# Cattaneo–type subdiffusion equation

Tadeusz Kosztołowicz<EMAIL_ADDRESS>1Institute of Physics,
Jan Kochanowski University,
Uniwersytecka 7, 25-406 Kielce, Poland
2Department of Radiological Informatics and Statistics,
Medical University of Gdańsk, Tuwima 15, 80-210 Gdańsk, Poland  Aldona
Dutkiewicz<EMAIL_ADDRESS>Faculty of Mathematics and Computer Science,
Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
Katarzyna D. Lewandowska<EMAIL_ADDRESS>Department of
Physics and Biophysics, Medical University of Gdańsk,
Dȩbinki 1, 80-211 Gdańsk, Poland

###### Abstract

The ordinary subdiffusion equation, with fractional time derivatives of at
most first order, describes a process in which the propagation velocity of
diffusing molecules is unlimited. To avoid this non-physical property the
Cattaneo diffusion equation has been proposed. Compared to the ordinary
subdiffusion equation, the Cattaneo equation contains an additional time
derivative of order greater than one and less than or equal to two. The
fractional order of the additional derivative depends on the subdiffusion
exponent. We study a Cattaneo-type subdiffusion equation (CTSE) that differs
from the ordinary subdiffusion equation by an additional integro–differential
operator (AO) which may be independent of subdiffusion parameters. The AO
describes processes affecting ordinary subdiffusion. The equation is derived
combining the modified diffusive flux equation with the continuity equation.
It can also be obtained within the continuous time random walk model with the
waiting time distribution for the molecule to jump controlled by the kernel of
AO. As examples, the CTSEs with the additional Caputo fractional time
derivative of the order independent of the subdiffusion exponent and with the
AO generated by a slowly varying function are considered. We discuss whether
the ordinary subdiffusion equation and CTSE provide qualitative differences in
the description of subdiffusion.

##  I Introduction

Subdiffusion occurs in media in which particle random walk is very hindered.
The examples are transport of molecules in viscoelastic chromatin network
lee2021 , porous media bijeljic2011 , living cells barkai2012 , transport of
sugars in agarose gel kdm , and antibiotics in bacterial biofilm km ; kmwa .
Subdiffusion is often defined as an anomalous diffusion process in which the
temporal evolution of mean square displacement (MSD) of a particle is
$\sigma^{2}(t)=2Dt^{\alpha}/\Gamma(1+\alpha)$, where $\alpha\in(0,1)$ is the
subdiffusion parameter (exponent), $D$ is a subdiffusion coefficient given in
units of ${\rm m}^{2}/{\rm s}^{\alpha}$. Frequently, ordinary subdiffusion is
described by a fractional differential equation wyss1986 ; hilferanton ;
compte ; mks ; mk ; barkai2000 ; skb ; klages2008 ; ks  with the Riemann-
Liouville fractional time derivative of $1-\alpha$ order or an equation with
the Caputo fractional time derivative of the order $\alpha$.

For the parabolic normal diffusion equation and subdiffusion equation with a
time derivative of at most first order the Green’s function $P(x,t)$, which is
the probability density of finding the molecule at position $x$ at time $t$,
is greater than zero for any $x$ and $t>0$. It means that the propagation
velocity of some particles is arbitrarily high. To avoid this non–physical
property, the Cattaneo normal diffusion equation has been proposed; the
equation was originally used to describe heat propagation cattaneo . The
Cattaneo normal diffusion equation involves a second-order time derivative
controlled by the parameter $\tau$. The derivation of the equation is based on
the assumption that the flux of diffusing molecules is delayed by time $\tau$
with respect to the concentration gradient. While the Cattaneo hyperbolic
normal diffusion equation is well defined, the Cattaneo subdiffusion equation
can take various non-equivalent forms, see for example Refs. compte1997 ;
fernandez ; qi ; liu ; roscani ; alegria ; luchko ; awad2021 ; awad2019 . It
contains a time derivative of the order greater than one and less than or
equal to two; the fractional order of this derivative depends on $\alpha$. The
Cattaneo equation with the Caputo and/or the Riemann–Liouville fractional
derivatives have been used to describe subdiffusion and subdiffusion with
reactions hamada ; metzler1999 ; metzler1998 ; gorska2020 ; gorska2021 ;
koszt2014 , in particular neutron transport inside the core of a nuclear
reactor vya , heat transport in porous media nikan , and in a system with
glass spheres in a tank filled with air moza . Cattaneo equations with
fractional derivatives other than those mentioned above have also been
considered. The examples are the equations with Caputo–Fabrizio fractional
time derivative liu2017 , with tempered Caputo derivative beghin , and with
Hilfer fractional derivative with respect to another function vieira .

We consider a Cattaneo-type subdiffusion equation (CTSE) which is defined here
as an equation that differs from the ordinary subdiffusion one by an
additional operator acting on the time variable. The operator is generated by
the constitutive equation describing the relation between the probability
density flux and the gradient of Green’s function. Combining this equation
with the continuity equation the CTSE is obtained. The equation can also be
derived within the continuous time random walk model, in which the
distribution of the waiting time for a molecule to jump is controlled by the
additional operator. The Cattaneo-type subdiffusion equation takes into
account a process, called the additional Cattaneo–type process (ACTP), that
change ordinary subdiffusion. An example is diffusion of antibiotics in
bacterial biofilms. In a dense biofilm subdiffusion of antibiotic molecules
may occur. However, bacteria can activate defense mechanisms against the
effects of antibiotics. Some of them additionally slow down subdiffusion or
temporarily trap antibiotic molecules aot ; mot .

