Generalized diffusion-wave equation with memory kernel

Before finding fundamental solution of the model proposed we recall definitions and some properties of the completely monotone (CM) and Bernstein functions (BF) [61], which are used to show the non-negativity of the fundamental solution.

2.1.1. Completely monotone functions.

Completely monotone function $\mathcal{M}(s)$ is defined in non-negative semi-axis and have a property that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{definition CM} (-1)^n \mathcal{M}^{(n)}(s) \geqslant 0 \quad {\rm for~ all}~ n\in {N}_{0} ~{\rm and}~ s\geqslant 0. \nonumber \end{align} \tag{ 6 }$$

According to the famous Bernstein theorem the completely monotone function can be represented as a Laplace transform of non-negative function $p(t)$ (see, for example, chapter XIII, section 4 in [16]), i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{CM NN} \mathcal{M}(s) = \int_0^\infty p(t){\rm e}^{-st}\,{\rm d}t. \nonumber \end{align} \tag{ 7 }$$

The following properties hold true for CM functions:

• (a)  
  Linear combination $a_1 \mathcal{M}_{1}(s)+a_2 \mathcal{M}_{2}(s)$ ($a_{1},a_{2}\geqslant0$) of completely monotone functions $\mathcal{M}_{1}(s)$ and $\mathcal{M}_{2}(s)$ is completely monotone function as well;
• (b)  
  The product $\mathcal{M}(s)=\mathcal{M}_1(s)\mathcal{M}_2(s)$ of completely monotone functions $\mathcal{M}_1(s)$ and $\mathcal{M}_2(s)$ is again completely monotone function.

An example of completely monotone function is $s^{\alpha}$, where $\alpha\leqslant0$.

2.1.2. Stieltjes functions.

2.1.3. Bernstein functions.

The Bernstein function is a non-negative function whose derivative is completely monotone, that is

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle (-1)^{(n-1)} \mathcal{B}^{(n)}(s) \geqslant 0 \quad {\rm for~ all} ~n=1,2,\dots.\nonumber \end{align*}$$

BFs have the following properties:

• (d)  
  Linear combination $a_1\mathcal{B}_1(s)+a_2\mathcal{B}_2(s)$ ($a_1,a_2 \geqslant 0$) of BFs $\mathcal{B}_1(s)$ and $\mathcal{B}_2(s)$ is a BF;
• (e)  
  A composition $\mathcal{M}(\mathcal{B}(s))$ of CM function $\mathcal{M}(s)$ and BF $\mathcal{B}(s)$ is CM function.

From these properties it follows that the function ${\rm e}^{-u\,\mathcal{B}(s)}$ is CM for $u>0$ if $\mathcal{B}(s)$ is a BF.

An example of a BF is $s^{\alpha}$, where $0\leqslant\alpha\leqslant1$.

2.1.4. Complete Bernstein functions.

At first let us consider the generalized diffusion-wave equation (2) with the memory kernel $\newcommand{\e}{{\rm e}} \eta(t)$ of a general form. The restrictions on the memory kernel are discussed below. In order to find the solution of equation (2) we use the Fourier $\newcommand{\im}{{\rm Im}} (\tilde{F}(\kappa)=\int_{-\infty}^{\infty}f(x){\rm e}^{\imath\kappa x}\,{\rm d}x)$ and Laplace $(\hat{f}(s)=\int_{0}^{\infty}f(t){\rm e}^{-st}\,{\rm d}t)$ transformations. For initial conditions $W(x,0)=\delta(x)$ and $\frac{\partial}{\partial t}W(x,0)=0$, and zero boundary conditions we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq Laplace-Fourier space M} \tilde{\hat{W}}(\kappa,s)=\frac{s\hat{\eta}(s)}{s^{2}\hat{\eta}(s)+\kappa^{2}}. \nonumber \end{align} \tag{ 8 }$$

By the inverse Fourier transformation of equation (8) we find the PDF in the Laplace space

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{PDF L} \hat{W}(x,s)=\frac{1}{2}\sqrt{\hat{\eta}(s)}\exp\left(-s\sqrt{\hat{\eta}(s)}|x|\right), \nonumber \end{align} \tag{ 9 }$$

from which one can find either the exact form of $W(x,t)$ or (at least) its asymptotic behavior by using Tauberian theorems [16]. From equation (9) one can easily conclude that the PDF is normalized to 1, i.e. $\int_{-\infty}^{\infty}W(x,t)\,{\rm d}x=1$, since $\int_{-\infty}^{\infty}\hat{W}(x,s)\,{\rm d}x=\frac{1}{s}$.

The non-negativity of the solution of the distributed order wave equation has been shown in [23] by using the properties of CM functions and BFs. In order to find the constraints for the memory kernel $\newcommand{\e}{{\rm e}} \eta(t)$ under which the considered generalized diffusion-wave equation (2) has a non-negative solution we use representation (9) in the Laplace space together with relation (7). The solution (9) can be considered as a product of two functions, $\newcommand{\e}{{\rm e}} \frac{1}{2}\sqrt{\hat{\eta}(s)}$ and $\newcommand{\e}{{\rm e}} \exp\left(-s\sqrt{\hat{\eta}(s)}|x|\right)$. In order to show non-negativity of the PDF $W(x,t)$ one should prove that its Laplace transform $\hat{W}(x,s)$ is a CM function. Thus, it is sufficient to prove that both functions

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sqrt{\hat{\eta}(s)} \quad {\rm and} \quad \exp\left(-s\sqrt{\hat{\eta}(s)}|x|\right)\nonumber \end{align*}$$

are CM (see property (b) in section 2.1.1.). Therefore, it is sufficient to show that $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is CM function, and $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}$ is BF, according to the property (e) in section 2.1.3. The non-negativity of the solution can also be shown by proving that the function $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is SF, which is again a CM function, or that $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}$ is CBF, which follows from the properties (f) and (g) in section 2.1.4.

