Scaling dependence on time and distance in nonlinear fractional diffusion equations and possible applications to the water transport in soils%
Kwok Sau Fa and E. K. Lenzi
Recently, fractional derivatives have been employed to analyze various systems in engineering, physics, finance and hidrology. For instance, they have been used to investigate anomalous diffusion processes which are present in different physical systems like: amorphous semicondutors, polymers, composite heterogeneous films and porous media. They have also been used to calculate the heat load intensity change in blast furnace walls, to solve problems of control theory \ and dynamic problems of linear and nonlinear hereditary mechanics of solids. In this work, we investigate the scaling properties related to the nonlinear fractional diffusion equations and indicate the possibilities to the applications of these equations to simulate the water transport in unsaturated soils. Usually, the water transport in soils with anomalous diffusion, the dependence of concentration on time and distance may be expressed in term of a single variable given by $\lambda _{q}=x/t^{q}.$ In particular, for $q=1/2$ the systems obey Fick's law and Richards' equation for water transport. We show that a generalization of Richards' equation via fractional approach can incorporate the above property.
cond-mat0404352.tex

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\begin{document}

\title{Scaling dependence on time and distance in nonlinear fractional
diffusion equations and possible applications to the water transport in soils%
}
\author{Kwok Sau Fa and E. K. Lenzi }
\affiliation{Departamento de F\'{\i}sica, Universidade Estadual de Maring\'{a},\\
Av. Colombo 5790, 87020-900, Maring\'{a}-PR, Brazil}

\begin{abstract}
Recently, fractional derivatives have been employed to analyze various
systems in engineering, physics, finance and hidrology. For instance, they
have been used to investigate anomalous diffusion processes which are
present in different physical systems like: amorphous semicondutors,
polymers, composite heterogeneous films and porous media. They have also
been used to calculate the heat load intensity change in blast furnace
walls, to solve problems of control theory \ and dynamic problems of linear
and nonlinear hereditary mechanics of solids. In this work, we investigate
the scaling properties related to the nonlinear fractional diffusion
equations and indicate the possibilities to the applications of these
equations to simulate the water transport in unsaturated soils. Usually, the
water transport in soils with anomalous diffusion, the dependence of
concentration on time and distance may be expressed in term of a single
variable given by $\lambda _{q}=x/t^{q}.$ In particular, for $q=1/2$ the
systems obey Fick's law and Richards' equation for water transport. We show
that a generalization of Richards' equation via fractional approach can
incorporate the above property.
\end{abstract}

\pacs{05.30.Pr, 46.40.Ff}
\maketitle

\section{Introduction}

\bigskip

In these last decades we have witnessed a growing use of the fractional
calculus to describe diverse systems in engineering, physics and finance. In
physics, the fractional diffusion equations have been used to describe
anomalous diffusion processes which are present, for instance, in amorphous
semiconductors \cite{scher}, polymers \cite{cates}, composite heterogeneous
films \cite{nikla} and porous media \cite{berko}. In engineering, for
example, the fractional derivative has been used in viscoelastic models
which generalize the classical mechanical models \cite{mainardi}. In finance
market, the Gaussian assumption for the probability distribution of returns
has been extended by using non-Gaussian assumption and the fractional
derivative approach, according to the empirical studies \cite{laskin}.

In hydrology, the description of the transport of water in many soils, in
horizontal column, is done by the nonlinear diffusion equation \cite{hillel}

\begin{equation}
\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial x}\left[
D_{1}(\theta )\frac{\partial \theta }{\partial x}\right] \text{ .}
\label{eq1}
\end{equation}%
In this equation presents a scaling dependence of concentration on time $t$
and distance $x$ given by

\begin{equation}
\lambda _{1/2}(\theta )=\frac{x}{t^{\frac{1}{2}}}\text{ .}  \label{eq2}
\end{equation}%
This is so called the Boltzmann variable. Eq. (\ref{eq1}) written in term of 
$\lambda _{1/2}$ is given by