In the case of passive subdiffusion, the solutions to the Cattaneo equation
and the ordinary subdiffusion one are usually not much different from each
other. Then, the latter equation, which appears to be easier to solve, is used
to describe diffusion. However, there are processes that both equations
provide qualitatively different results even for small $\tau$. Examples of
this are diffusive and subdiffusive impedance barbero ; kostrobij ; kl2009 ;
lk2008 . When considering subdiffusion described by a Cattaneo-type equation,
the question arises whether such an equation gives a significant difference
compared to the use of the ordinary subdiffusion equation. Therefore, we
consider functions describing how the ACTP affects ordinary subdiffusion.

The general subdiffusion equation controlled by a memory kernel (MK)
(equivalent to Eqs. (13)–(15) presented later), which is the basis for
deriving the Cattaneo-type equation, has been considered to derive various
types of anomalous diffusion equations. Examples are equations generated by
distributed order MK sandev2015 ; sandev2015a ; sandev2017 ; sandev2018 ;
sandev2019 , power–law MK sandev2015 , and truncated power law MK sandev2017 .
The equation with MK has been used to derive multi-fractional diffusion
equations metzler1998 ; sandev2015a ; sandev2018 ; sandev2019 . Generalized
diffusion equations with MK can also be derived by means of the subordination
method chechkin2021 .

The most commonly used function characterizing diffusion is a temporal
evolution of the mean square displacement of a diffusing particle. However,
MSD alone does not clearly determine the type of diffusion dybiec1 ; meroz2011
. We also consider the first passage time distribution of a particle through
an arbitrarily chosen point $F(t;x)$.

We consider subdiffusion in a one-dimensional homogeneous system with constant
parameters. We pay attention to the interpretation of ACTP and assesses how
this process affects subdiffusion. In Sec. II the subdiffusion equation
generated by the generalized flux equation controlled by a MK $\beta$ is
derived. In Sec. III we consider functions describing subdiffusion. The
explicit forms of the functions are given in terms of the Laplace transform.
In Sec. IV the function $\beta$ linear with respect to the parameter $\tau$ is
considered, this function generates a Cattaneo-type subdiffusion equation. The
CTSEs with the additional Caputo fractional time derivative of the order
independent of the subdiffusion exponent and with the additional operator
generated by a slowly varying function are studied in Secs. V and VI,
respectively. Final remarks are in Sec. VII.

##  II Subdiffusion equation generated by generalized flux equation

The subdiffusion equation can be obtained phenomenologically by a combination
of the continuity equation

$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial J(x,t)}{\partial x}.$ (1)

and the constitutive equation that defines the flux $J$. For ordinary
subdiffusion the flux is defined as

$J(x,t)=-D\frac{{}^{RL}\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial P(x,t)}{\partial x},$ (2)

where $\alpha\in(0,1)$, with the Riemann–Liouville fractional derivative

$\frac{{}^{RL}d^{\nu}f(t)}{dt^{\nu}}=\frac{1}{\Gamma(n-\nu)}\frac{d^{n}}{dt^{n}}\int_{0}^{t}(t-u)^{n-\nu-1}f(u)du,$
(3)

$n$ is a natural number, $n=[\nu]+1$ when $\nu\notin\mathcal{N}$ and
$n=\nu\in\mathcal{N}$. Throughout this paper we assume that the initial
particle position is $x_{0}=0$. Combining Eqs. (1) and (2) one gets

$\frac{\partial P(x,t)}{\partial
t}=D\frac{{}^{RL}\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial^{2}P(x,t)}{\partial x^{2}}.$ (4)

When deriving a diffusion equation, it is convenient to use the Laplace
transform

$\mathcal{L}[f(t)](s)=\int_{0}^{\infty}{\rm e}^{-st}f(t)dt\equiv\hat{f}(s).$
(5)

The Laplace transforms of derivatives are

$\mathcal{L}\Bigg{[}\frac{{}^{RL}d^{\alpha}_{g}f(t)}{dt^{\alpha}}\Bigg{]}(s)=s^{\alpha}\mathcal{L}\left[f(t)\right](s),\;\alpha\in(0,1),$
(6)
$\mathcal{L}\Bigg{[}\frac{d^{n}f(t)}{dt^{n}}\Bigg{]}=s^{n}\mathcal{L}[f(t)](s)-\sum_{i=0}^{n-1}s^{n-1-i}f^{(i)}(0),$
(7)

where $f^{(i)}(t)=d^{i}f(t)/dt^{i}$, $i$ and $n$ are natural numbers. The
Laplace transforms of Eqs. (1), (2), and (4) are, respectively,

$s\hat{P}(x,s)-P(x,0)=-\frac{\partial\hat{J}(x,s)}{\partial x},$ (8)
$\hat{J}(x,s)=-Ds^{1-\alpha}\frac{\partial\hat{P}(x,s)}{\partial x},$ (9)
$\displaystyle
s\hat{P}(x,s)-P(x,0)=Ds^{1-\alpha}\frac{\partial^{2}\hat{P}(x,s)}{\partial
x^{2}}.$ (10)

