omega }_{PL}\left(t\right) = \frac{1}{{\tau }_c}{t}^{\alpha - 1}{E}_{\alpha ,\alpha }\left(-\frac{t^{\alpha }}{{\tau }_c}\right)$$

(171)

Recall, that the asymptotic behavior of $E_{\alpha ,\alpha }\left(t\right)$, for $t\to \infty$, is a power-law, that is, we get the power-law waiting times of the anomalous diffusion (AD): ${\omega }_{AD}\left(t\right)\propto 1/t^{1 + \alpha }$.

Exponential kernel: With the Caputo–Fabrizio (CF) operator, the Laplace transform of the kernel is

$$K_{CF}\left(s\right) = \frac{b}{s + \frac{\alpha }{1 - \alpha }}$$

(172)

The probability distribution $\rho \left(x,t\right)$ through the Fourier–Laplace transform is

$${\rho }_{CF}\left(k,s\right) = \phi \left(k\right)\left(s + \frac{\alpha }{1 - \alpha }\right)\frac{1}{s\left(s + \frac{\alpha }{1 - \alpha } + Db{k}^2\right)}$$

(173)

In the space-time domain, with the Green function

$$G\left(x,t\right) = e^{-\left(\frac{\alpha }{1 - \alpha }t\right)}\frac{e - {\left(\frac{x}{2\sqrt{Dbt}}\right)}^2}{2\sqrt{\pi Dbt}}$$

(174)

we have

$${\rho }_{CF}\left(x,t\right) = \underset{-\infty }{\overset{\infty }{\int }}G\left(x - \tau \right)\phi \left(\tau \right)d\tau + \frac{\alpha }{1 - \alpha }\underset{0}{\overset{t}{\int }}\underset{-\infty }{\overset{\infty }{\int }}G\left(x - z,\tau \right)\phi \left(z\right)dzd\tau$$

(175)

The shape of the distribution (Figure 2b in [84]) resembles the Gaussian for small times, $t\to 0$, and “tent-type” for long times, but also (see the behavior of $G\left(x,t\right)$ and (175) there is a stationary behavior for $t\to \infty$ (after $t \ge 5$). This behavior is characteristic of confined diffusion processes. At this moment, recall that with the anomalous diffusion, described by CTRW, there is no stationarity. The mean square displacement, in this case, is [84])

$${\left(x^2\right)}_{CF} = 2Db\left(\frac{1 - \alpha }{\alpha }\right)\left(1 - e^{-\frac{\alpha }{1 - \alpha }t}\right)$$

(176)

From (176), for $t\to 0$, $(1 - e^{-\frac{\alpha }{1 - \alpha }t}) \approx \frac{\alpha }{1 - \alpha }t$ and consequently ${\left(x^2\right)}_{CF} \approx 2Dbt$, i.e., almost linear. For long times, $t\to \infty$, however, ${\left(x^2\right)}_{CF} \approx 2Db\left(\frac{1 - \alpha }{\alpha }\right) \approx const.$, i.e., corresponding to confined or hindered diffusions, processes different form the unrestricted jumps of CTRW (in its basic formulation).

As commented by Tateishi et al. [84]), an interesting feature of the non-local diffusion with CF kernel is the stochastic resetting. In their experiments, the diffusion equation can be re-written as

$$\frac{\partial {\rho }_{CF}\left(x,t\right)}{\partial t} = Db\frac{{\partial }^2{\rho }_{CF}\left(x,t\right)}{\partial x^2} - \frac{\alpha }{1 - \alpha }Db\underset{0}{\overset{t}{\int }}{e}^{-\frac{\alpha }{1 - \alpha }\left(t - \tau \right)}\frac{{\partial }^2{\rho }_{CF}\left(x,\tau \right)}{\partial x^2}d\tau$$

(177)

This is the same equation obtained in [85] for heat conduction, that results in

$$\frac{\partial T\left(x,t\right)}{\partial t} = A_1\frac{{\partial }^{2}T\left(x,t\right)}{\partial x^2} + A_2\left(1 - \alpha \right)\underset{0}{\overset{t}{\int }}{e}^{-\frac{\alpha }{1 - \alpha }\left(t - \tau \right)}\frac{d}{d\tau }\left[\frac{{\partial }^{2}T\left(x,\tau \right)}{\partial x^2}\right]d\tau$$

(178)

because the heat flux is constructed (constituted) following the fading memory approach as [86]

$$q\left(x,t\right) = -k_1\frac{\partial T\left(x,t\right)}{\partial x} - k_2\underset{0}{\overset{t}{\int }}{e}^{-\frac{\alpha }{1 - \alpha }\left(t - \tau \right)}\frac{\partial T\left(x,\tau \right)}{\partial x}d\tau$$

(179)

with thermal conductivities $k_1$ and $k_2$ (see details in [85]).

Then, after integration by parts in the second term of (179) and applying the continuity equation $\frac{\partial T}{\partial t} = -\frac{\partial q}{\partial x}$, we get (178). The last term in (178) is the Caputo–Fabrizio derivative of $\frac{{\partial }^{2}T\left(x,t\right)}{\partial x^2}$.

Therefore, as an experiment, we may present (constitute) the distribution ${\rho }_{CF}\left(x,t\right)$ through its diffusion flux as

$$q_{CF}\left(x,t\right) =-Db\frac{\partial {\rho }_{CF}\left(x,t\right)}{\partial x} - Db\underset{0}{\overset{t}{\int }}{e}^{-\frac{\alpha }{1 - \alpha }\left(t - \tau \right)}\frac{\partial {\rho }_{CF}\left(x,\tau \right)}{\partial x}d\tau$$

(180)

and, applying the technique used in [85] to obtain (177).

The deep thermodynamic sense of the fading memory formulations (179) and (180) is that the non-locality represented by the convolution term works for short times, while for long times, we get local diffusion flux, i.e., diffusion approaching the Gaussian behavior. Moreover, models constructed with the fading memory principle obey: the causality principle (through the convolution term), thermodynamic consistency, and model observability(objectivity) [87,88,89].

With the Mittag-Leffler function as a memory kernel, the waiting time distribution is [84]

$$\begin{array}{c}\hfill {\omega }_{AB}\left(t\right) = \frac{\xi b}{\pi }\frac{1}{{\tau }_c}sin\left(\pi \gamma \right)\underset{0}{\overset{t}{\int }}\frac{{\eta }^{\gamma }{e}^{-b\eta \left(1 - \alpha \right)\frac{t}{{\tau }_c}}}{{\left(1 - \eta \right)}^2 + 2\xi \left(1 - \eta \right){\eta }^{\gamma }cos\left(\pi \gamma \right) + {\eta }^{2\gamma }},\\ \hfill \gamma = 1 - \alpha ,\xi = b\frac{\alpha }{{\tau }_c}\end{array}$$

(181)

For small times, $t\to 0$, we get the stretched exponential

$${\omega }_{AB}\left(t\to 0\right)\propto E_{\alpha }\left(-\frac{\alpha }{1 - \alpha }{t}^{\alpha }\right) \approx exp\left(-\frac{\alpha }{1 - \alpha }\frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha \right)}\right)$$

(182)

For long times, $t\to \infty$, we have a power law as in the case of the Riemann–Liouville operator. Hence, the ABC operator works at a crossover between the stretched exponential and the power law, thus, spanning a wider range of distribution times than when only a power-law memory is assumed.

Further, the Laplace transform of the AB operator is

$$K_{AB} = b\frac{s^{\alpha - 1}}{s^{\alpha } + \frac{\alpha }{1 - \alpha }}$$

(183)

Then, through the Fourier–Laplace transform, the probability density function ${\rho }_{AB}\left(x,t\right)$ is

$${\rho }_{AB}\left(k,s\right) = \phi \left(k\right)\frac{s^{\alpha } + \frac{\alpha }{1 - \alpha }}{s^{\alpha }\left(s^{\alpha } + \frac{\alpha }{1 - \alpha }{s}^{1 - \alpha } + Db{k}^2\right)}$$

(184)

This yields (skipping cumbersome expressions) that for small times we have Gaussian-like behavior as in the case of the Caputo–Fabrizio operator, which is natural since for small values of the argument, the Mittag-Leffler function exhibits exponential behavior. However, for long times, the distribution ${\rho }_{AB}\left(x,t\right)$ gets a tent-type shape. It is important to say that ${\rho }_{AB}\left(x,t\right)$ has no stationary state, which is also natural, taking into account that upon such conditions, the Mittag-Leffler function exhibits power law asymptotics; that is, the operator approaches the behavior of the Riemann–Liouville or Caputo derivatives.