Remark 1. Here we note that the subordination approach, which was used to prove the non-negativity of the PDF in case of fractional and distributed order diffusion equations [23, 53], can not be applied in case of generalized diffusion-wave equations. Indeed, the solution (8) can be rewritten in a form of subordination integral as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tilde{\hat{W}}(k,s)=s\hat{\eta}(s)\int_{0}^{\infty}{\rm e}^{-u[s^{2}\hat{\eta}(s)+k^{2}]}\,{\rm d}u=\int_{0}^{\infty}{\rm e}^{-uk^{2}}\hat{G}(u,s)\,{\rm d}u, \nonumber \end{align} \tag{ 10 }$$

where

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{G}(u,s)=s\hat{\eta}(s){\rm e}^{-us^{2}\hat{\eta}(s)}.\nonumber \end{align*}$$

Therefore, the PDF $W(x,t)$ becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W(x,t)=\int_{0}^{\infty}\frac{{\rm e}^{-\frac{x^{2}}{4u}}}{\sqrt{4\pi u}}G(u,t)\,{\rm d}u. \nonumber \end{align} \tag{ 11 }$$

From here it follows that in order $W(x,t)$ to be non-negative, the function $G(u,t)$ should be non-negative, that is $\hat{G}(u,s)$ should be CM. For this it is sufficient to show that $\newcommand{\e}{{\rm e}} s\hat{\eta}(s)$ is CM and $\newcommand{\e}{{\rm e}} s^{2}\hat{\eta}(s)$ is a BF according to property $({{\rm \bf e}})$ in section 2.1.3. Such approach works well in case of fractional and distributed order diffusion equations but, as we will see later, not in case of generalized diffusion-wave equation. This was already noted in [23] while proving the non-negativity of the distributed order diffusion-wave equation that is a particular case of the generalized diffusion-wave equation.

From equation (8) we calculate the $q$th moment $\left\langle |x|^{q}(t)\right\rangle=\int_{-\infty}^{\infty}|x|^{q}W(x,t)\,{\rm d}x=2\int_{0}^{\infty}x^{q}W(x,t)\,{\rm d}x$ of the solution of equation (2), which is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{moments} \left\langle |x|^{q}(t)\right\rangle=\Gamma(q+1)\mathcal{L}^{-1}\left[\frac{s^{-1}}{\left(s^{2}\hat{\eta}(s)\right)^{q/2}}\right](t). \nonumber \end{align} \tag{ 12 }$$

For the second moment we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{second moment} \left\langle x^{2}(t)\right\rangle=\left.\left\{-\frac{\partial^{2}}{\partial\kappa^{2}}\mathcal{L}^{-1}\left[\tilde{\hat{W}}(\kappa,s)\right](\kappa,t)\right\}\right|_{\kappa=0}=2\,\mathcal{L}^{-1}\left[\frac{1}{s^{3}\hat{\eta}(s)}\right](t), \nonumber \end{align} \tag{ 13 }$$

from where one can analyze diffusion regimes for different forms of the memory function $\newcommand{\e}{{\rm e}} \eta(t)$.

Equation (4) for a Dirac delta memory kernel $\newcommand{\e}{{\rm e}} \eta(t)=\delta(t)$ yields the classical wave equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{classical cattaneo eq} \frac{\partial^{2}}{\partial t^{2}}W(x,t)=\frac{\partial^{2}}{\partial x^{2}}W(x,t). \nonumber \end{align} \tag{ 14 }$$

Its solution can be obtained from (8) in the d'Alembert form,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{PDF L delta} W(x,t)=\mathcal{F}^{-1}\left[\cos(t\,\kappa)\right]=\frac{1}{2}\left[\delta(x+t)+\delta(x-t)\right]. \nonumber \end{align} \tag{ 15 }$$

The non-negativity of the solution of standard wave equation (14) is obvious since $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}=1$ is CM (see the example in section 2.1.1), and $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}=s$ is a BF (see the example in section 2.1.3.).

Remark 2. Note that if one uses the sufficient conditions of the subordination approach (see remark 1), then $\newcommand{\e}{{\rm e}} s\eta(s)=s$ is CM, but $\newcommand{\e}{{\rm e}} s^{2}\eta(s)=s^{2}$ is not a BF, thus the sufficient conditions are not fulfilled. However, as we have shown above the solution is non-negative.

From equation (13) one easily finds the second moment,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{classical cattaneo eq MSD} \left\langle x^{2}(t)\right\rangle=t^{2}, \nonumber \end{align} \tag{ 16 }$$

which is an indication of the ballistic motion.

Let us take

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{power memory} \eta(t)=\frac{t^{1-\mu}}{\Gamma(2-\mu)}, \quad {\rm i.e.} \quad \hat{\eta}(s)=s^{\mu-2}, \nonumber \end{align} \tag{ 17 }$$

where $0<\mu<2$. We consider two cases. The case with $1<\mu<2$ corresponds to the fractional wave equation with Caputo fractional derivative (5) of the order $\mu$ [32],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{frac diff mu>1/2} {_{C}D_{t}^{\mu}}W(x,t)=\frac{\partial^{2}}{\partial x^{2}}W(x,t), \nonumber \end{align} \tag{ 18 }$$

whereas for the case with $0<\mu<1$ we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{frac diff mu<1/2} \frac{1}{\Gamma(2-\mu)}\int_{0}^{t}(t-t')^{1-\mu}\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}t'=\frac{\partial^{2}}{\partial x^{2}}W(x,t). \nonumber \end{align} \tag{ 19 }$$

We note that the left hand side of equation (19) cannot be written as a fractional derivative. The solution of equations (18) and (19) in the Fourier–Laplace space is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{fractional wave eq LF} \tilde{\hat{W}}(\kappa,s)=\frac{s^{\mu-1}}{s^{\mu}+\kappa^{2}}. \nonumber \end{align} \tag{ 20 }$$

The solution is non-negative since $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}=s^{\mu/2-1}$ is CM (see the example in section 2.1.1.) and $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}=s^{\mu/2}$ is a BF for $0<\mu<2$ (see the example in section 2.1.3.).

Remark 3. Similar to the previous case of the standard wave equation, here the sufficient conditions of the subordination approach require $\newcommand{\e}{{\rm e}} s\hat{\eta}(s)=s^{\mu-1}$ to be CM and $\newcommand{\e}{{\rm e}} s^{2}\hat{\eta}(s)=s^{\mu}$ to be BF. However, both conditions are not satisfied for $1<\mu<2$.