The assumption under which the Cattaneo normal diffusion equation was derived
is that the flux is delayed with respect to the concentration gradient.
However, in the case of subdiffusion, the relation between the flux and the
concentration gradient can be assumed in many ways, see Refs. compte1997 ;
awad2019 . We assume that

$\hat{\beta}(s;\tau)\hat{J}(x,s)=-Ds^{1-\alpha}\frac{\partial\hat{P}(x,s)}{\partial
x},$ (11)

$\beta$ is a memory kernel which depends on the parameter $\tau$. We also
assume that for $\tau=0$ we get ordinary subdiffusion equation, then

$\hat{\beta}(s,0)=1.$ (12)

Eqs. (8) and (11) provide

$\hat{\beta}(s;\tau)\big{[}s\hat{P}(x,s)-P(x,0)\big{]}=Ds^{1-\alpha}\frac{\partial^{2}\hat{P}(x,s)}{\partial
x^{2}}.$ (13)

For $\hat{\beta}(s;\tau)=1+\tau s$ and $\alpha=1$ we get the well–known
Cattaneo normal diffusion equation.

In the time domain Eq. (13) reads

$\mathcal{B}_{t;\tau}[P(x,t)]=D\frac{{}^{RL}\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial^{2}P(x,t)}{\partial x^{2}},$ (14)

where

$\mathcal{B}_{t;\tau}[f(t)]=\int_{0}^{t}\beta(t-t^{\prime};\tau)f^{(1)}(t^{\prime})dt^{\prime}.$
(15)

Constraints on the function $\beta$ can be derived from additional
considerations, see the comment under Eq. (29).

##  III Functions describing diffusion

The Green’s function is the solution to diffusion equation with the boundary
conditions

$P(\pm\infty,t)=0,$ (16)

and the initial condition

$P(x,0)=\delta(x),$ (17)

where $\delta$ is the delta–Dirac function. The above initial condition is
sufficient to determine the Laplace transform of a time derivative of at most
first order. When computing the transform of other operators additional
initial conditions may be needed. We assume that the only non-zero initial
condition is Eq. (17).

In terms of the Laplace transform the Green’s function of Eq. (14) is

$\hat{P}(x,s)=\frac{\sqrt{\hat{\beta}(s;\tau)}}{2\sqrt{D}s^{1-\alpha/2}}{\rm
e}^{-\frac{s^{\alpha/2}|x|\sqrt{\hat{\beta}(s;\tau)}}{\sqrt{D}}}.$ (18)

The Green’s function Eq. (18) is symmetrical, $P(x,t)=P(-x,t)$, this property
is used in deriving further equations. On the basis of the Green’s function,
other functions characterizing diffusion processes are determined. A
frequently used function is the temporal evolution of the mean square
displacement $\sigma^{2}$ of diffusing particle,

$\sigma^{2}(t)=2\int_{0}^{\infty}x^{2}P(x,t)dx.$ (19)

In terms of the Laplace transform we get

$\hat{\sigma^{2}}(s)=\frac{2D}{s^{1+\alpha}\hat{\beta}(s;\tau)}$ (20)

Another function that characterizes diffusion is the first passage time
distribution $F$. Assuming $x>0$, the distribution of time of the first
particle passing the point $x$ can be calculated by means of the formula

$F(t;x)=-\frac{\partial}{\partial t}\int_{-x}^{x}P(x^{\prime},t)dx^{\prime}.$
(21)

In terms of the Laplace transform the above equation reads

$\hat{F}(s;x)=1-s\int_{-x}^{x}\hat{P}(x^{\prime},s)dx^{\prime}.$ (22)

Putting Eq. (18) to Eq. (22) we get

$\hat{F}(s;x)={\rm
e}^{-\frac{xs^{\alpha/2}\sqrt{\hat{\beta}(s;\tau)}}{\sqrt{D}}}.$ (23)

The probability flux at $x$ is defined as

$J(x,t)=-\frac{\partial}{\partial
t}\int_{-\infty}^{x}P(x^{\prime},t)dx^{\prime}.$ (24)

The Laplace transform Eq. (24) is

$\hat{J}(x,s)=1-s\int_{-\infty}^{x}\hat{P}(x^{\prime},s)dx^{\prime}.$ (25)

For $x>0$ we obtain

$\hat{J}(x,s)=\frac{1}{2}{\rm
e}^{-\frac{xs^{\alpha/2}\sqrt{\hat{\beta}(s;\tau)}}{\sqrt{D}}}.$ (26)

The equations (23) and (26) provide the relation

$J(x,t)=\frac{1}{2}F(t;x),\;x>0.$ (27)

For $x<0$ the flux is negative, then $J(x,t)=-F(t;x)/2$.