\bigskip 
\begin{equation}
-\frac{\lambda _{1/2}}{2}\frac{\text{d}\theta }{\text{d}\lambda _{1/2}}=%
\frac{\text{d}}{\text{d}\lambda _{1/2}}\left[ D_{1}(\theta )\frac{\text{d}%
\theta }{\text{d}\lambda _{1/2}}\right] \text{ .}  \label{eq3}
\end{equation}%
However, in the soils like Columbia silt loam, Hesperia silt loam, Hesperia
sandy loam, Salkum silty clay loam and kaolinite-sand mixtures, deviations
from (\ref{eq2}) have been observed \cite{gardner}, and the dependence of
concentration on $t$ and $x$ \cite{pachep} is given by

\begin{equation}
\lambda _{q}(\theta )=\frac{x}{t^{q}}\text{ ,}  \label{eq4}
\end{equation}%
where $q$ can be greater or less than $1/2$. In order to obtain the
generalized scaling law (\ref{eq4}), based on the nonlinear diffusion
equation, further assumptions have been incorporated into Eq. (\ref{eq1}):
In \cite{pachep}, the diffusion coefficient is assumed to have an explicit
dependence on distance given by $x^{-\beta }D(\theta )$, while in \cite%
{guerrini} the diffusion coefficient is assumed to have an explicit
dependence on time given by $t^{2m-1}D(\theta )$, where $m$ is a positive
constant. Indeed, the above assumptions are phenomenological, and other
possibilities to obtain the scaling law (\ref{eq4}) exist. An interesting
possibility is to use the fractional approach. Fractional derivatives can be
defined in terms of integral, and due to this fact they are related to the
memory effects when they are employed in differential equations applied to
physical systems. In stochastic processes the fractional derivatives are
related to the non-Markovian \ nature of the systems, for instance, they can
provide a dynamic framework to describe long random walks and waiting time
between successive walks \cite{metzler}.

In this work, we investigate the scaling dependence on time and distance in
nonlinear fractional diffusion equations. To do so, we employ the
Riemann-Liouville and Caputo definitions for fractional derivatives. Indeed,
there are other definitions for fractional derivatives \cite{oldham}, but
the Riemann-Liouville and Caputo definitions are most often used in the
literature. According to the previous discussions, these nonlinear
fractional diffusion equations, with the scaling dependence on time and
distance, may be useful to describe the water transport in soils when the
memory effects are present in these systems. The presentation of this work
is outlined as follows. In section II we discuss on the scaling dependence
on time and distance in several nonlinear fractional diffusion equations by
using the fractional derivatives on distance and time. In section III, we
point out on some possible applications of these equations in the water
transport in soils. In section IV, we present our conclusions.

\section{Nonlinear fractional diffusion equations and scaling dependence on
time and distance}

Several physical systems are intrinsically structured with some fundamental
scaling relations. When they are revealed they can bring a beneficial
insight into a new analysis of the systems. Its presence in physical systems
indicates that the quantities involved in the scaling relation are more
rigidly connected, even with the changing of the parameters of the systems.
For instance, many systems of the water transport in soils are described by
Eq. (\ref{eq1}) which is related to the dependence of the concentration on
distance and time (\ref{eq3}). The propagator on fractals in Sierpinski
gasket also presents a scaling dependence on distance and time in its
asymptotic form \cite{metzler}. A remarkable presence of scaling relation
appears in the Gibbs potential \ on the region of the critical point \cite%
{callen}.

In order to investigate the scaling relations of nonlinear fractional
diffusion equations, we consider the Riemann-Liouville and Caputo
definitions for fractional derivatives which are given by

\begin{equation}
D_{RL(z)}^{\alpha }f(z)=\frac{1}{\Gamma (n-\alpha )}\frac{\text{d}^{n}}{%
\text{d}z^{n}}\int_{0}^{z}\frac{f(\tau )}{(z-\tau )^{\alpha +1-n}}\text{d}%
\tau  \label{eq5}
\end{equation}%
and

\begin{equation}
D_{C(z)}^{\alpha }f(z)=\frac{1}{\Gamma (n-\alpha )}\int_{0}^{z}\frac{%
f^{n}(\tau )}{(z-\tau )^{\alpha +1-n}}\text{d}\tau  \label{eq6}
\end{equation}%
where $D_{RL(z)}^{\alpha }$ indicates the Riemann-Liouville operator and $%
D_{C(z)}^{\alpha }$ indicates the Caputo operator. These two definitions are
connected by the following expression