By the inverse Fourier–Laplace transformation of equation (20) one finds the solution in terms of the Fox $H$-function,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{fractional wave eq LF one power} W(x,t)=\mathcal{F}^{-1}\left[E_{\mu}\left(-t^{\mu}\kappa^{2}\right)\right] =\frac{1}{2|x|}H_{1,1}^{1,0}\left[\left.\frac{|x|}{t^{\mu/2}}\right|\begin{array}{@{}l@{}}\displaystyle\left(1,\mu/2\right) \nonumber \\ (1,1)\end{array}\right]. \nonumber \end{align} \tag{ 21 }$$

Here $E_{\alpha}(z)$ is the one parameter M-L function [33]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\alpha}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+1)}, \quad \Re(\alpha)>0, \nonumber \end{align} \tag{ 22 }$$

and $H_{p,q}^{m,n}(z)$ is the Fox $H$-function which is given in terms of the Mellin transform as [39]

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\im}{{\rm Im}} \displaystyle \label{H_integral} H_{p,q}^{m,n}(z)&=H_{p,q}^{m,n}\left[z\left|\begin{array}{@{}l@{}} (a_1,A_1),...,(a_p,A_p) \nonumber \\ (b_1,B_1),...,(b_q,B_q) \end{array}\right.\right]=H_{p,q}^{m,n}\left[z\left|\begin{array}{@{}l@{}} (a_p,A_p) \nonumber \\ (b_q,B_q) \end{array}\right.\right]\nonumber \\&=\frac{1}{2\pi\imath}\int_{\Omega}\theta(s)z^{-s}{\rm d}s, \nonumber \end{align} \tag{ 23 }$$

with

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \theta(s)=\frac{\prod_{j=1}^{m}\Gamma(b_j+B_js)\prod_{j=1}^{n}\Gamma(1-a_j-A_js)}{\prod_{j=m+1}^{q}\Gamma(1-b_j-B_js)\prod_{j=n+1}^{\,p}\Gamma(a_j+A_js)},\nonumber \end{align*}$$

$0\leqslant n\leqslant p$, $1\leqslant m\leqslant q$, $a_i,b_j \in \mathrm{C}$, $A_i,B_j\in\mathrm{R}^{+}$, $i=1,...,p$, $j=1,...,q$. The contour of integration $\Omega$ starts at $\newcommand{\im}{{\rm Im}} c-\imath\infty$ and finishes at $\newcommand{\im}{{\rm Im}} c+\imath\infty$ separating the poles of the function $\Gamma(b_j+B_js)$, $j=1,...,m$ with those of the function $\Gamma(1-a_i-A_is)$, $i=1,...,n$.

Remark 4. For $\mu=1$ we recover the Gaussian PDF

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{gaussian pdf} W(x,t)=\frac{1}{2|x|}H_{1,1}^{1,0}\left[\left.\frac{|x|}{t^{1/2}}\right|\begin{array}{@{}l@{}}\displaystyle\left(1,1/2\right) \nonumber \\ (1,1)\end{array}\right]=\frac{1}{\sqrt{4\pi t}}{\rm e}^{-\frac{x^{2}}{4t}}. \nonumber \end{align} \tag{ 24 }$$

The second moment for the fractional diffusion-wave equations (18) and (19) is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{MSD 1} \left\langle x^{2}(t)\right\rangle=2\,\frac{t^{\mu}}{\Gamma(1+\mu)}. \nonumber \end{align} \tag{ 25 }$$

Thus, the generalized diffusion-wave equation (2) with the memory kernel (17) is able to describe both superdiffusive and subdiffusive processes since $0<\mu<2$. The case $\mu=1$ reduces to the classical diffusion equation for the Brownian motion, i.e. $\left\langle x^{2}(t)\right\rangle=2\,t$, whereas the case with $\mu=2$ yields ballistic diffusion, $\left\langle x^{2}(t)\right\rangle=t^{2}$.

Here we consider bi-fractional diffusion-wave equation with the memory kernel of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{power memory bi} \eta(t)=\alpha_{1}\frac{t^{1-\mu_{1}}}{\Gamma(2-\mu_{1})}+\alpha_{2}\frac{t^{1-\mu_{2}}}{\Gamma(2-\mu_{2})}, \quad \alpha_{1}+\alpha_{2}=1, \nonumber \end{align} \tag{ 26 }$$

and $\newcommand{\e}{{\rm e}} \hat{\eta}(s)=\alpha_{1}s^{\mu_{1}-2}+\alpha_{2}s^{\mu_{2}-2}$. Again we consider two cases. The case with $1<\mu_1<\mu_2<2$ yields

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{two power law} \alpha_{1}{_C}D_{0+}^{\mu_{1}}W(x,t)+\alpha_{2}{_C}D_{0+}^{\mu_{2}}W(x,t)=\frac{\partial^{2}}{\partial x^{2}}W(x,t), \nonumber \end{align} \tag{ 27 }$$

where ${_C}D_{0+}^{\mu_{j}}$ is the Caputo fractional derivative (5) of the order $1<\mu_{j}<2$, whereas the case $0<\mu_{1}<\mu_{2}<1$ yields an equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq memory p 2 delta} &\frac{\alpha_{1}}{\Gamma(2-\mu_{1})}\int_{0}^{t}(t-t')^{1-\mu_{1}}\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}t'\nonumber \\&+\frac{\alpha_{2}}{\Gamma(2-\mu_{2})}\int_{0}^{t}(t-t')^{1-\mu_{2}}\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}t'=\frac{\partial^{2}}{\partial x^{2}}W(x,t). \nonumber \end{align} \tag{ 28 }$$

The solutions of both equations are non-negative. Let us show this. Since

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\alpha_{1}s^{\mu_1-2}+\alpha_{2}s^{\mu_2-2}\nonumber \end{align*}$$

is a SF for $1<\mu_{1}<\mu_{2}<2$ as a linear combination of two SFs (see property (c) in section 2.1.2.), then $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is a SF as a composition of CBF and SF (see property (g) in section 2.1.4.), which means that the function $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is CM (SFs are subclass of CM functions, see section 2.1.2.). From here we may conclude that $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}$ is a CBF (see property (h) in section 2.1.4.), and hence the solution of equation (27) is non-negative. On the other hand, for $0<\mu_{1}<\mu_{2}<1$, equation (28), we use the subordination approach (see remark 1), that is $\newcommand{\e}{{\rm e}} s\hat{\eta}(s)=\alpha_{1}s^{\mu_1-1}+\alpha_{2}s^{\mu_2-1}$ is CM (see property (a) and the example in section 2.1.1.), and $\newcommand{\e}{{\rm e}} s^{2}\hat{\eta}(s)=\alpha_{1}s^{\mu_1}+\alpha_{2}s^{\mu_2}$ is a BF (see property (d) and the example in section 2.1.3.).