The equation (14) can also be derived by changing the probability distribution
$\psi$ of the waiting time for a molecule to jump within the continuous time
random walk model. The model provides the diffusion equation given in terms of
the Laplace transform, see Ref. koszt2023 ,

$s\hat{P}(x,s)-P(x,0)=\frac{\epsilon^{2}s\hat{\psi}(s)}{2[1-\hat{\psi}(s)]}\frac{\partial^{2}\hat{P}(x,s)}{\partial
x^{2}},$ (28)

where $\epsilon$ is the mean value of a single jump length. Comparing Eqs.
(13) and (28) we obtain

$\hat{\psi}(s)=\frac{1}{1+\lambda s^{\alpha}\hat{\beta}(s;\tau)},$ (29)

where $\lambda=\epsilon^{2}/2D$. The parameter describing subdiffusion is $D$,
$\epsilon$ is a scale-dependent parameter. We further assume that $\epsilon$
is small, which provides $\lambda\ll 1$. Due to the normalization of $\psi$,
$\hat{\psi}(0)=1$, $\hat{\beta}$ meets the condition
$s^{\alpha}\hat{\beta}(s;\tau)\rightarrow 0$ when $s\rightarrow 0$.

The question arises whether the Cattaneo subdiffusion equation brings a new
quality in comparison with the ordinary subdiffusion equation. We compare the
functions $F$ and $\sigma^{2}$ derived for $\tau=0$ and for $\tau\neq 0$. The
functions $F_{R}$ and $\sigma^{2}_{R}$ show the time evolution of the relative
change of first passage time distribution and MSD, respectively,

$F_{R}(t;x)=\frac{F_{\tau=0}(t;x)-F(t;x)}{F_{\tau=0}(t;x)},$ (30)
$\sigma^{2}_{R}(t)=\frac{\sigma^{2}_{\tau=0}(t)-\sigma^{2}(t)}{\sigma^{2}_{\tau=0}(t)},$
(31)

in Eq. (30) $x$ is a parameter. Additionally, we consider the relative change
of the Green’s function

$P_{R}(x,t)=\frac{P_{\tau=0}(x,t)-P(x,t)}{P_{\tau=0}(x,t)},$ (32)

here $x$ is a random variable, $t$ is a parameter.

##  IV Cattaneo–type subdiffusion equation

Let us assume that

$\hat{\beta}(s;\tau)=1+\tau\hat{\gamma}(s).$ (33)

Combining Eqs. (13) and (33) we obtain

$\displaystyle(1+\tau\hat{\gamma}(s))\big{[}s\hat{P}(x,s)-P(x,0)\big{]}$ (34)
$\displaystyle=Ds^{1-\alpha}\frac{\partial^{2}\hat{P}(x,s)}{\partial x^{2}}.$

The inverse Laplace transform of Eq. (34) provides the Cattaneo–type
subdiffusion equation

$\displaystyle\tau\mathcal{D}_{t}[P(x,t)]+\frac{\partial P(x,t)}{\partial
t}=D\frac{{}^{RL}\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial^{2}P(x,t)}{\partial x^{2}},$ (35)

where

$\mathcal{D}_{t}[f(t)]=\int_{0}^{t}\gamma(t-t^{\prime})f^{(1)}(t^{\prime})dt^{\prime}.$
(36)

The function $\gamma$ is determined by the ACTP. We note that for some
$\gamma$ the operator $\mathcal{D}_{t}$ can be interpreted as a fractional
time derivative.

The Laplace transforms of Green’s function and first passage time distribution
are

$\hat{P}(x,s)=\frac{\sqrt{1+\tau\hat{\gamma}(s)}}{2\sqrt{D}s^{1-\alpha/2}}{\rm
e}^{-\frac{s^{\alpha/2}|x|\sqrt{1+\tau\hat{\gamma}(s)}}{\sqrt{D}}},$ (37)
$\hat{F}(s;x)={\rm
e}^{-\frac{xs^{\alpha/2}\sqrt{1+\tau\hat{\gamma}(s)}}{\sqrt{D}}}.$ (38)

Assuming $\tau\hat{\gamma}(s)<1$, the above functions can be presented as
power series with respect to $\tau$,

$\hat{P}(x,s)=\frac{1}{2\sqrt{D}s^{1-\alpha/2}}{\rm
e}^{-\frac{|x|s^{\alpha/2}}{\sqrt{D}}}\sum_{i=0}^{\infty}\tau^{i}a_{i}(x,s)\hat{\gamma}^{i}(s),$
(39) $\hat{F}(s;x)={\rm
e}^{-\frac{|x|s^{\alpha/2}}{\sqrt{D}}}\sum_{i=0}^{\infty}\tau^{i}b_{i}(x,s)\hat{\gamma}^{i}(s).$
(40)

The functions $a_{i}(x,s)$ and $b_{i}(x,s)$ are determined applying the
following series
$\sqrt{1+u}=1+u/2+\sum_{n=2}^{\infty}(-1)^{n-1}(2n-1)!!u^{n}/2^{n}n!$,
$|u|<1$, and ${\rm e}^{-u}=\sum_{n=0}^{\infty}(-1)^{n}u^{n}/n!$. In the
following, we use the approximation of $\hat{P}$ and $\hat{F}$ including four
leading terms in the series. For $0\leq i\leq 3$ we have