6.2. Anomalous Diffusion via the Langevin Equation with Mittag-Leffler Kernels

At this point, we consider the generalized Langevin equation, which is non-local, and in cases when deterministic forces are absent, it can be expressed in a convolution form as [27,30,90]

$$\ddot{X}\left(t\right) = \underset{0}{\overset{t}{\int }}\gamma \left(t - t^{\prime }\right)\dot{X}\left(t\right) = F\left(t\right)$$

(185)

where $X\left(t\right)$ is the position of the diffusion particle with a mass m (assumed for the same of simplicity as $m = 1$, without loss of generality), $\gamma \left(t\right)$ is a dissipative memory kernel.

The forces $F\left(t\right)$ a zero-centered Gaussian and stationary random force cite Vinales2007 satisfying the fluctuation-dissipation theorem

$$⟨F\left(t\right)F\left(t^{\prime }\right)⟩ = C\left(t - t^{\prime }\right) = k_{B}T\gamma \left(\left|t - t^{\prime }\right|\right)$$

(186)

where $k_B$ is the Boltzmann constant, and T is the absolute temperature of the medium where the diffusion process takes place.

Note: Hereafter, all expressions in this section are in the original notations used in [27,30,90] with one exception:the fractional order$\lambda$ used in these studies is replaced by$\alpha$ that allows a more homogenous presentation.

The solution of (185) through the Laplace transforms is [90]

$$X\left(t\right) = ⟨X\left(t\right)⟩ + \underset{0}{\overset{t}{\int }}G\left(t - t^{\prime }\right)F\left(t^{\prime }\right)d{t}^{\prime }$$

(187)

$$\dot{X}\left(t\right) = ⟨\dot{X}\left(t\right)⟩ + \underset{0}{\overset{t}{\int }}g\left(t - t^{\prime }\right)F\left(t^{\prime }\right)d{t}^{\prime },g\left(t\right) = \frac{d}{dt}G\left(t\right)$$

(188)

where with the initial position $x_0 = X\left(t = 0\right)$ and initial velocity $\dot{X}\left(t = 0\right) = v_0$ we have

$$⟨X\left(t\right)⟩ = x_0 + v_{0}G\left(t\right),⟨\dot{X}\left(t\right)⟩ = v_{0}g\left(t\right)$$

(189)

The Laplace transforms of the kernels $G\left(t\right)$ and $g\left(t\right)$ are

$$\overline{G}\left(s\right) = \frac{1}{s^2 + \overline{\gamma }\left(s\right)s},\overline{g}\left(s\right) = \frac{1}{s + \overline{\gamma }\left(s\right)}$$

(190)

From (187) and (188) it follows that for $s = 0$ (that is for $t\to \infty$) we have $G\left(0\right) = 0$ and $g\left(0\right) = 1$

The explicit variances are [27]:

$${\sigma }_{xx}\left(t\right) = k_{B}T\left[2I\left(t\right) - G^2\left(t\right)\right],I\left(t\right) = \underset{0}{\overset{t}{\int }}G\left(t^{\prime }\right)d{t}^{\prime }$$

(191)

$${\sigma }_{vv}\left(t\right) = k_{B}T\left[1 - g^2\left(t\right)\right]$$

(192)

$${\sigma }_{xv}\left(t\right) = k_{B}TG\left(t\right)\left[1 - g\left(t\right)\right]$$

(193)

In this context, the second moments can be presented alternatively as [90]

$$⟨X^2\left(t\right)⟩ = x_0^2 + \left(v_0^2 - k_{B}T\right)G^2\left(t\right) + k_{B}TI\left(t\right) + 2x_0{v}_{0}G\left(t\right)$$

(194)

$$⟨{\dot{X}}^2\left(t\right)⟩ = k_{B}T + \left(v_0 - k_{B}T\right)g^2\left(t\right)$$

(195)

with a general correlation function $C\left(t\right)$ based on the Mittag-Leffler function [90]

$$C\left(t\right) = C_0\left(\alpha \right)\frac{1}{{\tau }^{\alpha }}{E}_{\alpha }\left[-{\left(\frac{\left|t\right|}{\tau }\right)}^{\alpha }\right],0 < \alpha < 2$$

(196)

where $\tau$ is a characteristic memory time-scale, while $C_0\left(\alpha \right)$ is a time-independent proportionality coefficient; $E_{\alpha }\left(·\right)$ is one-parameter Mittag-Leffler function.

The limit case $\tau \to 0$ reduces (196) to a power-law because in such a case $\left|t\right|/\tau \to \infty$ (long times) and $E_{\alpha }\left(t^{-\lambda }\right)\sim t^{-\lambda }$ that is [90]

$$C{\left(t\right)}_{\tau \to 0} = C_0\left(\alpha \right)\frac{t^{-\alpha }}{\mathrm{\Gamma }\left(1 - \alpha \right)}$$

(197)

Moreover, asymptotically for $\alpha \ne 1$ and $t\to 0$ (short times) we get a stretched exponential. For $\alpha \to 1$ the correlation function reduces to $C{\left(t\right)}_{\alpha \to 1} = C_0\delta \left(t\right)$ and corresponds to Brownian diffusion.

From (186) the memory kernel can be expressed as [90]

$$\gamma \left(t\right) = {\gamma }_{\alpha }\frac{1}{{\tau }^{\alpha }}{E}_{\alpha }\left[-{\left(\frac{\left|t\right|}{\tau }\right)}^{\alpha }\right]$$

(198)

and its Laplace transformation is

$$\overline{\gamma }\left(s\right) = \frac{{\gamma }_{\alpha }{s}^{\alpha - 1}}{1 + s^{\alpha }{\tau }^{\alpha }}$$

(199)

Thus, the Laplace transform of the kernel integral $I\left(t\right) = \underset{0}{\overset{t}{\int }}G\left(t^{\prime }\right)d{t}^{\prime }$ is

$$\overline{I}\left(s\right) = \frac{\overline{G}\left(s\right)}{s} = {\overline{I}}_0\left(s\right) + {\overline{I}}_1\left(s\right),{\overline{I}}_0\left(s\right) = \frac{s^{-1}{\tau }^{\alpha }}{{\tau }^{\alpha }{s}^2 + s^{2 - \alpha } + {\gamma }_{\alpha }},{\overline{I}}_1\left(s\right) = \frac{1}{{\tau }^{\alpha }}\frac{1}{s^{\alpha }}{\overline{I}}_0\left(s\right)$$

(200)

Consequently, in the time domain, we have [90]

$$I_0\left(t\right) = \sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}{\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)}^k{t}^{2\left(k + 1\right)}{E}_{\alpha ,3 + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]$$

(201)

$$\begin{array}{cc}\hfill & I_1\left(t\right) = {\left(\frac{t}{\tau }\right)}^{\alpha }{I}_0\left(t\right),\hfill \\ \hfill & I_1\left(t\right) = {\left(\frac{t}{\tau }\right)}^{\alpha }\sum _{u = k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}{\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)}^k{t}^{2\left(k + 1\right)}{E}_{\alpha ,3 + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]\hfill \end{array}$$

(202)

In (201) and (202), $E_{p,q}\left(z\right)$ is a two-parameter Mittag-Leffler function: $E_{p,q}\left(z\right) = \sum _{k = 0}^{\infty }\frac{z^k}{\mathrm{\Gamma }\left(pk + q\right)}$, $p > 0$, $q > 0$. $E_{p,q}^{\left(k\right)}\left(z\right)$ is a derivative of $E_{p,q}\left(z\right)$. As a result, the memory kernels are [90]: $G\left(t\right) = G_0\left(t\right) + G_1\left(t\right)$ and $g\left(t\right) = g_0\left(t\right) + g_1\left(t\right)$, where [90]

$$G_0\left(t\right) = \sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)t^{2k + 1}{E}_{\alpha ,2 + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]$$

(203)

$$G_1\left(t\right) = {\left(\frac{t}{\tau }\right)}^{\alpha }\sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)t^{2k + 1}{E}_{\alpha ,2 + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]$$