The solution of equations (27) and (28) can be found in terms of infinite series of the Fox $H$-function by using the series expansion approach [50], as it is done for the bi-fractional diffusion equation in [56]. Therefore, the exact solution becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed2 PDF two delta} P(x,t)&=\frac{1}{\sqrt{\frac{4\pi}{B_2}t^{\mu_2}}} \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\left(\frac{\alpha_1}{\alpha_2}\right)^n t^{\left(\mu_2-\mu_1\right)n}\nonumber \\&\times\left\{H_{1,2}^{2,0}\left[\frac{x^2}{\frac{4}{\alpha_2}t^{\mu_2}}\left| \begin{array}{@{}l@{}}\left([\mu_2-\mu_1]n+1-\mu_2/2,\mu_2\right) \nonumber \\ (0,1),(n+1/2,1)\end{array}\right.\right]\right.\nonumber \\ &\left.+\frac{\alpha_1}{\alpha_2}t^{\mu_2-\mu_1}H_{1,2}^{2,0}\left[\frac{x^2}{\frac{4}{\alpha_2}t^{\mu_2}}\left|\begin{array}{@{}l@{}}\left([\mu_2-\mu_1](n+1)+1-\mu_2/2,\mu_2\right) \nonumber \\ (0,1),(n+1/2,1)\end{array}\right.\right]\right\}, \nonumber \end{align} \tag{ 29 }$$

while the second moment for both cases, equations (27) and (28), becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msdPower2} \left\langle x^{2}(t)\right\rangle=2\,\mathcal{L}^{-1}\left[\frac{s^{-1}}{\alpha_{1}s^{\mu_{1}}+\alpha_{2}s^{\mu_{2}}}\right](t)=\frac{2}{\alpha_{2}}t^{\mu_{2}}E_{\mu_2-\mu_1,\mu_2+1}\left(-\frac{\alpha_{1}}{\alpha_{2}}t^{\mu_{2}-\mu_{1}}\right),\nonumber \\ \nonumber \end{align} \tag{ 30 }$$

where $E_{\alpha,\beta}(z)$ is the two parameter M-L function [33]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ml2} E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+\beta)}, \quad \Re(\alpha)>0. \nonumber \end{align} \tag{ 31 }$$

From equation (31) and by using the formula

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\alpha,\beta}(-z)\simeq-\sum_{n=1}^{N}\frac{(-z)^{-n}}{\Gamma(\beta-\alpha n)}, \quad z\gg1, \nonumber \end{align} \tag{ 32 }$$

we obtain the following asymptotics:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd bi-frac asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} \frac{2}{\alpha_2}\frac{t^{\mu_2}}{\Gamma(1+\mu_2)}, \quad & t\ll1 , \nonumber \\ \frac{2}{\alpha_1}\frac{t^{\mu_1}}{\Gamma(1+\mu_1)}, \quad & t\gg1 , \end{array}\right. \nonumber \end{align} \tag{ 33 }$$

which means decelerating superdiffusion for $1<\mu_{1}<\mu_{2}<2$, including crossover from superdiffusion to normal diffusion in the case $1=\mu_1<\mu_2<2$, and decelerating subdiffusion for $0<\mu_{1}<\mu_{2}<1$, including crossover from normal diffusion to subdiffusion for the case $0<\mu_1<\mu_2=1$. Graphical representation of the second moment (30) is given in figure 1. Such crossover, for example, from superdiffusion to normal diffusion has been observed in Hamiltonian systems with long-range interactions [28], and different biological systems [8, 70].

Zoom In Zoom Out Reset image size

Remark 5. We note that for the case $0<\mu_1<1$ and $1<\mu_2<2$ we are not able to prove the non-negativity of the solution. Thus, whether such diffusion-wave equation for the PDF may exist or not is still an open question.

A natural generalization of the previous case is a memory kernel of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{memory N} \eta(t)=\sum_{j=1}^{N}\alpha_{j}\frac{t^{1-\mu_{j}}}{\Gamma(2-\mu_{j})}, \nonumber \end{align} \tag{ 34 }$$

where $0<\mu_1<\mu_2<\dots<\mu_N<2$. From (34) we find $\newcommand{\e}{{\rm e}} \hat{\eta}(s)=\sum_{j=1}^{N}\alpha_{j}s^{\mu_{j}-2}$. Thus, from equation (2) with kernel (34) and $1<\mu_{j}<2$ we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq memory p N delta} \sum_{j=1}^{N}\alpha_{j}{_C}D_{0+}^{\mu_{j}}W(x,t)=\frac{\partial^{2}}{\partial x^{2}}W(x,t), \nonumber \end{align} \tag{ 35 }$$

where ${_C}D_{0+}^{\mu_{j}}$ is the Caputo fractional derivative (5) of order $1<\mu_{j}<2$. For $0<\lambda<1$, $0<\mu_{j}<1$, $j=1,2,\dots,N$ we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq memory p N delta2} \sum_{j=1}^{N}\alpha_{j}\int_{0}^{t}\frac{(t-t')^{1-\mu_{j}}}{\Gamma(2-\mu_{j})}\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}t'=\frac{\partial^{2}}{\partial x^{2}}W(x,t). \nonumber \end{align} \tag{ 36 }$$

The solutions of equation (35) with $1<\mu_1<\mu_2<\dots<\mu_N<2$ is non-negative since

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\sum_{j=1}^{N}\alpha_{j}s^{\mu_j-2}\nonumber \end{align*}$$

is a SF (see property (c) in section 2.1.2.), that is $\newcommand{\e}{{\rm e}} \sqrt{\eta(s)}$ a SF too (see property (g) in section 2.1.4.), which means that $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is CM. Thus, $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}$ is a CBF (see property (h) in section 2.1.4.), with which we complete the proof of non-negativity of the solution. For the case with $0<\mu_1<\mu_2<\dots<\mu_N<1$ the non-negativity of the solution of equation (36) can be shown by using the subordination approach (see remark 1), that is $\newcommand{\e}{{\rm e}} s\hat{\eta}(s)=\sum_{j=1}^{N}\alpha_{j}s^{\mu_j-1}$ a CM, and $\newcommand{\e}{{\rm e}} s^{2}\hat{\eta}(s)=\sum_{j=1}^{N}\alpha_{j}s^{\mu_j}$ is a BF.