$\displaystyle
a_{0}(x,s)=1,\;a_{1}(x,s)=\frac{1}{2}\Big{(}1-\frac{|x|s^{\alpha/2}}{\sqrt{D}}\Big{)},$
(41) $\displaystyle
a_{2}(x,s)=\frac{1}{8}\Big{(}-1-\frac{|x|s^{\alpha/2}}{\sqrt{D}}+\frac{|x|^{2}s^{\alpha}}{D}\Big{)},$
(42) $\displaystyle
a_{3}(x,s)=\frac{1}{16}\Big{(}1+\frac{|x|s^{\alpha/2}}{\sqrt{D}}-\frac{|x|^{3/2}s^{3\alpha/2}}{3D^{3/2}}\Big{)},$
(43) $\displaystyle
b_{0}(x,s)=1,\;b_{1}(x,s)=-\frac{|x|s^{\alpha/2}}{2\sqrt{D}},$ (44)
$\displaystyle
b_{2}(x,s)=\frac{|x|s^{\alpha/2}}{8\sqrt{D}}\Big{(}1+\frac{|x|s^{\alpha/2}}{\sqrt{D}}\Big{)},$
(45) $\displaystyle
b_{3}(x,s)=\frac{-|x|s^{\alpha/2}}{16\sqrt{D}}\Big{(}12+\frac{|x|s^{\alpha/2}}{\sqrt{D}}+\frac{|x|^{2}s^{\alpha}}{3D}\Big{)}.$
(46)

From Eqs. (20), (29), and (33) we have

$\hat{\sigma}^{2}(s)=\frac{2D}{s^{1+\alpha}(1+\tau\hat{\gamma}(s))},$ (47)
$\hat{\psi}(s)=\frac{1}{1+\lambda s^{\alpha}+\lambda\tau
s^{\alpha}\hat{\gamma}(s)}.$ (48)

The series representations of the above functions are

$\hat{\sigma^{2}}(s)=\frac{2D}{\Gamma(1+\alpha)s^{1+\alpha}}\sum_{i=0}^{\infty}\tau^{i}(-\hat{\gamma}(s))^{i}.$
(49)
$\hat{\psi}(s)=\sum_{i=0}^{\infty}\tau^{i}\hat{\gamma}^{i}(s)\sum_{n=i}^{\infty}{n\choose
i}(-\lambda)^{n}s^{n\alpha}.$ (50)

The interpretation of ACTP uses the function $\psi$. When $\lambda\ll 1$ we
have

$\hat{\psi}(s)\approx\hat{\psi}_{\tau=0}(s)\hat{\psi}_{\gamma}(s),$ (51)

where

$\hat{\psi}_{\tau=0}(s)=\frac{1}{1+\lambda s^{\alpha}},$ (52)
$\hat{\psi}_{\gamma}(s)=\frac{1}{1+\lambda\tau s^{\alpha}\hat{\gamma}(s)}.$
(53)

In the time domain Eq. (51) reads

$\psi(t)\approx\int_{0}^{t}\psi_{\tau=0}(t-t^{\prime})\psi_{\gamma}(t^{\prime})dt^{\prime}.$
(54)

The function $\psi_{\tau=0}$ is the distribution of the waiting time for a
molecule to jump in a subdiffusive medium characterized by the parameters
$\alpha$ and $D$, $\psi_{\gamma}$ is the distribution of the waiting time when
an additional mechanism disturbing the jump is turned on.

##  V CTSE with the additional Caputo fractional time derivative

Let

$\hat{\gamma}(s)=s^{\kappa},\;0<\kappa\leq 1.$ (55)

In terms of the Laplace transform we get

$(\tau
s^{\kappa}+1)\big{[}s\hat{P}(x,s)-P(x,0)\big{]}=Ds^{1-\alpha}\frac{\partial^{2}\hat{P}(x,s)}{\partial
x^{2}}.$ (56)

Using the relation

$\mathcal{L}\Bigg{[}\frac{{}^{C}\partial^{\alpha}f(t)}{\partial
t^{\alpha}}\Bigg{]}(s)=s^{\alpha}\hat{f}(s)-\sum_{i=0}^{n-1}s^{\alpha-1-i}f^{(i)}(0),$
(57)

where

$\frac{{}^{C}d^{\alpha}f(t)}{dt^{\alpha}}=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-t^{\prime})^{n-\alpha-1}f^{(n)}(t^{\prime})dt^{\prime}$
(58)

is the Caputo fractional derivative (the natural number $n$ is defined as
under Eq. (3)), we get

$\tau\frac{{}^{C}\partial^{1+\kappa}P(x,t)}{\partial
t^{1+\kappa}}+\frac{\partial P(x,t)}{\partial
t}=D\frac{{}^{RL}\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial^{2}P(x,t)}{\partial x^{2}},$ (59)

the initial conditions are $P(x,0)=\delta(x)$ and $P^{(1)}(x,0)=0$.