(204)

$$g_0\left(t\right) = \sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)t^{2k}{E}_{\alpha ,1 + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]$$

(205)

$$g_1\left(t\right) = {\left(\frac{t}{\tau }\right)}^{\alpha }\sum _{k = 0}^{\infty }\frac{{\left(-1\right)}^k}{k!}\left(\frac{{\gamma }_{\alpha }}{{\tau }^{\alpha }}\right)t^{2k}{E}_{\alpha ,1 + \alpha + \left(2 - \alpha \right)k}^{\left(k\right)}\left[-{\left(\frac{t}{\tau }\right)}^{\alpha }\right]$$

(206)

Now, using the basic properties of the Mittag-Leffler function and its derivatives in asymptotic situations, we have asymptotic behaviors. For long times, $t \gg \tau$ we have

$$E_{p,q}\left(-z\right)\sim \frac{1}{z\mathrm{\Gamma }\left(q - p\right)},E_{p,q}^{\left(k\right)}\left(-z\right)\sim {\left(-1\right)}^k\frac{k!}{z^{k + 1}\mathrm{\Gamma }\left(q - p\right)},\frac{t}{\tau } \gg 1$$

(207)

Consequently, we get [90]:

$$I\left(t\right) \approx t^2{E}_{2 - \alpha ,3}\left[{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right] + \frac{{\tau }^{\alpha }}{{\gamma }_{\alpha }}\left\{1 - E_{2 - \alpha }\left[{\left(-\omega t\right)}^{2 - \alpha }\right]\right\}$$

(208)

$$G\left(t\right) = t{E}_{2 - \alpha ,3}\left[ - {\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right] - \frac{{\tau }^{\alpha }}{{\gamma }_{\alpha }}\left\{1 - E_{2 - \alpha }\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right]\right\}$$

(209)

$$g\left(t\right) = E_{2 - \alpha }\left[ - {\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right] - \frac{{\tau }^{\alpha }}{{\gamma }_{\alpha }}\frac{d^2}{d{t}^2}\left\{1 - E_{2 - \alpha }\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right]\right\}$$

(210)

with ${\left({\omega }_{\alpha }\right)}^{2 - \alpha } = {\gamma }_{\alpha }$.

In (208)–(210), the first three terms correspond to the exact power-law behaviors of the kernels, while the second group of three terms in the same expansions work at intermediate times $\tau < t < 1/{\omega }_{\alpha }$, where $1/{\omega }_{\alpha }$ defines the characteristic time scale of the process.

With these results the second moment $⟨X^2\left(t\right)⟩$ is [90]

$$⟨X^2\left(t\right)⟩ = \left(2k_{B}T\right)t^2{E}_{2 - \alpha ,3}\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right] + 2k_{B}T\frac{{\tau }^{\alpha }}{{\gamma }_{\alpha }}\left\{1 - E_{2 - \alpha }\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right]\right\}$$

(211)

Then the diffusion coefficient is defined as $D\left(t\right) = \frac{1}{2}\frac{d}{dt}⟨X^2\left(t\right)⟩$ is

$$D\left(t\right) \approx \left(k_{B}Tt\right)E_{2 - \alpha ,3}\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right] - 2k_{B}T\frac{{\tau }^{\alpha }}{{\gamma }_{\alpha }}\frac{d}{dt}{E}_{2 - \alpha }\left[-{\left({\omega }_{\alpha }t\right)}^{2 - \alpha }\right]$$

(212)

For cases when the time is much larger than the characteristic process time-scale, that is, when ${\omega }_{\alpha }t \gg 1$ based on the asymptotic expansions of the Mittag-Leffler function, we have the following approximations [90]:

$$I\left(t\right) \approx \frac{1}{{\gamma }_{\alpha }\mathrm{\Gamma }\left(\alpha + 1\right)}{t}^{\alpha },G\left(t\right) \approx \frac{1}{{\gamma }_{\alpha }\mathrm{\Gamma }\left(\alpha \right)}{t}^{\alpha - 1},g\left(t\right) \approx \frac{1}{{\gamma }_{\alpha }\mathrm{\Gamma }\left(\alpha - 1\right)}{t}^{\alpha - 2}$$

(213)

and the approximations of $⟨X^2\left(t\right)⟩$ and $D\left(t\right)$ are [90]

$$⟨X^2\left(t\right)⟩ \approx k_{B}T\frac{2}{{\gamma }_{\alpha }\mathrm{\Gamma }\left(\alpha + 1\right)}{t}^{\alpha },D\left(t\right) \approx k_{B}t\frac{1}{{\gamma }_{\alpha }\mathrm{\Gamma }\left(\alpha \right)}{t}^{\alpha - 1},{\omega }_{\lambda }t \gg 1$$

(214)

Therefore, for ${\omega }_{\alpha }t \gg 1$ there is a subdiffusive transport with $0 < \alpha < 1$ and superdiffusive transport with $1 < \alpha < 2$. It is worth noting that these results confirm the earlier estimations in [27] (we skip the discussion on this work here since there are too common elements with the analysis performed above). A similar study has been developed in [91] on anomalous diffusion of harmonic operators with a Mittag-Leffler noise. With a close ideology, a generalized Langevin equation with a three-parameter Mittag-Leffler noise has been studied in [92].

6.3. Diffusion with AB Derivative: Front Propagation and First Passage Time

This section discusses the formulation of a diffusion equation with the Atangana–Baleanu derivative, where the kernel is the one-parameter Mittag-Leffler function. The main efforts are oriented toward correct diffusion equation formulation (Section 6.3.1) with flux constituted as the associated AB fractional integral, following the structure of the fading memory formalism [86,93,94,95]. As a result, the approach applied to study the time evolution of the front solution (which can be considered as a solution propagator, analog of the mean squared displacement) is based on the approximate solution by applying the integral-balance technique (Section 6.3.2 also allowing to define the first passage time (Section 6.3.3).

6.3.1. Diffusion Equation with AB Derivative: Model Formulation

Now, we consider the diffusion flux constructed as (see (Appendix A41)) [96]

$${ }^{AB}{I}_{a + }^{\alpha }u\left(t\right) = \left(1 - \alpha \right)u\left(x,t\right) + {\alpha }^{RL}{I}_{a + }^{\alpha }u\left(x,t\right) = f\left(x,t\right)$$

(215)

or equivalently

$${ }^{AB}{I}_{a + }^{\alpha }u\left(t\right) = m\left(\alpha \right)u\left(x,t\right) + \lambda {\left(\alpha \right)}^{RL}{I}_{a + }^{\alpha }u\left(x,t\right)$$

(216)

The constructions of (215) and (216) are the same as that of the fading memory concept [86] (see for example (179); following the Boltzmann linear superposition functional [93] and the formulations in [94,95] with a time-dependent memory function) $R\left(t,z\right)$, namely

$$\phi \left(x,t\right) = m\left[v_x\left(x,t\right)\right] + \lambda \underset{0}{\overset{t}{\int }}R\left(t,z\right)v_x\left(z\right)dz$$

(217)

The memory integral (the 2nd terms in (215) and (216)) is the standard Riemann–Liouville fractional integral ${ }^{RL}{I}_{a + }^{\alpha }u\left(x,t\right)$. In (217) $v_x\left(z\right) = \nabla v\left(z\right)$ and $\nabla \cdot \nabla = \mathsf{\Delta }$ is the Laplacian. The coefficients m and $\lambda$ are weighting functions (transport coefficients) depending on the character of the modeled diffusion process (heat or mass) and are explained in the next point. As it follows from (215) and (216), the case $\alpha = 1$ recovers the Fick (Fourier) law and the Gaussian diffusion Now, let us turn on to the basic formulation of the rate equation (see Equation (A38)) [96]

$${ }_0^{ABR}{D}_t^{\alpha }\left[f\left(t\right)\right] = u\left(t\right)$$

(218)

with a unique solution (see in the appendix Equation (A39)), expressed through either ABR or ABC derivative [14,97] allowing the time-fractional derivative to be related to the flux as a function of the gradient $\partial f/\partial x$.