For the MSD we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msdPowerN} &\left\langle x^{2}(t)\right\rangle=2\,\mathcal{L}^{-1}\left[\frac{s^{-1}}{\sum_{j=1}^{N}\alpha_{j}s^{\mu_{j}}}\right](t)\nonumber \\&=\frac{2}{\alpha_{N}}t^{\mu_{N}}E_{(\mu_N-\mu_1,\dots,\mu_N-\mu_{N-1}),\mu_N+1}\left(-\frac{\alpha_{1}}{\alpha_{N}}t^{\mu_{N}-\mu_{1}},\dots,-\frac{\alpha_{N-1}}{\alpha_{N}}t^{\mu_{N}-\mu_{N-1}}\right),\nonumber \\ \nonumber \end{align} \tag{ 37 }$$

where $E_{(a_1,a_2,\dots,a_N),b}(z)$ is the multinomial M-L function [25, 30], defined by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\left(a_{1},a_{2},\ldots,a_{N}\right),b}\left(z_{1},z_{2},\ldots,z_{N}\right)=& \sum_{j=1}^{\infty}\sum_{k_{1}\geqslant0,k_{2}\geqslant0,\ldots,k_{N}\geqslant0}^{k_{1}+k_{2}+ \ldots+k_{N}=j}\left(\begin{array}{@{}c@{}}j \nonumber \\k_{1} \quad k_{2} \quad \ldots \quad k_{N} \end{array}\right)\nonumber \\&\times\frac{\prod_{i=1}^{N}\left(z_{i}\right)^{k_{i}}}{\Gamma\left(b+ \sum_{i=1}^{N}a_{i}k_{i}\right)}, \nonumber \end{align} \tag{ 38 }$$

where

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \left(\begin{array}{@{}c@{}}j \nonumber \\k_{1} \quad k_{2} \quad ... \quad k_{N}\end{array}\right)= \frac{j!}{k_{1}!k_{2}!...k_{N}!}\nonumber \end{align*}$$

are the so-called multinomial coefficients. For the asymptotic behavior of the second moment we get

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd N-frac asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} \frac{2}{\alpha_N}\frac{t^{\mu_N}}{\Gamma(1+\mu_N)}, \quad & t\ll1 \nonumber \\ \frac{2}{\alpha_1}\frac{t^{\mu_1}}{\Gamma(1+\mu_1)}, \quad & t\gg1 \end{array}\right. \nonumber \end{align} \tag{ 39 }$$

i.e. decelerating superdiffusion for $1<\mu_1<\mu_2<\dots<\mu_N<2$, and decelerating subdiffusion for $0<\mu_1<\mu_2<\dots<\mu_N<1$. We note that the multinomial M-L function plays an important role in the analysis of the generalized Langevin equation [58] and distributed order diffusion equations [53, 56].

Graphical representation of the second moment in case of three power-law memory kernels ($N=3$) is given in figure 2. It is seen that the second moment in the short time limit behaves as power-law with exponent $\mu_3$, which turns to power-law behavior with an exponent $\mu_1$. In the intermediate time the behavior is represented by the multinomial M-L function. By introducing more than two power-law memory kernels it is possible to achieve better fit in the intermediate time domain between short and long time asymptotics.

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Next we consider uniformly distributed memory kernel

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \eta(t)=\int_{1}^{2}\frac{t^{1-\lambda}}{\Gamma(2-\lambda)}\,{\rm d}\lambda,\nonumber \end{align*}$$

that corresponds to $p(\lambda)=1$ in equation (4). Equation (2) then reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq memory p=1} \int_{0}^{t}\int_{1}^{2}\frac{(t-t')^{1-\lambda}}{\Gamma(2-\lambda)}\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}\lambda {\rm d}t'=\frac{\partial^{2}}{\partial x^{2}}W(x,t), \nonumber \end{align} \tag{ 40 }$$

i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{distributed order wave eq memory p=1 2} \int_{1}^{2}{_C}D_{0+}^{\lambda}W(x,t)\,{\rm d}\lambda=\frac{\partial^{2}}{\partial x^{2}}W(x,t), \nonumber \end{align} \tag{ 41 }$$

where ${_C}D_{0+}^{\lambda}$ is the Caputo fractional derivative of the order $1<\lambda<2$.

Let us show the non-negativity of the solution. The function

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\int_1^2p(\lambda)s^{\lambda-2}\,{\rm d}\lambda\nonumber \end{align*}$$

is a SF since the function $\sum_{j}p_{j}s^{\lambda_{j}-2}$ with $p_{j}\geqslant0$ and $1<\lambda_{j}\leqslant2$ is a SF (see property (c) in section 2.1.2.), and a pointwise limit of this linear combination $\newcommand{\e}{{\rm e}} \eta(s)=\int_0^1p(\lambda)s^{\lambda-2}\,{\rm d}\lambda$ is a SF too. Therefore, $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is a SF as well, since a composition of CBF and SF is a SF (see property (g) in section 2.1.4.), and thus it is CM. Therefore, $\newcommand{\e}{{\rm e}} s\sqrt{\hat{\eta}(s)}$ is CBF (if $c(x)$ is a CBF, then $c(x)/x$ is a SF (see property (h) in section 2.1.4.), which means that the solution of the equation (41) is non-negative.

From the solution in Laplace space,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{asympt pdf} \hat{W}(x,s)=\frac{1}{2s}\left(s\sqrt{\frac{s-1}{s\log{s}}}\right)\exp\left(-s\sqrt{\frac{s-1}{s\log{s}}}|x|\right), \nonumber \end{align} \tag{ 42 }$$

by applying the Tauberian theorem for slowly varying functions [16] we can obtain the asymptotic behavior of $W(x,t)$ in the short and long time limit. This theorem states that if some function $r(t)$, $t\geqslant 0$, has the Laplace transform $\hat{r}(s)$ whose asymptotics behaves as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tauber7} \hat{r}(s)\simeq s^{-\rho}L\left(\frac{1}{s}\right), \quad s\rightarrow0, \quad \rho\geqslant0, \nonumber \end{align} \tag{ 43 }$$

then

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tauber8} r(t)=\mathcal{L}^{-1}\left[\hat{r}(s)\right]\simeq \frac{1}{\Gamma(\rho)}t^{\rho-1}L(t), \quad t\rightarrow\infty. \nonumber \end{align} \tag{ 44 }$$

Here $L(t)$ is a slowly varying function at infinity, i.e.

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \lim_{t\rightarrow\infty}\frac{L(at)}{L(t)}=1,\nonumber \end{align*}$$

for any $a>0$. The theorem is also valid if $s$ and $t$ are interchanged, that is $s\rightarrow\infty$ and $t\rightarrow0$ [16]. The value of $\rho$ is equal to $0$ and $1/2$ for short and long time limit respectively. Therefore, from equation (42) we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W(x,t)\simeq\left\lbrace \begin{array}{@{}c l@{}} \frac{1}{\sqrt{4t^{2}\log{\frac{1}{t}}}}\exp\left(-\frac{|x|}{\sqrt{t^{2}\log{\frac{1}{t}}}}\right), \quad & t\ll1, \nonumber \\ \frac{1}{\sqrt{4t\log{t}}}\exp\left(-\frac{|x|}{\sqrt{t\log{t}}}\right), \quad & t\gg1. \end{array}\right. \nonumber \end{align} \tag{ 45 }$$

The MSD reads

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msdD} \left\langle x^{2}(t)\right\rangle=2\,\mathcal{L}^{-1}\left[\frac{\log{s}}{s^{2}(s-1)}\right](t). \nonumber \end{align} \tag{ 46 }$$

From here, by applying the Tauberian theorem for slowly varying functions, we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd distr asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} 2\,\mathcal{L}^{-1}\left[s^{-3}\log{s}\right](t)\simeq t^{2}\log\frac{1}{t}, \quad & t\ll1, \nonumber \\ 2\,\mathcal{L}^{-1}\left[s^{-2}\log\frac{1}{s}\right](t)=2\, t\left(-1+\gamma+\log{t}\right)\simeq2\,t\log{t}, \quad & t\gg1. \end{array}\right. \nonumber \end{align} \tag{ 47 }$$

Therefore, the behavior of the MSD is weakly superballistic at short times and weakly superdiffusive at long times. Interestingly, the MSD (47) for the diffusion–wave equation (41) in comparison to the MSD for the diffusion equation with uniformly distributed order memory kernel [9, 10], has an additional multiplicative term $t$ for both short and long times.