\begin{equation}
D_{RL(z)}^{\alpha }f(z)=D_{C(z)}^{\alpha
}f(z)+\sum_{k=0}^{n-1}f^{(k)}(0^{+})\Phi _{(k-\alpha +1)}(z)\text{ \ ,}
\label{eq7}
\end{equation}%
where

\begin{equation}
\Phi _{\beta }(z)=\left( 
\begin{array}{c}
\frac{z^{\beta -1}}{\Gamma (\beta )}\text{, \ \ \ }z\geqslant 0 \\ 
0\text{, \ \ \ \ \ \ \ \ \ \ }z<0%
\end{array}%
\right) \text{ \ \ .}  \label{eq8}
\end{equation}

First, we consider the Riemann-Liouville fractional derivative on distance
applied to Eq. (\ref{eq1})

\begin{equation}
\frac{\partial \theta }{\partial t}=D_{RL(x)}^{\alpha }\left[ D_{\alpha
}(\theta )\frac{\partial \theta }{\partial x}\right] ,\text{ \ \ }0<\alpha <1%
\text{.}  \label{eq9}
\end{equation}%
This equation, with $D_{\alpha }(\theta )=D$, has been used to study
anomalous diffusion for L\'{e}vy flights. For $D_{\alpha }(\theta )=D\theta
^{\alpha },$ it has been studied in \cite{bologna}. By using the generalized
scaling variable (\ref{eq4}) we obtain from Eq. (\ref{eq9}) the following
result

\begin{equation}
-q\frac{\lambda _{q}}{t}\frac{\text{d}\theta }{\text{d}\lambda _{q}}=\frac{1%
}{\Gamma (1-\alpha )}\frac{\partial }{\partial x}\int_{0}^{x}\frac{D_{\alpha
}(\theta )\partial _{\tau }\theta }{(x-\tau )^{\alpha }}\text{d}\tau \text{ .%
}  \label{eq10}
\end{equation}%
Introducing the variable $u=\tau /t^{q}$ into Eq. (\ref{eq10}) yields

\begin{equation}
-q\lambda _{q}\frac{\text{d}\theta }{\text{d}\lambda _{q}}=\text{ }%
D_{RL(\lambda _{q})}^{\alpha }\left[ D_{\alpha }(\theta )\frac{\text{d}%
\theta }{\text{d}\lambda _{q}}\right]   \label{eq11}
\end{equation}%
with $q=1/(1+\alpha )$. We see that Eq. (\ref{eq9}) can be expressed as a
function of generalized scaling relation (\ref{eq4}). Note that, for $\alpha
=1,$ we recover the Boltzmann variable.

For Caputo operator on distance, we also use the above procedure and the
result is similar to Eq. (\ref{eq11}) that is

\begin{equation}
-q\lambda _{q}\frac{\text{d}\theta }{\text{d}\lambda _{q}}=D_{C(\lambda
_{q})}^{\alpha }\left[ D_{\alpha }(\theta )\frac{\text{d}\theta }{\text{d}%
\lambda _{q}}\right] \text{ ,}  \label{eq12}
\end{equation}%
with $q=1/(1+\alpha )$.

For fractional derivative on time, we consider the Caputo operator acting on
the left side of Eq. (\ref{eq1}),

\begin{equation}
D_{C(t)}^{\alpha }\theta =\frac{\partial }{\partial x}\left[ D_{\alpha
}(\theta )\frac{\partial \theta }{\partial x}\right] \text{ , \ \ \ }%
0<\alpha <1\text{.}  \label{eq13}
\end{equation}%
This equation has been used to study anomalous diffusion for subdiffusive
regimes \cite{metzler}. Introducing the variable $u=\frac{x}{t^{1+q}}\tau $
into Eq. (\ref{eq13}) and using the variable (\ref{eq4}) we obtain