Assuming that in (218) $u\left(x,t\right) = \left(-\partial f/\partial x\right)$, without loss of the generality of this equation, and applying ${ }^{AB}{I}_{a+}^{\alpha }u\left(t\right)$ expressed as (215) we have [96]

$${ }^{AB}{I}_{a + }^{\alpha }\left[u\left(x,t\right)\right] = { }^{AB}{I}_{a+}^{\alpha }\left[-\frac{\partial f\left(x,t\right)}{\partial x}\right] = -\left\{m\left(\alpha \right)\frac{\partial f\left(x,t\right)}{\partial x} + \lambda \left(\alpha \right){ }^{RL}{I}_{a+}^{\alpha }\left[\frac{\partial f\left(x,t\right)}{\partial x}\right]\right\}$$

(219)

The weighting functions $m\left(\alpha \right) = 1 - \alpha$ and $\lambda \left(\alpha \right) = \alpha$ depend on the degree of fractionality of the modeled diffusion process. For $\alpha = 1$ we get $m\left(\alpha \right) = 0$ and $\lambda \left(\alpha \right) = 1$. Hence, we may rewrite (218) as a continuity equation in terms of ABR derivative

$$\frac{{\partial }^{\alpha }f\left(x,t\right)}{\partial t^{\alpha }} = -\frac{d}{dx}{j}_{\alpha }\left(x,t\right)$$

(220)

Differentiation concerning x yields

$$-\frac{d}{dx}{j}_{\alpha }\left(x,t\right){ = }^{{ }^{AB}}{I}_{a+}^{\alpha }\left[D_0\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right] = D_0\left\{m\left(\alpha \right)\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2} + \lambda \left(\alpha \right){ }^{RL}{I}_{a+}^{\alpha }\left[\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right]\right\}$$

(221)

Hence, using the above conjecture (219) the diffusion equation takes the form

$${ }^{ABR}{D}_{a+}^{\alpha }f\left(x,t\right) = D_0\left[{ }^{AB}{I}_{a+}^{\alpha }\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right]$$

(222)

Explicitly, setting the lower terminal of the ABR derivative and the Riemann–Liouville integral as $a = 0$, we get

$${ }_0^{ABR}{D}_t^{\alpha }f\left(x,t\right) = D_0\left\{m\left(\alpha \right)\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2} + \lambda \left(\alpha \right)0^{RL}{I}_t^{\alpha }\left[\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right]\right\}$$

(223)

As commented above, for $\alpha = 1$ we have $m\left(\alpha \right) = 0$ and $\lambda \left(\alpha \right) = 1$, and the memory integral in (223) becomes

$${\left[{ }_0^{AB}{I}_t^{\alpha }\left(\frac{{\partial }^{2}u\left(x,\tau \right)}{\partial x^2}\right)\right]}_{\alpha = 1} = \frac{{\partial }^{2}u\left(x,t\right)}{\partial x^2}$$

(224)

Hence, $\alpha = 1$ in (220) recovers the Fourier (Fick) law $j_{\alpha = 1} = -D_0\left(\partial f/\partial x\right)$ and Equation (223) reduces to the classical diffusion equation.

Alternatively, for the sake of clarity of this statement, we may express (223) as (taking into account that $m\left(\alpha \right) = 1 - \alpha$ and $\lambda \left(\alpha \right) = \alpha$)

$${ }_0^{ABR}{D}_t^{\alpha }f\left(x,t\right) = D_0\left\{\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2} + \alpha \left[{ }_0^{RL}{I}_t^{\alpha }\left(\frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right) - \frac{{\partial }^{2}f\left(x,t\right)}{\partial x^2}\right]\right\}$$

(225)

When $\alpha \to 1$, then the last term in the squared brackets tends to zero, and we recover the classical diffusion equation.

Moreover, we can see that when the power-law kernel controls the memory integral (see Equation (223)), then the term $m\left(\alpha \right)\to 0$ and $\lambda \left(\alpha \right)\to 1$. However, this is a formal comparison that cannot be related to $\alpha = 1$ because the singular power-law kernel and the non-singular one based on the Mittag-Leffler function describe different relaxation processes and model different physical phenomena, depending on the time scale.

6.3.2. Approximate Solution and Front Propagation Scaling

Now, we refer to an approach based on approximate analytical solutions of diffusion models (a Dirichlet problem, for instance, see below) with a finite speed of solution relevant to the non-local concept. The finite speed is modeled by a front ${\delta }_D\left(t\right)$ separating the medium into two parts: disturbed for $0\le x\le {\delta }_D$, where $f\left(x,t\right) \ne 0$, and a virgin section for ${\delta }_D \le x \le \infty$, with $f\left(x,t\right) = 0$. The concept of the front, which is a physically motivated approach, means that at $x = {\delta }_D$ we have two boundary conditions $f\left({\delta }_D\left(t\right)\right) = \partial f\left({\delta }_D\left(t\right)\right)/\partial x = 0$ and, therefore, the flux is also zero, i.e., $j_a\left({\delta }_D\right) = 0$. More details on how this approach is applied to fractional diffusion models are available elsewhere [98]. It is worth noting again the analogy between ${\delta }_D\left(t\right)$ and $⟨x\left(t\right)⟩$, and ${\delta }_D^2\left(t\right)$ as an analog of $⟨x^2\left(t\right)⟩$.

The common approach of this method, known as the integral-balance method [98], is the use of an assumed profile (distribution), which is then optimized through the solution. In what follows, we will demonstrate the main idea of kinetic analysis by application of a parabolic profile with an unspecified exponent [96,98]

$$f_a = f_s{\left(1 - \frac{x}{\delta }\right)}^n$$

(226)

For the sake of simplicity, we consider the Dirichlet problem, which means $f_s = f\left(0,t\right) = 1$. Moreover, we will skip some solution steps (see [96,98]) and will focus the attention on the time evolution (propagation) of the front.

Now, the equation about ${\delta }_D\left(t\right)$ is [96])

$${ }_0^{ABR}{D}_t^{\alpha }\left[{\delta }_D^2\right] = D_{0}N\left[\left(1 - \alpha \right) + \alpha \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)}\right],N = \left(n + 1\right)\left(n + 2\right)$$

(227)

Equation (227) should have an initial condition $\delta \left(0\right) = 0$ since there is no diffusant penetration at $t = 0$. Moreover, the solution of (227) with respect to ${\delta }_D^2$ follows directly from the definition of ${ }^{AB}{I}_0^{\alpha }f\left(t\right)$. Then, we have two equivalent forms of ${\delta }_D^2$ (the differences are in the last terms)

$${\delta }_D^2 = D_{0}N\left\{\left(1 - \alpha \right) + \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)} + {\alpha }^2\frac{\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(2\alpha + 1\right)}{t}^{2\alpha }\right\}$$

(228)

$${\delta }_D^2 = D_{0}N\left\{\left(1 - \alpha \right) + \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)} + \frac{\alpha }{2}\frac{\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(2\alpha \right)}{t}^{2\alpha }\right\}$$

(229)

From (229) we get

$${\delta }_D^2 = D_{0}N{t}^{2\alpha }\left\{\frac{\left(1 - \alpha \right)}{t^{2\alpha }} + \frac{1}{\mathrm{\Gamma }\left(\alpha + 1\right)}\frac{1}{t^{\alpha }} + \frac{\alpha }{2}\frac{\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(2\alpha \right)}\right\}$$

(230)

Then, for long times ${\delta }_D^2\sim D_{0}N{t}^{2\alpha }\frac{\alpha }{2}\frac{\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(2\alpha \right)}$, that is ${\delta }_D\sim t^{\alpha }$ indicating a subdiffusive motion of the front.