The following formula plays a key role in calculating the inverse Laplace
transforms tkoszt2004

$\displaystyle\mathcal{L}^{-1}\left[s^{\nu}{\rm
e}^{-as^{\beta}}\right](t)\equiv f_{\nu,\beta}(t;a)$ (60)
$\displaystyle=\frac{1}{t^{1+\nu}}\sum_{j=0}^{\infty}\frac{1}{j!\Gamma(-\nu-\beta
j)}\left(-\frac{a}{t^{\beta}}\right)^{j},$

$a,\beta>0$, $f_{\nu,\beta}$ is a special case of the H–Fox function. From
Eqs. (39)–(46) and Eq. (60) we get

$\displaystyle
P(x,t)=\frac{1}{2\sqrt{D}}\Bigg{[}\xi_{-1+\alpha/2}(|x|,t)+\frac{\tau}{2}\Big{(}\xi_{-1+\kappa+\alpha/2}(|x|,t)-\frac{|x|}{\sqrt{D}}\xi_{-1+\kappa+\alpha}(|x|,t)\Big{)}$
(61)
$\displaystyle+\frac{\tau^{2}}{8}\Big{(}-\xi_{-1+2\kappa+\alpha/2}(|x|,t)-\frac{|x|}{\sqrt{D}}\xi_{-1+2\kappa+\alpha}(|x|,t)+\frac{|x|^{2}}{D}\xi_{-1+2\kappa+3\alpha/2}(|x|,t)\Big{)}$
$\displaystyle+\frac{\tau^{3}}{16}\Big{(}\xi_{-1+3\kappa+\alpha/2}(|x|,t)+\frac{|x|}{\sqrt{D}}\xi_{-1+3\kappa+\alpha}(|x|,t)-\frac{|x|^{3}}{3D^{3/2}}\xi_{-1+3\kappa+2\alpha}(|x|,t)\Big{)}\Bigg{]},$

$\displaystyle
F(t;x)=\xi_{0}(|x|,t)-\frac{\tau|x|}{2\sqrt{D}}\xi_{\kappa+\alpha/2}(|x|,t)+\frac{\tau^{2}|x|}{8\sqrt{D}}\Big{(}\xi_{2\kappa+\alpha/2}(|x|,t)+\frac{|x|}{\sqrt{D}}\xi_{2\kappa+\alpha}(|x|,t)\Big{)}$
(62)
$\displaystyle-\frac{\tau^{3}|x|}{16\sqrt{D}}\Big{(}12\xi_{3\kappa+\alpha/2}(|x|,t)+\frac{|x|}{\sqrt{D}}\xi_{3\kappa+\alpha}(|x|,t)+\frac{|x|^{2}}{3D}\xi_{3\kappa+3\alpha/2}(|x|,t)\Big{)},$

where

$\xi_{\nu}(x,t)\equiv f_{\nu,\alpha/2}\left(t;\frac{x}{\sqrt{D}}\right).$ (63)

A special case of this function for $\nu=-1+\alpha/2$ is called the Mainardi
function.

Assuming $\tau s^{\kappa}<1$ we obtain

$\hat{\sigma^{2}}(s)=\frac{2D}{s^{1+\alpha}}\sum_{i=0}^{\infty}(-\tau)^{i}s^{\kappa
i}.$ (64)

It is not possible to calculate the inverse Laplace transform of the series in
the above equation term by term. Therefore, we calculate the transform of
${\rm e}^{-as^{\mu}}\hat{\sigma^{2}}(s)$, $a,\mu>0$, using Eq. (60), next the
limit of the obtained function when $a\rightarrow 0^{+}$ is calculated; the
result is independent of the parameter $\mu$. We obtain

$\sigma^{2}(t)=2Dt^{\alpha}\sum_{i=0}^{\infty}\frac{(-\tau)^{i}}{\Gamma(1+\alpha-i\kappa)t^{i\kappa}}.$
(65)

In the long time limit the functions Eqs. (62) and (65) generate the following
relations

$F_{R}(t\rightarrow\infty;x)=A_{\kappa}\frac{\tau}{t^{\kappa}},$ (66)
$\sigma^{2}_{R}(t\rightarrow\infty)=B_{\kappa}\frac{\tau}{t^{\kappa}},$ (67)

where $A_{\kappa}=-\Gamma(-\alpha/2)/[2\Gamma(-\kappa-\alpha/2)]$ and
$B_{\kappa}=\Gamma(1+\alpha)/\Gamma(1+\alpha-\kappa)$.

For a sufficiently small $s$ the series expansion of the function $\hat{\psi}$
with respect to $\tau$ is

$\hat{\psi}(s)=\sum_{n=0}^{\infty}\tau^{n}\sum_{i=n}^{\infty}(-\lambda)^{i}{i\choose
n}s^{\alpha i+\kappa n}.$ (68)

Using the same procedure as for calculating Eq. (65), we get

$\psi(t)=\sum_{n=0}^{\infty}\tau^{n}\sum_{i=n}^{\infty}{i\choose
n}\frac{(-\lambda)^{i}}{\Gamma(-\alpha i-\kappa n)t^{1+\alpha i+\kappa n}}.$
(69)

In the limit of long time we have

$\displaystyle\psi(t\rightarrow\infty)=\frac{-\lambda}{t^{1+\alpha}}\Bigg{[}\frac{1}{\Gamma(-\alpha)}$
(70)
$\displaystyle-\frac{\lambda}{\Gamma(-2\alpha)t^{\alpha}}+\frac{\tau}{\Gamma(-\alpha-\kappa)t^{\kappa}}\Bigg{]}.$

The equation (69) provides

$\psi_{\tau=0}(t)=\sum_{i=1}^{\infty}\frac{(-\lambda)^{i}}{\Gamma(-\alpha
i)t^{1+\alpha i}}.$ (71)

Based on Eqs. (53), (55), and (71) we get

$\psi_{\gamma}(t)=\sum_{i=1}^{\infty}\frac{(-\lambda\tau)^{i}}{\Gamma(-(\alpha+\kappa)i)t^{1+(\alpha+\kappa)i}}.$
(72)