\begin{equation}
\frac{\lambda _{q}^{\alpha }}{t^{\alpha }}\frac{1}{\Gamma (1-\alpha )}%
\int_{0}^{\lambda _{q}}\frac{\partial _{u}\theta }{(\lambda _{q}-u)^{\alpha }%
}\text{d}u=\frac{1}{t^{2q}}\frac{\text{d}}{\text{d}\lambda _{q}}\left[
D_{\alpha }(\theta )\frac{\text{d}\theta }{\text{d}\lambda _{q}}\right] 
\text{ .}  \label{eq14}
\end{equation}%
By setting $\alpha =2q$, we have

\begin{equation}
\lambda _{q}^{\alpha }O_{C(t)}^{\alpha }\theta =\frac{\text{d}}{\text{d}%
\lambda _{q}}\left[ D_{\alpha }(\theta )\frac{\text{d}\theta }{\text{d}%
\lambda _{q}}\right] \text{ .}  \label{eq15}
\end{equation}%
Note that the singularity which appears in the variable $u=\frac{x}{t^{1+q}}%
\tau $ \ for $t\rightarrow 0$ does not invalidate the integration of Eq. (%
\ref{eq13}) due to the fact that the limits of the corresponding integral
tend to the same values. Therefore, Eq. (\ref{eq14}) is valid for $t>0$. The
scaling relation (\ref{eq4}) related to Eq. (\ref{eq13}) is also
corroborated by the analytical result obtained in \cite{metzler}. An
observation on the integral of the left side of Eq. (\ref{eq14}); It does
not represent the usual definition of fractional derivative (\ref{eq5}) due
to the fact that the variable $\theta $ depends on both $\lambda _{q}$ and
integration variable $u$. For this reason, we have employed the operator $%
O_{C(t)}^{\alpha }$ instead of $D_{C(t)}^{\alpha }$.

For Riemann-Liouville operator on time acting on the left side of Eq. (\ref%
{eq1}), the corresponding fractional equation related to the scaling
variable (\ref{eq4}) is given by

\begin{equation}
\frac{(1-\alpha )(2+\alpha )\lambda _{q}^{\alpha -1}}{2\Gamma (1-\alpha )}%
\int_{0}^{\lambda _{q}}\frac{\theta }{(\lambda _{q}-u)^{\alpha }}\text{d}u-%
\frac{\alpha }{2}\lambda _{q}^{\alpha }O_{RL(\lambda _{q})}^{\alpha }\theta =%
\frac{\text{d}}{\text{d}\lambda _{q}}\left[ D_{\alpha }(\theta )\frac{\text{d%
}\theta }{\text{d}\lambda _{q}}\right]  \label{eq16}
\end{equation}%
where $q=\alpha /2$. In the case of the Riemann-Liouville operator on time
acting on the right side of Eq. (\ref{eq1}), it can be replaced by the
Caputo operator on time acting on the left side (\ref{eq13}), see \cite%
{metzler}. The corresponding fractional equation is

\begin{equation}
\frac{\partial \theta }{\partial t}=D_{RL(t)}^{\alpha }\left[ D_{\alpha
}(\theta )\frac{\partial \theta }{\partial x}\right] =\frac{1}{\Gamma
(1-\alpha )}\frac{\partial }{\partial t}\int_{0}^{t}\frac{\partial _{x}\left[
D_{\alpha }(\theta )\partial _{x}\theta \right] }{(t-\tau )^{\alpha }}\text{d%
}\tau \text{ .}  \label{eq17}
\end{equation}

\bigskip

\section{Water transport in soils and nonlinear fractional diffusion
equations}

\bigskip

As we have previously mentioned that many systems with the transport of
water in soils, in horizontal column, are simulated by Eq. (1) and scaling
relation (\ref{eq2}). However, the scaling relation (\ref{eq2}) can not
provide a good agreement for all the experimental results \cite%
{guerrini,pachep}.