Let now see what the asymptotic behaviours of ${\delta }_D^2$ for $\alpha \to 0$ and $\alpha \to 1$ are [96]). For $\alpha \to 0$ we have $\left(1 - \alpha \right)\to 1$,$\mathrm{\Gamma }\left(\alpha + 1\right)\to 1$, $\mathrm{\Gamma }\left(2\alpha + 1\right)\to 1$, and ${\alpha }^2$ could be neglected. Then, the approximation is (in two forms)

$${\delta }_{D2\alpha \to 0}^2 \equiv D_{0}N\left\{1 + \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)}\right\},{\delta }_{D2\alpha \to 0}^2 \equiv D_{0}N{t}^{\alpha }\left\{\frac{1}{\mathrm{\Gamma }\left(\alpha + 1\right)} + \frac{1}{t^{\alpha }}\right\}$$

(231)

Therefore, for small $\alpha$ (strong fractionality) and long times the front exhibits power-law (subdiffusion) behavior and ${\delta }_{\alpha \to 0}\left(t\to \infty \right) \equiv \sqrt{D_{0}N{t}^{\alpha }}$ as in the cases solved in [96,98]). The first version of the estimation (231) can be presented as

$${\delta }_{D2\alpha \to 0}^2 \equiv D_{0}N\left\{\frac{{\left(t^{\alpha }\right)}^0}{\mathrm{\Gamma }\left(\alpha ·0 + 1\right)} + \frac{{\left(t^{\alpha }\right)}^1}{\mathrm{\Gamma }\left(\alpha ·1 + 1\right)}\right\} \equiv D_{0}N\sum _0^1\frac{{\left(t^{\alpha }\right)}^k}{\mathrm{\Gamma }\left(\alpha k + 1\right)}$$

(232)

Thus, for $\alpha \to 0$, the front propagation is modeled by the first two terms of $E_{\alpha }\left(t^{\alpha }\right)$. Recall that for $\alpha \to 0$, the relaxation is extremely slow (not superslow !), and it could be assumed that it practically does not happen. For long times and $\alpha \to 0$ we get from the second form of (231) that ${\delta }_{D\alpha \to 0}^2\sim t^{\alpha }$, i.e., subdiffusive behavior is modeled, as in the case of the Riemann–Liouville derivative [98].

For $\alpha \to 1$ we have $\left(1 - \alpha \right)\to 0$,$\mathrm{\Gamma }\left(\alpha + 1\right)\to 1$, $\mathrm{\Gamma }\left(2\alpha + 1\right)\to 2$, and ${\alpha }^2\to 1$. Thus, we have the approximation

$${{\delta }^2}_{\alpha \to 1} \equiv D_{0}N\left\{t^{\alpha } + \frac{{\alpha }^2}{2}{t}^{2\alpha }\right\} \equiv D_{0}N{t}^{2\alpha }\left\{\frac{{\alpha }^2}{2} + \frac{1}{t^{\alpha }}\right\}$$

(233)

The approximation (233) indicates that for long times the movement is subdiffusive, that is

$${\delta }_{D\alpha \to 1}^2 \equiv \sqrt{D_{0}N}{t}^{\alpha }$$

(234)

For $t = 0$, we have ${\delta }_D = 0$, which means that there is no diffusant penetration into the medium, which is physically correct. Moreover, for short times and $\alpha \to 1$, we get ${\delta }_{D\alpha \to 1}^2\sim \sqrt{D_{0}N{t}^{\alpha }}$, a solution known from subdiffusive cases when the fractional derivative is Riemann–Liouville [98]).

In the context of the above expressions about ${\delta }_D^2$ we may rearrange (229) as

$${\delta }_D^2 = D_{0}N\left\{\sum _0^1\frac{{\left(\alpha t^{\alpha }\right)}^k}{\mathrm{\Gamma }\left(\alpha + 1\right)} - D_{0}N\left[-\alpha + \left(1 - \alpha \right)\frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)}\right]\right\}$$

(235)

where the sum refers to only the first two terms of the Mittag-Leffler function.

6.3.3. The First Passage Time

The first passage time distribution (FPT) distribution is defined as $F\left(t\right) = d{f}_{fpt}/dt$ where $f_{fpt} = \underset{0}{\overset{L}{\int }}f\left(x,t\right)dx$. With the concept of the finite penetration depth, the upper terminal is $L = {\delta }_D$.

Then, with the assumed profile (226) we have

$$f_{fpt} = \underset{0}{\overset{\delta }{\int }}{\left(1 - \frac{x}{{\delta }_D}\right)}^{n}dx = \frac{{\delta }_D\left(t\right)}{n + 1}\Rightarrow F_{\delta } = \frac{d}{dt}\left[\frac{{\delta }_D\left(t\right)}{n + 1}\right]$$

(236)

Further, the mean FPT (MFPT) is defined as

$$T_{FPT} = \underset{0}{\overset{\infty }{\int }}{f}_{fpt}dt = \underset{0}{\overset{\infty }{\int }}\left[\underset{0}{\overset{\delta }{\int }}{\left(1 - \frac{x}{{\delta }_D}\right)}^{n}dx\right]dt$$

(237)

Taking into account the results (228) and (229) about ${\delta }_D^2$ and the asymptotic estimations, it is not possible to get a simple expression about ${\delta }_D$ allowing easy integration in (237). Because of that, we will use the squared value of FPT termed here as Squared first passage time distribution (SFPT), $F^2\left(t\right) = {\left(d{f}_{fpt}/dt\right)}^2$ which in therm of ${\delta }_D$ can be expressed as

$$F_{\delta }^2 = \frac{d}{dt}\left[\frac{{\delta }_D^2\left(t\right)}{{\left(n + 1\right)}^2}\right]$$

(238)

Consequently, we have

$${\left(T_{FPT}\right)}^2 = \underset{0}{\overset{\infty }{\int }}\frac{D_{0}N}{\left(n + 1\right)}\left\{\left(1 - \alpha \right) + \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha + 1\right)} + \frac{\alpha }{2}\frac{\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(2\alpha \right)}{t}^{2\alpha }\right\}dt$$

(239)

The integration in (239) (for any $0 < \alpha < 1$) yields ${\left(T_{FPT}\right)}^2\to \infty$ and consequently $T_{FPT}\to \infty$. This result is consistent with the results of Fa and Lenzi [99] that MFPT for subdiffusive transport is infinite. To be precise, this statement is correct for any $\alpha < 0.5$. Moreover, the result should also be valid for valid for $0.5 < \alpha < 1$ because if ${\delta }_D^2\sim t^{2\alpha }$, as it follows from the second term of (228), we may estimate that ${\delta }_D\sim t^{\alpha }$ and, therefore, for such a case $T_{FPT}\to \infty$.

6.4. Some Comments on Dynamics (Kinetics) with Non-Singular Memories

In all of the issues discussed here, which were all concerned with non-local kinetics, fractional calculus with singular and non-singular memory was intimately engaged in the key aspects of the solutions. We can see from the collection that non-local problems cover a wide range of topics. They are frequently not mentioned together, making it challenging to recognize the relationships between them. According to a literature search, issues with statistical physics and irreversible thermodynamics are the main topics of fractional kinetic research. On the other hand, fractional calculus evolves in a world unrelated to the fundamental issues underlying the non-localities of modeled events. We hope that this study will build a bridge between these areas and, more importantly, will demonstrate to the fractional calculus community the importance of the fundamental background and how it cannot be neglected in favor of calculation methods because the solution must be logical and realistic, and interpretable.

7. Final Comments and Outlines

We can see certain common issues arising at the end of this fascinating tour through the non-local kinetic “jungle,” which was not previously compiled in the manner described in this article; among them:

• The common problem in the established models is the non-locality that manifests itself at different stages of implementation, sometimes mechanistically and other times logically.
• Moreover, there is no common point of view, up to this text compilation, we guess, which could allow better seeing the correct approach to fractional modeling implementation
• There are frequent attempts to use various problem-solving strategies without making any effort to interpret the results that are obtained. We did not place a great deal of emphasis on these publications because, in our opinion, they add more noise to the scientific world than actual values.
• An important moment is the application of fractional kinetics to material processing. However, in such a case, we have to stress attention to the big gap between fractional calculus modelers and material processing scientists due to the different levels in phenomena analyses and interpretations. Only people working at the interfaces of material physics and mathematical modeling could resolve the emerging problems. We tried to emphasize such few studies, thus, demonstrating that there is a large area where fractional modeling of kinetic dynamics is not applied yet.
• The new moments highlighted in this article are the fractional kinetics based on non-singular kernels and the emphasis on fractional allometry as a potential area for future studies.
• Kinetics based on non-local fractional operators covers an area of phenomena where the classical fractional calculus can model only some extreme cases, as is shown in this article, such as for long-time approximation range, where the power-law is valid. However, intermediate ranges are successfully covered by non-singular memories adequately implemented in kinetic models. Moreover, non-local models based on non-singular memories cover cases of confined diffusion and diffusion with resetting, as it was highlighted in this article.
• Correct applications of fractional kinetics based on the fading memory formalism, which is thermodynamically consistent, and ensures model observability, show that the results can be adequately interpreted, thus, providing a secure approach to modeling kinetic problems related to various real-world diffusion problems.