For a short time (which corresponds to large $s$ in the Laplace transform) we
have $\psi_{\tau=0}(t)\sim 1/t^{1-\alpha}$ and $\psi_{\gamma}(t)\sim
t^{\kappa}/t^{1-\alpha}$, in the long time limit there is
$\psi_{\tau=0}(t)\sim 1/t^{1+\alpha}$ and $\psi_{\gamma}(t)\sim
1/t^{1+\alpha+\kappa}$. This suggests that for relatively short times the ACTP
process is dominant over ordinary subdiffusion, the opposite is true for a
long time. The function $\psi$ is a convolution of $\psi_{\tau=0}$ and
$\psi_{\gamma}$.

Figure 1: Time evolution of the relative MSD for the parameters given in the
legend, $\tau=0.1$ and $D=5$. Figure 2: Time evolution of the relative first
passage time for the parameters given in the legend, $x=10$, $\tau=0.1$ and
$D=5$. Figure 3: The relative Green’s function $P_{R}$ for $\alpha$ and
$\kappa$ given in the legend, $t=20$, $\tau=0.1$, and $D=10$.

The figures 1–3 show the plots of the relative functions $\sigma^{2}_{R}$,
$F_{R}$, and $P_{R}$, respectively. Values of variables and parameters are
given in arbitrarily chosen units. When ACTP has a large impact on
subdiffusion the relative functions take large values and when the impact is
small the functions are close to zero. Figs. 1 and 2 suggest that the ACTP
disappears over time, the main influence on the functions is the parameter
$\kappa$. In Fig. 3 we see that the ACTP increases with moving away from the
initial position of a particle. This effect causes that the probability of
finding the particle far from its initial position is much lower when the
process is described by the Cattaneo-type subdiffusion equation.

##  VI CTSE with an additional operator generated by a slowly varying function

A slowly varying function $\mathcal{R}(t)$ at infinity fulfils the relation

$\frac{\mathcal{R}(at)}{\mathcal{R}(t)}\rightarrow 1$ (73)

when $t\rightarrow\infty$ for any $a>0$. Functions that have finite limits
when $t\rightarrow\infty$ and logarithmic functions are examples of slowly
varying functions.

Let $\hat{\gamma}$ be a slowly varying function. In the following, we use the
strong Tauberian theorem to determine the inverse Laplace transform in the
long-time limit hughes : if $\phi(t)\geq 0$, $\phi(t)$ is ultimately monotonic
like $t\rightarrow\infty$, $\mathcal{R}$ is slowly varying at infinity, and
$0<\rho<\infty$, then each of the relations

$\hat{\phi}(s)\approx\frac{\mathcal{R}(1/s)}{s^{\rho}}$ (74)

as $s\rightarrow 0$ and

$\phi(t)\approx\frac{\mathcal{R}(t)}{\Gamma(\rho)t^{1-\rho}}$ (75)

as $t\rightarrow\infty$ implies the other.

In the following $\mathcal{R}(1/s)\equiv\hat{\gamma}(s)$ and $\gamma(t)$
denotes the exact inverse Laplace transform of $\hat{\gamma}(s)$ ($R(t)$ in
the long time limit and $\gamma(t)$ can be expressed by different formulas).

When $s\rightarrow 0$, the Green’s function Eq. (37) is approximated as

$\displaystyle\hat{P}(x,s)=\frac{\sqrt{1+\tau\mathcal{R}(1/s)}}{2\sqrt{D}s^{1-\alpha/2}}$
(76)
$\displaystyle\times\Bigg{[}1-\frac{s^{\alpha/2}|x|\sqrt{1+\tau\mathcal{R}(1/s)}}{\sqrt{D}}\Bigg{]}.$

Using the strong Tauberian theorem we get in the long time limit

$\displaystyle
P(x,t)=\frac{\sqrt{1+\tau\mathcal{R}(t)}}{2\sqrt{D}\Gamma(1-\alpha/2)t^{\alpha/2}}$
(77)
$\displaystyle\times\Bigg{[}1-\frac{\Gamma(1-\alpha/2)\sqrt{1+\tau\mathcal{R}(t)}|x|}{\sqrt{D}\Gamma(1-\alpha)t^{\alpha/2}}\Bigg{]}.$

Putting the above equation to Eq. (21) we get

$\displaystyle
F(t\rightarrow\infty;x)=\frac{|x|}{2\sqrt{D}\Gamma(1-\alpha/2)\sqrt{1+\tau\mathcal{R}(t)}t^{\alpha/2}}$
(78)
$\displaystyle\times\Bigg{[}\frac{1+\tau\mathcal{R}(t)}{t}-\tau\mathcal{R}^{(1)}(t)\Bigg{]}.$

The equation (78) provides

$\displaystyle F_{R}(t\rightarrow\infty;x)=1-\sqrt{1+\tau\mathcal{R}(t)}$ (79)
$\displaystyle+\frac{\tau
t\mathcal{R}^{(1)}(t)}{\sqrt{1+\tau\mathcal{R}(t)}}.$