Next we present results in case of a truncated power-law memory kernel of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tr kernel} \eta(t)={\rm e}^{-bt}\frac{t^{1-\mu}}{\Gamma(2-\mu)}, \nonumber \end{align} \tag{ 48 }$$

where $b>0$, and $1\leqslant\mu<2$. Its Laplace transform is given by $\newcommand{\e}{{\rm e}} \hat{\eta}(s)=(s+b)^{\mu-2}$, where we use the shift rule of the Laplace transform

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{shift rule} \mathcal{L}\left[\,f(t){\rm e}^{-at}\right](s)=\hat{F} (s+a), \quad {\rm with} \quad \mathcal{L}\left[\,f(t)\right](s)=\hat{F}(s). \nonumber \end{align} \tag{ 49 }$$

The solution of the diffusion-wave equation with the truncated memory kernel (48) is non-negative for $1\leqslant\mu<2$ since $\newcommand{\e}{{\rm e}} \hat{\eta}(s)=(s+b)^{\mu-2}$ is a SF as a composition of SF ($s^{\mu-2}$, $1\leqslant\mu<2$) and CBF ($s+b$) (see property (f) in section 2.1.4.). Therefore $\newcommand{\e}{{\rm e}} \sqrt{\eta(s)}$ is a SF too, with which we complete the proof.

From the solution in Laplace space we find the exact solution in the following form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle W(x,t)&=\mathcal{L}^{-1}\left[\frac{1}{2s}(s+b)^{\mu/2}\exp\left(-(s+b)^{\mu/2}|x|\right)\right](x,t)\nonumber \\&={_{RL}}I_{t}^{1}\left(\frac{{\rm e}^{-bt}}{2|x|t}H_{1,1}^{1,0}\left[\frac{|x|}{t^{\mu/2}}\left|\begin{array}{@{}c l@{}} (0,\mu/2) \nonumber \\ (0,1) \end{array}\right.\right]\right), \nonumber \end{align} \tag{ 50 }$$

where we use the relation between the exponential and the Fox $H$-function [39]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{exp H} {\rm e}^{-z}=H_{0,1}^{1,0}\left[z\left|\begin{array}{@{}l@{}} - \nonumber \\ (0,1) \end{array}\right.\right], \nonumber \end{align} \tag{ 51 }$$

the shift rule (49), and the Laplace transform formula [39]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Laplace H} \mathcal{L}^{-1}\left[s^{-\rho}H_{p,q}^{m,n}\left[zs^{\sigma}\left|\begin{array}{@{} l@{}} (a_p,A_p) \nonumber \\ (b_q,B_q) \end{array}\right.\right]\right](t)=t^{\rho-1}H_{p+1,q}^{m,n}\left[zt^{-\sigma}\left|\begin{array}{@{} l@{}} (a_p,A_p),(\rho,\sigma) \nonumber \\ (b_q,B_q) \end{array}\right.\right],\nonumber \\ \nonumber \end{align} \tag{ 52 }$$

where $\sigma>0$, $\Re(s)>0$, $\Re\left(\rho+\sigma \max_{1\leqslant j\leqslant n}\left(\frac{1-a_j}{A_j}\right)\right)>0$, $|{\rm arg}(z)| < \pi\theta_{1}/2$, $\theta_1>0$, $\theta_1=\theta-a$.

From equation (13) for the second moment we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\langle x^{2}(t)\right\rangle=2\,\mathcal{L}^{-1}\left[s^{-3}(s+b)^{2-\mu}\right](t)=2\,{_{RL}}I_{t}^{3}\left({\rm e}^{-bt}\frac{t^{-3+\mu}}{\Gamma(-2+\mu)}\right), \nonumber \end{align} \tag{ 53 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{RLintegral} {_{RL}I_t^{\alpha}}f(t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-t')^{\alpha-1}f(t')\,{\rm d}t' \nonumber \end{align} \tag{ 54 }$$

is the Riemann–Liouville fractional integral, whose Laplace transform is given by $\mathcal{L}\left[{_{RL}I_t^{\alpha}}f(t)\right](s)=s^{-\alpha}\mathcal{L}\left[\,f(t)\right](s)$ [50]. For the asymptotic behavior of the second moment we use the Tauberian theorem [16], which states that if the asymptotic behavior of $r(t)$ for $t\rightarrow\infty$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tauber1} r(t)\simeq t^{-\alpha}, \quad t\rightarrow\infty, \nonumber \end{align} \tag{ 55 }$$

then, the corresponding Laplace pair $\hat{r}(s)=\mathcal{L}[r(t)](s)$ has the following behavior for $s\rightarrow0$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tauber2} \hat{r}(s)\simeq \Gamma(1-\alpha)s^{\alpha-1}, \quad s\rightarrow0, \nonumber \end{align} \tag{ 56 }$$

and vice-versa, ensuring that $r(t)$ is non-negative and monotone function at infinity [16]. Thus, we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd trunc asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} 2\,\frac{t^{\mu}}{\Gamma(1+\mu)}, \quad & t\ll1, \nonumber \\ b^{2-\mu}t^{2}, \quad & t\gg1, \end{array}\right. \nonumber \end{align} \tag{ 57 }$$

which means that the process switches from superdiffusive behavior to ballistic motion in the case with $1<\mu<2$, and from normal diffusion to ballistic motion in the case with $\mu=1$. The fact that we encounter accelerating superdiffusion caused by truncation of the power law kernel in the diffusion-wave equation is not surprising. Indeed, it is analogous to the transition from subdiffusion to normal diffusion described by the fractional diffusion equation with truncated power law kernel in the Caputo form [53, 57]. As in diffusion equation the truncation naturally leads to a normal diffusion at long times, in diffusion-wave equation the truncation results in the long-time ballistic behavior.