We believe that the attempts to create this compilation of results in non-local kinetic modeling are, to a greater extent, successful. However, this article also focuses on emerging problems and, in this context, addresses new areas where non-local kinetics can be explored, or at least may provoke ideas to apply it to physical objects not covered here.

Appendix A.1. Time-Fractional Integral and Derivatives with Singular Kernels

Here we give some basic definitions of the fractional integral and the Riemann–Liouville and Caputo fractional derivatives, which are relevant to the problems where power-law memories are applied.

Appendix A.1.1. Fractional Integral

Following the Riemann–Liouville approach to the fractional calculus, the fractional integral of order $\mu > 0$ is a natural result of Cauchy’s formula reducing calculations of the m-fold primitive of a function $f\left(t\right)$ to a single integral of convolution type [100]

$${ }_0{I}^{n}f\left(t\right): = f_n\left(t\right) = \frac{1}{\left(n - 1\right)!}\underset{0}{\overset{t}{\int }}{\left(t - \tau \right)}^{n - 1}f\left(\tau \right)d\tau ,t > 0,n\in \mathbb{N}$$

(A1)

where $\mathbb{N}$ is a set of positive integers.

The definition (A1) implies that $f_n\left(t\right)$ and its derivatives of order $1,2,\cdots ,n - 1$ vanish for $t < 0$. Taking into account that $\left(n - 1\right) = \mathrm{\Gamma }\left(n\right)$, then for an arbitrary positive number $\mu > 0$ we may define the fractional integral of order $\mu > 0$

$${ }_0{I}^{\mu }f\left(t\right): = \frac{1}{\mathrm{\Gamma }\left(\mu \right)}\underset{0}{\overset{t}{\int }}{\left(t - \tau \right)}^{\mu - 1}f\left(\tau \right)d\tau ,t > 0,n\in {\mathbb{R}}^{+}$$

(A2)

where ${\mathbb{R}}^{+}$ is a set of positive real numbers. For the sake of convenience, we will also use the notation ${ }_0{D}^{-\mu }f\left(t\right)$ for ${ }_0{I}^{\mu }f\left(t\right)$.

Further, the law of exponents for fractional integrals means

$${ }_0{D}^{-\mu }{ }_0{D}^{-\gamma }f\left(t\right) = { }_0{D}^{-\mu - \gamma }f\left(t\right) = { }_0{D}^{-\gamma }{ }_0{D}^{-\mu }f\left(t\right)$$

(A3)

The Laplace transform of the fractional integral is straightforwardly defined by the convolution theorem as

$$L\left[{ }_0{D}_t^{-\mu }f\left(t\right)\right] = L\left[\frac{t^{\mu - 1}}{\mathrm{\Gamma }\left(\mu \right)}\right]L\left[f\left(t\right);s\right] = s^{-\mu }F\left(s\right)$$

(A4)

where $\Re \left(s\right) > 0,\Re \left(\mu \right) > 0$ and $F\left(s\right)$ is the Laplace transform of $f\left(t\right)$

Appendix A.1.2. Riemann–Liouville Fractional Derivative

The fractional derivative $D^{n}f\left(t\right)$ with $n\in \mathbb{N}$ can be defined by the relations [100] as

$${ }_0{D}^n{ }_0{I}^n = I,{ }_0{I}^n{ }_0{D}^n \ne I,n\in \mathbb{N}$$

(A5)

It is worth noting that $D^n$ is left-inverse, but not right-inverse, to the integral operator $I^n$ because

$${ }_0{I}^n{ }_0{D}^{n}f\left(t\right) = f\left(t\right) - \sum _{k = 0}^{n - 1}{f}^{\left(k\right)}\left(0^{+}\right)\frac{t^k}{k!},t > 0$$

(A6)

Hence, since $D^n$ is left-inverse to $I^n$ introducing a positive integer m such that $m - 1 < \mu \le m$ the natural definition of the Riemann–Liouville (left-sided) fractional derivative of order $\mu > 0$ is

$$\begin{array}{c}\hfill { }_0{D}^{\mu }f\left(t\right): ={ }_0{D}^m{ }_0{I}^{m - \mu }f\left(t\right) = \frac{1}{\mathrm{\Gamma }\left(m - \mu \right)}\frac{d^m}{d{t}^m}\underset{0}{\overset{t}{\int }}\frac{f\left(t\right)}{{\left(t - \tau \right)}^{\mu + 1 - m}},\\ \hfill m - 1 < \mu \le m,m\in \mathbb{N}\end{array}$$

(A7)

Consequently, it follows that $D^0 = I^0 = I$, that is $D^{\mu }{I}^{\mu } = I$ for $\mu \ge 0$. In addition, the fractional derivative of a power-law function, frequently used in this article, is

$${ }_0{D}^{\mu }{t}^{\beta } = \frac{\mathrm{\Gamma }\left(\beta + 1\right)}{\mathrm{\Gamma }\left(\beta + 1 - \mu \right)}{t}^{\beta - \mu },\mu > 0,\beta > -1,t > 0$$

(A8)

A noteworthy fact related to the Riemann–Liouville fractional derivative is that $D^{\mu }f\left(t\right) \ne 0$ when $f\left(t\right) = const.$, that is

$${ }_0{D}^{\mu }C = C\frac{t^{-\mu }}{\mathrm{\Gamma }\left(1 - \mu \right)},{ }_0{D}^{\mu }1 = \frac{t^{ - \mu }}{\mathrm{\Gamma }\left(1 - \mu \right)},C = 1$$

(A9)

If $\mu \in \mathbb{N}$, then $D^{\mu }C \equiv 0$ due to the poles of the Gamma function in the points $0,-1,-2$

Similarly, the fractional integrals of the power-law function and a constant are

$${ }_0{D}^{-\mu }{t}^{\beta } = \frac{\mathrm{\Gamma }\left(\beta + 1\right)}{\mathrm{\Gamma }\left(\beta + 1 + \mu \right)}{t}^{\mu + \beta },{ }_0{D}^{-\mu }C = \frac{C}{\mathrm{\Gamma }\left(1 + \mu \right)}{t}^{\mu },\mu \ne 1,2,...$$

(A10)

The Laplace transform of the Riemann–Liouville fractional derivative for $m\in \mathbb{N}$ is

$$L\left[\frac{d^m}{d{t}^m}f\left(t\right);s\right] = s^{m}F\left(s\right) - \sum _{k = 0}^{m - 1}{s}^{m - k - 1}{f}^{\left(m\right)}\left(0+\right) = s^{m}F\left(s\right) - \sum _{k = 1}^m{s}^{k - 1}{D}_t^{\mu - k}f\left(0+\right)$$

(A11)

where $\Re \left(s\right) > 0,\Re \left(\mu \right) > 0$ and $m - 1 \le \mu < m$.

Appendix A.1.3. Caputo Fractional Derivative

The Caputo derivative of a casual function $f\left(t\right)$, i.e., $f\left(t\right) = 0$ for $t < 0$, is defined [100,101] as

$${ }^C{D}_t^{\mu }f\left(t\right) ={ }_0{I}^{m - \mu }\frac{d^m}{d{t}^m}f\left(t\right) = { }_0{D}_t^{-\left(m - \mu \right)}{f}^{\left(m\right)}\left(t\right) = \frac{1}{\mathrm{\Gamma }\left(m - \mu \right)}\underset{0}{\overset{t}{\int }}\frac{f^{\left(m\right)}\left(t\right)}{{\left(\tau - 1\right)}^{\mu + 1 - m}}dt$$

(A12)

where $m\in \mathbb{N}$ and $m - 1 < \mu < m$

The Laplace transform of the Caputo derivative is

$$L\left[{ }^C{D}_t^{\mu }f\left(t\right);s\right] = s^{\mu }F\left(s\right) - \sum _{k = 0}^{m - 1}{f}^{\left(m\right)}\left(0\right)s^{\mu - k - 1}$$

(A13)

Caputo derivative of a constant is zero, i.e., that matches the common knowledge we have from the integer order calculus, and because if the notion of Riemann-–Liouville derivative is the preferred fractional derivative among mathematicians, while Caputo fractional derivative is the preferred one among engineers [102].