From Eqs. (31), (47), and the strong Tauberian theorem we obtain

$\sigma^{2}(t\rightarrow\infty)=\frac{2Dt^{\alpha}}{\Gamma(1+\alpha)(1+\tau\mathcal{R}(t))},$
(80)
$\sigma_{R}^{2}(t\rightarrow\infty)=\frac{\tau\mathcal{R}(t)}{1+\tau\mathcal{R}(t)}.$
(81)

As an example we consider the CTSE with

$\hat{\gamma}(s)\equiv\mathcal{R}(1/s)=c\;{\rm
log}\frac{s^{2}+as+b}{s^{2}-as+b},$ (82)

then

$\mathcal{R}(t)=c\;{\rm log}\frac{bt^{2}+at+1}{bt^{2}-at+1},$ (83)

where the parameters $a$ and $c$ are given in the units of $1/{\rm s}$, $b$ of
$1/{\rm s}^{2}$. The inverse Laplace transform of $\hat{\gamma}(s)$ is ob

$\gamma(t)=4ct^{-1}{\rm sinh}(at/2){\rm cos}(t\sqrt{b-a^{2}/4}).$ (84)

In the limit of long time we get

$F_{R}(t\rightarrow\infty;x)=A\frac{\tau}{t},$ (85)
$\sigma_{R}^{2}(t\rightarrow\infty)=B\frac{\tau}{t},$ (86)

where $A=-2ac/[b\alpha(2+\alpha)]$ and $B=2ac/b$.

The process is described by Eq. (35), where the kernel Eq. (84) of the
operator $\mathcal{D}_{t}$ has an oscillatory component. To find the solutions
to the equation in the long time limit the strong Tauberian theorem has been
used. In this solution, the oscillatory component does not occur. This is
because the ACTP decays for long times.

##  VII Final remarks

In our considerations, the constitutive equation relating the flux and the
concentration gradient is modified by the integral operator with kernel
$\beta(t;\tau)$. In the time domain the equation reads

$\int_{0}^{t}\beta(t-t^{\prime};\tau)J(x,t^{\prime})dt^{\prime}=-D\frac{\partial^{1-\alpha}}{\partial
t^{1-\alpha}}\frac{\partial P(x,t)}{\partial x}.$ (87)

Combining the above equation with the continuity equation we get the
generalized subdiffusion equation. When
$\hat{\beta}(s;\tau)=1+\tau\hat{\gamma}(s)$,
$\beta(t;\tau)=\delta(t)+\tau\gamma(t)$ in the time domain, we have the
Cattaneo–type subdiffusion equation that differs from the ordinary
subdiffusion one by an integro–differential operator with kernel $\gamma(t)$.
When $\hat{\gamma}(s)=s^{\kappa}$ with $0<\kappa<1$, which corresponds to
$\gamma(t)\sim 1/t^{1+\kappa}$, the operator is the Caputo fractional time
derivative of the order $1+\kappa$. The functions $\sigma^{2}_{R}$, $F_{R}$,
and $P_{R}$ can be interpreted as measures of the influence of ACTP on
subdiffusion.

For the process described by the CTSE with additional operator being the
Caputo fractional time derivative there is $\sigma^{2}_{R}\sim F_{R}\sim
1/t^{\kappa}$, $0<\kappa<1$, while for the equation with additional operator
generated by a slowly varying function $\hat{\gamma}$ we have
$\sigma^{2}_{R}\sim F_{R}\sim 1/t$ when $t\rightarrow\infty$. This suggests
that the ACTP described by a slowly variable function decays more quickly over
time.

The persistent random walk model has been used to derive the Cattaneo
subdiffusion equation masoliver ; masoliver2016 . The CTSE can be derived
within the “ordinary” continuous time random walk model with the waiting time
of a molecule to jump $\psi$ Eq. (48). This distribution has a heavy tail
modified by ACTP, see Eq. (70). It can be interpreted as a subdiffusion model
which includes an “additional” process that affects the kinetics of
subdiffusion. The additional process may be independent of the process
generating subdiffusion. In our considerations, we have assumed that the
parameters $\kappa$ and $\alpha$ in Eq. (59) are independent of each other. As
we have mentioned in Sec. I, an example would be subdiffusion (or normal
diffusion when $\alpha=1$) of antibiotics in a bacterial biofilm. The defense
mechanisms of bacteria against the action of antibiotics can change the random
walk of antibiotic molecules. In our opinion, the Cattaneo-type equation can
be used to describe such processes. It may be interesting to use the Cattaneo-
type equation in modeling the spread of an epidemic. The first arrival of an
infected object to some point may create new source of infection. Since ACTP
changes the distribution of $F$, the use of Eqs. (4) and (59) (with possibly
added reaction terms in both equations) in epidemic modeling may give
qualitatively different results.

When $\hat{\beta}(s\rightarrow 0;\tau)\rightarrow 1$, which corresponds to
$\hat{\gamma}(s)\rightarrow 0$, the effect generated by the function $\beta$
disappears over time, $F_{R}(t,x)\rightarrow 0$ and
$\sigma^{2}_{R}(t)\rightarrow 0$ when $t\rightarrow\infty$. In order to make
the additional Cattaneo-type process noticeable also for long time, it can be
assumed that the parameter $\tau$ increases over time. This case is met when
$\mathcal{L}[\tau\gamma(t)](s)$ does not goes to zero when $s\rightarrow 0$.

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