Here we consider tempered M-L memory kernel of form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{gamma tempered ml} \eta(t)={\rm e}^{-bt}t^{\beta-1}E_{\alpha,\beta}^{\delta}\left(-\nu t^{\alpha}\right), \nonumber \end{align} \tag{ 58 }$$

where $\alpha\delta<\beta<1$, and $E_{\alpha,\beta}^{\delta}(z)$ is the three parameter M-L function [51] (for details on three parameter M-L function we refer to [17, 66])

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{three ml} E_{\alpha,\beta}^{\delta}(z)=\sum_{k=0}^{\infty}\frac{(\delta)_k}{\Gamma(\alpha k+\beta)}\frac{z^k}{k!}, \nonumber \end{align} \tag{ 59 }$$

$\beta, \delta, z \in \mathrm{C}$, $\Re(\alpha)>0$, and $(\delta)_{k}$ is the Pochhammer symbol $(\delta)_{0}=1$, $(\delta)_{k}=\frac{\Gamma(\delta+k)}{\Gamma(\delta)}$. The asymptotic behavior of the three parameter M-L function for large arguments follows from the formula [17, 56]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{ml three asympt} E_{\alpha,\beta}^{\delta}(-t^{\alpha})=\frac{t^{-\alpha\delta}}{\Gamma(\delta)}\sum_{n=0}^{\infty}\frac{\Gamma(\delta+n)}{\Gamma(\beta-\alpha(\delta+n))}\frac{(-t^{\alpha})^{-n}}{n!}, \quad t\gg1, \nonumber \end{align} \tag{ 60 }$$

for $0<\alpha<2$. The Laplace transform of the memory kernel becomes

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\frac{(s+b)^{\alpha\delta-\beta}}{\left[(s+b)^{\alpha}+\nu\right]^{\delta}},\nonumber \end{align*}$$

where we use the Laplace transform formula [51]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Laplace ML3_1} \mathcal{L}\left[t^{\beta-1}E_{\alpha,\beta}^{\delta}\left(-\lambda t^{\alpha}\right)\right](s)=\frac{s^{\alpha\delta-\beta}}{(s^\alpha+\lambda)^\delta}, \quad |\lambda/s^{\alpha}|<1. \nonumber \end{align} \tag{ 61 }$$

Therefore, the generalized diffusion-wave equation has the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{wave tempered ml} \int_{0}^{t}(t-t')^{\beta-1}E_{\alpha,\beta}^{\delta}\left(-\nu[t-t']^{\alpha}\right)\frac{\partial^{2}}{\partial t'^{2}}W(x,t')\,{\rm d}t'=\frac{\partial^2}{\partial x^2}W(x,t). \nonumber \end{align} \tag{ 62 }$$

In order to find the parameters' constraints, for which the solution of equation (62) is non-negative, we notice that $\newcommand{\e}{{\rm e}} \hat{\eta}(s)$ is a SF (and thus CM) if both functions $(s+b)^{-(\beta-\alpha\delta)}$ and $\left[(s+b)^{\alpha}+\nu\right]^{-\delta}$ are SFs. The first one is a SF if $0<\beta-\alpha\delta<1$ and the second one—if $0<\alpha\delta<1$. Therefore, we use $0<\alpha\delta<\beta<1$.

For the MSD we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{second moment tempered ml} \left\langle x^{2}(t)\right\rangle&=2\,\mathcal{L}^{-1}\left[s^{-3}\frac{(s+b)^{-\alpha\delta+\beta}}{\left[(s+b)^{\alpha}+\nu\right]^{-\delta}}\right](t)\nonumber \\ &=2\,{_{RL}}I_{t}^{3}\left({\rm e}^{-bt}t^{-\beta-1}E_{\alpha,-\beta}^{-\delta}\left(-\nu t^{\alpha}\right)\right). \nonumber \end{align} \tag{ 63 }$$

Again, by using the Tauberian theorem [16], for the short and long time limit we find

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd distr ml asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} 2\,t^{2-\beta}/\Gamma(3-\beta), \quad & t\ll1, \nonumber \\ b^{-\alpha\delta+\beta}\left(b^{\alpha}+\nu\right)^{\delta}t^{2}, \quad & t\gg1. \end{array}\right. \nonumber \end{align} \tag{ 64 }$$

Therefore, truncation (tempering) of the M-L kernel causes ballistic motion in the long time limit, which follows the superdiffusive initial behavior. Here we note that tempered M-L memory kernel has been introduced in the analysis of the generalized Langevin equation [29], where the crossover from initial subdiffusion to normal diffusion is observed. The case with no truncation ($b=0$) for the second moment yields $\left\langle x^{2}(t)\right\rangle=2t^{2-\beta}E_{\alpha,3-\beta}^{-\delta}\left(-\nu t^{\alpha}\right)$, which in the long time limit behaves as $\left\langle x^{2}(t)\right\rangle\simeq t^{2+\alpha\delta-\beta}$.

Graphical representation of the second moment (63) is given in figure 3. The influence of tempering on particle behavior is clearly observed. Comparison with the asymptotic behaviors in the short and long time limit is given in figure 4.

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Zoom In Zoom Out Reset image size

Let us now consider the regularized Prabhakar derivative defined by [18]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{prabhakar derivative} {_C}\mathcal{D}_{\rho,-\nu,t\,}^{\delta,\mu}f(t)=\left(\mathcal{E}_{\rho,m-\mu,-\nu,t\,}^{-\delta}\frac{{\rm d}^{m}}{{\rm d}t^{m}}f\right)(t), \nonumber \end{align} \tag{ 65 }$$

where $\mu, \nu, \delta, \rho \in C$, $\Re(\mu)>0$, $\Re(\rho)>0$, $m=[\mu]+1$, and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{prabhakar integral} \left(\mathcal{E}_{\rho,m-\mu,-\nu,t\,}^{-\delta}f\right)(t)=\int_{0}^{t}(t-t')^{m-\mu-1}E_{\rho,m-\mu}^{-\delta}\left(-\nu(t-t')^{\rho}\right)\,f(t')\,{\rm d}t' \nonumber \end{align} \tag{ 66 }$$