If $f\left(0\right) = f^{\prime }\left(0\right) = f^{″}\left(0\right) = ... = f^{\left(m - 1\right)}\left(0\right) = 0$, then both Riemann–Liouville and Caputo derivatives coincide. In particular for $\mu \in \left(0.1\right)$ and $f\left(0\right) = 0$ one has ${ }^C{D}_t^{\mu }f\left(t\right) = { }^{RL}{D}_t^{\mu }f\left(t\right)$. This article uses the notations ${ }^{RL}{D}_t^{\mu }f\left(t\right)$ and ${ }^C{D}_t^{\mu }f\left(t\right)$ to discriminate the effects on the solutions when both derivatives are used.

Appendix A.1.4. Mittag-Leffler Function: Properties

The two-parameter of Mittag-Leffler type is defined as a series expansion [102]

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

(A14)

It is an entire function of the variable z for any $\alpha ,\beta \in C$, $Re\left(\alpha > 0\right)$, of order $\rho = 1/\alpha$ and converges for each finite z and, therefore, its sum is an analytic function in the whole complex plane. From this definition, it follows that

$$E_{1,1}\left(z\right) = \sum _{k = 0}^{\infty }\frac{z^k}{\mathrm{\Gamma }\left(k + 1\right)} = \sum _{k = 0}^{\infty }\frac{z^k}{k!} = e^z$$

(A15)

$$E_{1,2} = \sum _{k = 0}^{\infty }\frac{z^k}{\mathrm{\Gamma }\left(k + 2\right)} = \sum _{k = 0}^{\infty }\frac{z^k}{(k + 1)!} = \frac{1}{z}\sum _{k = 0}^{\infty }\frac{z^{k + 1}}{(k + 1)} = \frac{e^z - 1}{z}$$

(A16)

Integration of the Mittag-Leffler-Function

$$\underset{0}{\overset{z}{\int }}{E}_{\alpha ,\beta }\left(\lambda t^{\alpha }\right)t^{\beta - 1} = z^{\beta }{E}_{\alpha ,\beta + 1}\left(\lambda t^{\alpha }\right),\beta > 0$$

(A17)

Derivatives of the Mittag-Leffler-Function

By the Riemann–Liouville definition of fractional derivative, the fractional differentiation of (A14) is

$${ }^{RL}{D}_t^{\gamma }\left[t^{\alpha k + \beta - 1}{E}_{\alpha ,\beta }^{\left(k\right)}\left(\lambda t^{\alpha }\right)\right] = t^{\alpha k + \beta - \gamma - 1}{E}_{\alpha ,\beta - \gamma }^{\left(k\right)}\left(\lambda t^{\alpha }\right)$$

(A18)

When, $k = 0$, $\lambda = 1$ and $\gamma = m$ is an integer we get

$${\left(\frac{d}{dt}\right)}^m\left[t^{\beta - 1}{E}_{\alpha ,\beta }\left(t^{\alpha }\right)\right] = t^{\beta - m - 1}{E}_{\alpha ,\beta - m}\left(t^{\alpha }\right),m = 1,2,3,\cdots$$

(A19)

For $m = 1$ we have

$$\frac{d}{dt}\left[t^{\beta - 1}{E}_{\alpha ,\beta }\left(t^{\alpha }\right)\right] = t^{\beta - 2}{E}_{\alpha ,\beta - 1}\left(t^{\alpha }\right)$$

(A20)

When $\beta = 1$ (one parameter Mittag-Leffler function (A20) yields

$$\frac{d}{dt}\left[E_{\alpha ,1}\left(t^{\alpha }\right)\right] = \frac{E_{\alpha ,0}\left(t^{\alpha }\right)}{t}$$

(A21)

For $\lambda \ne 1$

$$\frac{d}{dt}\left[E_{\alpha ,1}\left(\lambda t^{\alpha }\right)\right] = \frac{E_{\alpha ,0}\left(\lambda t^{\alpha }\right)}{t}$$

(A22)

Inverse Mittag-Leffler function The generalized inverse Mittag-Leffler function $E_{\alpha ,\beta }^{-1}\left(z\right)$ is a solution of the equation [103]

$$E_{\alpha ,\beta }^{-1}\left[E_{\alpha ,\beta }\left(z\right)\right] = z$$

(A23)

The ability to calculate $E_{\alpha ,\beta }\left(z\right)$ allows to evaluate $E_{\alpha ,\beta }^{ - 1}\left(z\right)$ by a numerical solution of (A23) and it can be determined if 3 conditions are satisfied [103]: (1) The function $E_{\alpha ,\beta }^{-1}\left(z\right)$ is single-valued and well defined on its principle branch; (2) The principle branch reduces to the principle branch of logarithm for $\alpha \to 1$. (3) The principle branch is a simply connected subset of the complex plane.

Appendix B.1. Caputo-Fabrizio Fractional Operator

The Caputo-Fabrizio derivative is defined as [13]

$${ }_{CF}{D}_t^{\alpha }f\left(t\right) = \frac{M\left(\alpha \right)}{1 - \alpha }\underset{0}{\overset{t}{\int }}exp\left[-\frac{\alpha \left(t - s\right)}{1 - \alpha }\right]\frac{df\left(t\right)}{dt}ds$$

(A24)

The normalization function $M\left(\alpha \right)$ should satisfy the conditions $M\left(0\right) = M\left(1\right) = 1$.

According to the definition (A24) if $f\left(t\right) = C = const.$, then ${ }_{CF}{D}_t^{\alpha }f\left(t\right) = 0$, an expected results as in the classical Caputo derivative [102]. The analysis of Losada and Nieto [104] on the associate integral and the behaviour of $M\left(\alpha \right)$ leads to the definition in the form (A25)

$${ }_{CF}{D}_t^{\alpha }f\left(t\right) = \frac{1}{1 - \alpha }\underset{0}{\overset{t}{\int }}exp\left[-\frac{\alpha \left(t - s\right)}{1 - \alpha }\right]\frac{df\left(t\right)}{dt}ds$$

(A25)

Then, the associated integral of (A25) is [104]

$${ }_{LN}^{cf}{I}^{\alpha }f\left(t\right) = \left(1 - \alpha \right)u\left(t\right) + \alpha \underset{0}{\overset{t}{\int }}u\left(s\right)ds,t \ge 0$$

(A26)

while the Laplace transform is

$$L_T\left[{ }_{cf}^c{D}_t^{\alpha }f\left(t\right)\right] = \frac{p{L}_T\left[f\left(t\right) - f\left(0\right)\right]}{p + \alpha \left(1 - p\right)}$$

(A27)

Appendix B.1.1. Caputo-Fabrizio Fractional Operator: The Fractional Parameter

In the definition of the Caputo-Fabrizio fractional operator, the stretched time is multiplied by a dimensional factor $\alpha /\left(1 - \alpha \right)$ which should have a dimension $s^{-1}$ while actually, it should be dimensionless because physically $\alpha$ is a dimensionless parameter. By a nondimesionalization of the exponential function with help of characteristic time of the relaxation process $t_0$, namely

$$exp\frac{\left(t - s\right)}{\tau } = exp\frac{\left(t/t_0 - s/t_0\right)}{\tau /t_0} = exp\frac{\left(\overline{t} - \overline{s}\right)}{\overline{\tau }}$$

(A28)

This nondimesionalization does not change the meaning of the exponential relaxation function but avoids any doubts about the definition of the fractional order $\alpha$ as [105].

$$\frac{1 - \alpha }{\alpha } = \frac{\tau }{t_0}\Rightarrow \alpha = \frac{1}{1 + \tau /t_0}$$

(A29)

Appendix B.1.2. Relationship of Caputo-Fabrizio Operator to the Caputo Derivative of Distributed Order

Atanackovic et al. [106] demonstrated that the operator of type (A26) expressed as (A30) is a fractional operator, namely

$$\begin{array}{c}\hfill { }^{CF}{D}_t^{\alpha ,\beta }y\left(t\right) = {\int }^{\beta \left[\frac{U_0}{T}\frac{dY\left(\overline{s}\right)}{d\overline{s}}\right]\left(Td\overline{s}\right)}{{ }_{\alpha }}^{CF}{D}_t^{\gamma }y\left(t\right)d\gamma = \\ \hfill = {\int }_{\alpha }^{\beta }\frac{1}{1 - \gamma }{\int }_0^{t}exp\left(-\frac{\gamma }{1 - \gamma }\right)\frac{dy\left(s\right)}{ds}d\gamma \end{array}$$

(A30)

with $\gamma \in \left(\alpha ,\beta \right),t\in \left[0,\infty \right),t > 0$. More details and analyzes are available in [106].