We now consider a generalization of Eq. (\ref{eq1}) which has been
introduced in \cite{guerrini} with the diffusion coefficient dependent of
the time given by

\begin{equation}
\frac{\partial \theta }{\partial t}=\frac{\partial }{\partial x}\left[
t^{-\beta }D_{\beta }(\theta )\frac{\partial \theta }{\partial x}\right] 
\text{ ,}  \label{eq18}
\end{equation}%
where $\beta $ is a real parameter and $\theta $ is the volumetric water
content. In particular, Eq. (\ref{eq18}), for $D_{\beta }(\theta )=D$ and $%
\beta =-2$, introduced by Batchelor, has been used to simulate turbulent
flows \cite{klafter}. For water transport in soils, the boundary conditions
are usually chosen as follows \cite{hillel}:

\begin{equation}
x>0,\text{ \ \ \ }\theta =\theta _{i}\text{, \ }t=0  \label{eq19}
\end{equation}

\begin{equation}
x=0,\text{ \ \ }\theta =\theta _{0},\text{ \ }t>0  \label{eq20}
\end{equation}
where $\theta _{i}$ is the initial water content along the soil column and $%
\theta _{0}$ is the constant water content at $x=0$. It has been shown that
Eq. (\ref{eq18}) is related to scaling relation (\ref{eq4}), and it has also
been used to simulate the single-energy gamma-ray attenuation data for
Salkum silty clay loam \cite{guerrini}.

Now, we employ an anternative route to analyze Eq. (\ref{eq18}) in
comparison with the fractional equation (\ref{eq17}). Integrating Eq. (\ref%
{eq18}) on time we have

\begin{equation}
\theta (x,t)-\theta (x,0)=\frac{1}{\Gamma (1-\beta )}\int_{0}^{t}\frac{%
\partial _{x}\left[ D_{\beta }(\theta )\partial _{x}\theta \right] }{\tau
^{\beta }}\text{d}\tau \text{ .}  \label{eq21}
\end{equation}%
The constant $\Gamma (1-\beta )$ was introduced by convenience. \
Introducing the variable $\tau =t-u$ yields

\begin{equation}
\theta (x,t)-\theta (x,0)=\frac{1}{\Gamma (1-\beta )}\int_{0}^{t}\frac{%
\partial _{x}\left[ D_{\beta }(\theta )\partial _{x}\theta \right] }{%
(t-u)^{\beta }}\text{d}u\text{ .}  \label{eq22}
\end{equation}%
Applying the derivative on time we obtain

\begin{equation}
\frac{\partial \theta }{\partial t}=\frac{1}{\Gamma (1-\beta )}\frac{%
\partial }{\partial t}\int_{0}^{t}\frac{\partial _{x}\left[ D_{\beta
}(\theta )\partial _{x}\theta \right] }{(t-u)^{\beta }}\text{d}u\text{ .}
\label{eq23}
\end{equation}%
It is interesting to note that this last equation looks like Eq. (\ref{eq17}%
), but they are not the same. In Eq. (\ref{eq23}) the variable $\theta $
depends on $x,t$ and $u$, while in Eq. (\ref{eq17}) the variable $\theta $
depends only on $x$ and $\tau $. Operationally speaking, the denominator of
the integrand of Eq. (\ref{eq17}), $t-\tau $, runs from $t$ to zero, while
the denominator of the integrand of Eq. (\ref{eq21}), $\tau $, runs from
zero to $t$; They are the inverse of each other. Eq. (\ref{eq17}) and Eq. (%
\ref{eq21}) are the same for a specific case, i.e., if the numerators of the
integrands of these equations are symmetrical with relation to the point $t/2
$ for all the instant $t$.

An other generalization of Eq. (\ref{eq1}) has been introduced \ in \cite%
{pachep} with the diffusion coefficient dependent on the distance

\begin{equation}
\frac{\partial \theta }{\partial t}=-\frac{\partial }{\partial x}\left[ J%
\right] =\frac{\partial }{\partial x}\left[ x^{-\beta }D_{\beta }(\theta )%
\frac{\partial \theta }{\partial x}\right] \text{ ,}  \label{eq24}
\end{equation}%
where $J$ is the flux. In particular, Eq. (\ref{eq24}), for $D_{\beta
}(\theta )=D$ and $\beta =-4/3$, introduced by Richardson, has been used to
simulate turbulent flows \cite{sokolov}. Eq. (\ref{eq9}) can be obtained
from Eq. (\ref{eq24}) by taking the early contributions of the flux.