is the Prabhakar integral [51]. Therefore, for $1<\mu<2$ ($m=2$) we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{prabhakar derivative m=2} {_C}\mathcal{D}_{\rho,-\nu,t\,}^{\delta,\mu}f(t)&=\left(\mathcal{E}_{\rho,2-\mu,-\nu,t\,}^{-\delta}\frac{{\rm d}^{2}}{{\rm d}t^{2}}f\right)(t)=\int_{0}^{t}(t-t')^{1-\mu}E_{\rho,2-\mu}^{-\delta}\left(-\nu(t-t')^{\rho}\right)\frac{{\rm d}^2}{{\rm d}t'^2}f(t')\,{\rm d}t'\nonumber \\&=\int_{0}^{t}\eta(t-t')\frac{{\rm d}^2}{{\rm d}t'^2}f(t')\,{\rm d}t', \nonumber \end{align} \tag{ 67 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{gamma prabhakar} \eta(t)=t^{1-\mu}E_{\rho,2-\mu}^{-\delta}\left(-\nu t^{\rho}\right), \quad 1<\mu<2, \nonumber \end{align} \tag{ 68 }$$

that is the kernel on the left hand side of equation (2). This resembles the case considered in section 3.7, see equation (58), if $b=0$, however, in that previous case the MSD asymptotics at long times is determined by $b$, see equation (64). This is why the kernel (68) is considered separately. We also use that $0<\rho<1$ and $0<\delta<1$. The generalized diffusion-wave equation then becomes

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{wave Pr} {_C}\mathcal{D}_{\rho,-\nu,t\,}^{\delta,\mu}W(x,t)=\frac{\partial^2}{\partial x^2}W(x,t). \nonumber \end{align} \tag{ 69 }$$

The Prabhakar derivative has been introduced and applied to fractional Poisson processes in [18], and further used in viscoelastic theory [19–21], fractional differential filtration dynamics [7], and generalized Langevin equation [52]. Moreover, the fractional diffusion equation with regularized Prabhakar derivative has been recently derived from the continuous time random walk theory [54].

The non-negativity of the solution of equation (69) can be proven as follows. The function

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\left(s^{\frac{\mu-2}{\delta}}+\nu s^{\frac{\mu-2}{\delta}-\rho}\right)^{\delta}\nonumber \end{align*}$$

is a SF if $s^{(\mu-2)/\delta}$ and $s^{(\mu-2)/\delta-\rho}$ are SFs, and $0<\delta<1$ (composition of a CBF and SF is a SF, see property (g) in section 2.1.4.). Therefore, we obtain that $0<2-\mu<\delta$ and $2-\delta<\mu-\rho\delta<2$. For these values of parameters $\newcommand{\e}{{\rm e}} \sqrt{\hat{\eta}(s)}$ is a SF as well, with which we show the non-negativity of the solution.

Thus, for the second moment we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{second moment Pr} \left\langle x^{2}(t)\right\rangle&=2\,\mathcal{L}^{-1}\left[\frac{s^{-3}}{s^{-\rho\delta+\mu-2}(s^{\rho}+\nu)^{\delta}}\right](t) =2\,\mathcal{L}^{-1}\left[\frac{s^{-1+\rho\delta-\mu}}{(s^{\rho}+\nu)^{\delta}}\right](t)\nonumber \\&=2\,t^{\mu}E_{\rho,\mu+1}^{\delta}\left(-\nu t^{\rho}\right). \nonumber \end{align} \tag{ 70 }$$

From here, by using equation (60), we find the asymptotic behavior in the short and long time limit,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd prabh asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} 2\,t^{\mu}/\Gamma(1+\mu), \quad & t\ll1, \nonumber \\ 2\,t^{\mu-\rho\delta}/\Gamma(1+\mu-\rho\delta), \quad & t\gg1, \end{array}\right. \nonumber \end{align} \tag{ 71 }$$

which implies that the random motion shows decelerating superdiffusive behavior.

Further, for the first time in the literature we introduce the distributed order M-L memory kernel of the form

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{gamma prabhakar distributed} \eta(t)=\int_{1}^{2}t^{1-\mu}E_{\rho,2-\mu}^{-\delta}\left(-\nu t^{\rho}\right)\,{\rm d}\mu, \nonumber \end{align} \tag{ 72 }$$

where $0<\mu,\rho,\delta<1$, $0<2-\mu<\delta$ and $2-\delta<\mu-\rho\delta<2$. The generalized diffusion-wave equation becomes distributed order wave equation with regularized Prabhakar derivative,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{wave Pr distributed} \int_{1}^{2}{_C}\mathcal{D}_{\rho,-\nu,t\,}^{\delta,\mu}W(x,t)\,{\rm d}\mu=\frac{\partial^2}{\partial x^2}W(x,t). \nonumber \end{align} \tag{ 73 }$$

Note that for $\delta=0$ equation (73) is reduced to the uniformly distributed order wave equation (41).

From the memory kernel (72) we find that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\frac{s-1}{s\log{s}}\left(1+\frac{\nu}{s^{\rho}}\right)^{\delta}. \nonumber \end{align} \tag{ 74 }$$

In order to check the non-negativity of the solution of equation (73), we should analyze the function

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \hat{\eta}(s)=\int_1^2\left(s^{\frac{\mu-2}{\delta}}+\nu s^{\frac{\mu-2}{\delta}-\rho}\right)^{\delta}\,{\rm d}\mu.\nonumber \end{align*}$$

It is a SF for the same restrictions of parameters as those for the diffusion-wave equation with regularized Prabhakar derivative since a pointwise limit of the linear combinations of SFs,

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle \sum_{j}\left(s^{\frac{\mu_j-2}{\delta}}+\nu s^{\frac{\mu_j-2}{\delta}-\rho}\right)^{\delta}\nonumber \end{align*}$$

is also SF.

For the MSD we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{second moment Pr distributed} \left\langle x^{2}(t)\right\rangle=2\,\mathcal{L}^{-1}\left[\frac{\log{s}}{s^2(s-1)}\left(1+\frac{\nu}{s^{\rho}}\right)^{-\delta}\right](t). \nonumber \end{align} \tag{ 75 }$$

Therefore, from the Tauberian theorem the asymptotics at short and long times read

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{msd distr prabh asympt} \langle x^{2}(t)\rangle\simeq\left\lbrace \begin{array}{@{}c l@{}} 2\,\mathcal{L}^{-1}\left[\frac{\log{s}}{s^3}\right](t)\simeq t^{2}\log\frac{1}{t}, \quad & t\ll1, \nonumber \\ \frac{2}{\nu^\delta}\mathcal{L}^{-1}\left[\frac{\log{\frac{1}{s}}}{s^{2-\rho\delta}}\right](t)\simeq\frac{2}{\nu^\delta}\frac{t^{1-\rho\delta}}{\Gamma(2-\rho\delta)}\log{t}, \quad & t\gg1, \end{array}\right. \nonumber \end{align} \tag{ 76 }$$

and thus, the random motion is weakly superballistic at short times and weakly subdiffusive at long times since $0<1-\rho\delta<1$.