Further, taking from [107] that the Caputo fractional derivative can be expressed as

$${ }^C{D}_t^{\alpha }y\left(t\right) = {\int }_0^{\infty }\frac{1}{\mathrm{\Gamma }(1 - \alpha )\mathrm{\Gamma }\left(\alpha \right){\zeta }^{\alpha + 1}}\left[{\int }_0^{t}exp-\left(\frac{t - s}{\zeta }\right)\frac{dy\left(s\right)}{ds}\right]d\zeta ,t \ge 0$$

(A31)

and combining (A31) with the basic definition (A24) it is proved that [106])

$$\begin{array}{c}\hfill { }^C{D}_t^{\alpha }y\left(t\right) = \frac{1}{\mathrm{\Gamma }(1 - \alpha )\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^t\left[{ }^{CF}{D}_t^{\gamma y}y\left(t\right)\mu \left(t\right)\right]d\gamma ,\\ \hfill \mu \left(\gamma \right) = \frac{1}{{(1 - \gamma )}^{\alpha }{\gamma }^{(1 - \alpha )}},\gamma \in (0,1),t \ge 0\end{array}$$

(A32)

From this result, the classical Caputo derivative could be treated as a distributed order Caputo–Fabrizio operator with a weighting function $\mu \left(\gamma \right)$ [106].

Appendix B.2. Derivatives with Non-Singular Mittag-Leffler Kernels

The Atangana–Baleanu derivatives (AB derivatives) have two basic definitions

Riemann–Liouville sense (ABR derivative)

$${ }^{ABR}{D}_{a + }^{\alpha }f\left(t\right) = \frac{B\left(\alpha \right)}{1 - \alpha }\frac{d}{dt}\underset{0}{\overset{z}{\int }}f\left(z\right)E_{\alpha }\left[\frac{-\alpha }{1 - \alpha }{\left(t - z\right)}^{\alpha }\right]dz$$

(A33)

with $0 < \alpha < 1$, $a < t < b$ and $f\left(t\right)\in L^1\left[a,b\right]$.

Caputo sense (ABC derivative)

$${ }^{ABC}{D}_{a + }^{\alpha }f\left(t\right) = \frac{B\left(\alpha \right)}{1 - \alpha }\underset{0}{\overset{z}{\int }}\frac{df\left(z\right)}{dz}{E}_{\alpha }\left[\frac{-\alpha }{1 - \alpha }{\left(t - z\right)}^{\alpha }\right]dz$$

(A34)

with $0 < \alpha < 1$, $a < t < b$ and $f\left(x,t\right)$ is differentiable function on $\left[a,b\right]$ such that $df/dt\in L^1\left[a,b\right]$.

The normalization function $B\left(\alpha \right)$ can be any function satisfying the conditions $B\left(0\right) = B\left(1\right) = 1$. In these definitions $E_{\alpha }$ is one-parameter Mittag-Leffler function $E_{\alpha } = {\displaystyle \sum _0^{\infty }}\frac{z^k}{\mathrm{\Gamma }\left(\alpha k + 1\right)}$ [102].

The Laplace transform of ABR derivative is [14]

$$L\left\{{ }_0^{ABR}{D}_t^{\alpha }\left[f\left(t\right)\right]\right\}\left(p\right) = \frac{B\left(\alpha \right)}{1 - \alpha }\frac{p^{\alpha }}{p^{\alpha } + \frac{\alpha }{1 - \alpha }}L\left\{f\left(t\right)\right\}\left(p\right)$$

(A35)

Similarly for the ABC derivative

$$L\left\{{ }_0^{ABC}{D}_t^{\alpha }\left[f\left(t\right)\right]\right\}\left(p\right) = \frac{B\left(\alpha \right)}{1 - \alpha }\frac{p^{\alpha }}{p^{\alpha } + \frac{\alpha }{1 - \alpha }}\left[L\left\{f\left(t\right)\right\}\left(p\right) - p^{\alpha - 1}f\left(0\right)\right]$$

(A36)

The relation between ABR and ABC is [14]

$${ }_0^{ABC}{D}_t^{\alpha }\left[f\left(t\right)\right] = { }_0^{ABR}{D}_t^{\alpha }\left[f\left(t\right)\right] - \frac{B\left(\alpha \right)}{1 - \alpha }f\left(0\right)E_{\alpha }\left(-\frac{\alpha }{1 - \alpha }{t}^{\alpha }\right)$$

(A37)

Hence, with zero initial conditions, both derivatives are identical, a property already known from the classical Riemann–Liouville and Caputo-Liouville derivatives [102].

As was demonstrated by Atangana and Baleanu [14], the following fractional differential equation

$${ }_0^{ABR}{D}_t^{\alpha }\left[f\left(t\right)\right] = u\left(t\right)$$

(A38)

has a unique solution

$$f\left(t\right) = \frac{1 - \alpha }{B\left(\alpha \right)}u\left(t\right) + \frac{\alpha }{B\left(\alpha \right)}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^{t}u\left(\tau \right){(t - \tau )}^{\alpha - 1}d\tau$$

(A39)

where for $\alpha = 0$ we recover the initial function, while for $\alpha = 1$ we get the ordinary Riemann integral.

The ABR fractional derivative can be expressed as [97]

$${ }_0^{ABR}{D}_t^{\alpha }\left[f\left(t\right)\right] = \frac{B\left(\alpha \right)}{1 - \alpha }\sum _{k = 0}^{\infty }{\left(\frac{ - \alpha }{1 - \alpha }\right)}^k\frac{d}{dt}\left[{ }^{RL}{I}_{a + }^{\alpha k+1}f\left(t\right)\right]$$

(A40)

The AB fractional integral operator ${ }^{AB}{I}_{a+}^{\alpha }$ follows directly from the solution (A39) and can be precisely defined as [97]

$${ }^{AB}{I}_{a+}^{\alpha }f\left(t\right) = \frac{1 - \alpha }{B\left(\alpha \right)}f\left(t\right) + {\frac{\alpha }{B\left(\alpha \right)}}^{RL}{I}_{a+}^{\alpha }f\left(t\right)$$

(A41)

Where ${ }^{RL}{I}_{a+}^{\alpha k + 1}f\left(t\right)$ is the Riemann–Liouville fractional integral [102]. The relation (A41) can be easily developed by applying the Laplace transform to equation (A38) as it was demonstrated by Baleanu and Fernandez [97]. Further, we have the following left and right inverse properties [97]

$${ }^{AB}{I}_{a+}^{\alpha }\left[{ }^{ABR}{D}_{a + }^{\alpha }f\left(t\right)\right] = f\left(t\right)$$

(A42)

$${ }^{ABR}{D}_{a+}^{\alpha }\left[{ }^{AB}{I}_{a+}^{\alpha }f\left(t\right)\right] = f\left(t\right)$$

(A43)

and the commutative properties for $\beta \in (0,1)$

$${ }^{ABR}{D}_{a+}^{\alpha }\left[{ }^{ABR}{D}_{a+}^{\beta }f\left(t\right)\right] ={ }^{ABR}{D}_{a+}^{\beta }\left[{ }^{ABR}{D}_{a+}^{\alpha }f\left(t\right)\right]$$

(A44)

$${ }^{AB}{I}_{a+}^{\alpha }\left[{ }^{AB}{I}_{a+}^{\beta }f\left(t\right)\right] ={ }^{AB}{I}_{a+}^{\beta }\left[{ }^{AB}{I}_{a+}^{\alpha }f\left(t\right)\right]$$

(A45)

$${ }^{ABR}{D}_{a+}^{\alpha }\left[{ }^{AB}{I}_{a+}^{\alpha }f\left(t\right)\right] ={ }^{AB}{I}_{a+}^{\alpha }\left[{ }^{ABR}{D}_{a+}^{\alpha }f\left(t\right)\right]$$

(A46)

For the sake of the simplicity, assuming hereafter $B\left(\alpha \right) = 1$, we get from (A41)

$${ }^{AB}{I}_{a+}^{\alpha }u\left(t\right) = (1 - \alpha )u(x,t)+{\alpha }^{RL}{I}_{a+}^{\alpha }u(x,t) = f\left(t\right)$$

(A47)