We should mention that the complexities of the soil structures are not
considered in the descriptions of the water transport in soils by using
diffusion equations. In these descriptions are considered the changing of
porous media and bulk density, and the systems may be categorized as follows 
\cite{guerrini}: (1) rigid porous media with no change in the bulk density.
(2) semirigid porous media with no change in the bulk density and (3)
swelling porous media with the change in the bulk density. The first
category is usually described by Eq. (\ref{eq1}) with the diffusion
coefficient depends only on the volumetric water content $\theta $. For the
second and third categories, with the possibilities of changing in the
structures of porous media and bulk density, the descriptions of these
systems may be Eqs. (\ref{eq18}), (\ref{eq24}) or nonlinear fractional
diffusion equations described in the text, with the possibilities of using
more complex diffusivity functions. We note that for the fractional
diffusion equation on time the value of $q=\alpha /2$ is less than $1/2$,
while for the fractional diffusion equation on distance the value of $%
q=1/(1+\alpha )$ is greater than $1/2$, for $0<\alpha <1$. These both
equations can cover the range $0<q<1$.

\bigskip

\section{Conclusion}

\bigskip

We have investigated the scaling relation (\ref{eq4}) for several types of
nonlinear fractional diffusion equations. This relation indicates that the
nonlinear fractional diffusion equations may play important role to the
transport of water in soils when the Boltzmann scaling is violated. Further,
we have shown that the usual nonlinear diffusion equation with the diffusion
coefficient dependent on the time (\ref{eq18}) is closely related to the
nonlinear fractional diffusion equation (\ref{eq17}). We hope that this work
may contribute to reveal some aspects related to the fractional diffusion
equations and the interplay between fractional and usual diffusion equations.

\newpage

\begin{thebibliography}{99}
\bibitem{scher} H. Scher and E. Montroll, Phys. Rev. B 12, 2455 (1975); U.
Even et al., Phys. Rev. Lett. 52, 2164 (1984).

\bibitem{cates} M. Cates, Phys. Rev. Lett. 53, 926 (1984); H. Schiessel and
A. Blumen, Fractals 3, 483 (1995).

\bibitem{nikla} G. Niklasson and C. Granqvist, Phys. Rev. Lett. 56, 256
(1986).

\bibitem{berko} B. Berkowitz and H. Scher, Phys. Rev. Lett. 79, 4038 (1997);
A. Klemm, H. M\"{u}ller and R. Kimmih, Phys. Rev. E 55, 4413 (1997).

\bibitem{mainardi} F. Mainardi, Fractinal calculus: Some basic problems in
continuum and statistical mechanics, in : A. Carpinteri, F. Mainardi (Eds.),
Fractals and Fractional Calculus in Continnum Mechanics, Springer-Verlag,
NY, 1997, pp. 291-348; T.M. Atanackovic and B. Stankovic, Journal of sound
and vibration (2004), in press.

\bibitem{laskin} N. Laskin, Physica A 287, 482 (2000).

\bibitem{hillel} D. Hillel, Soil and Water: Physical Principles and
Processes, 1971, Academic Press, NY.

\bibitem{gardner} W. Gardner and J.A. Widtsoe, Soil Sci. 11, 215 (1921);
Nielsen et al., Soil Sci. Soc. Am. Proc. 26, 107 (1962); S.I. Rawlins and
W.H. Gardner, Soil Sci. Soc. Am. Proc. 27, 507 (1963); H. Ferguson and W.R.
Gardner, Soil Sci. Soc. Am. Proc. \ 27, 243 (1963).

\bibitem{pachep} Y. Pachepsky and D. Timlin, J. Hydrology 204, 98 (1998).

\bibitem{guerrini} I.A. Guerrini and D. Swartzendruber, Soil Sci. Soc. Am.
J. 56, 335 (1992).

\bibitem{metzler} R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).

\bibitem{oldham} K. B. Oldham and J. Spannier, The fractional calculus
(Academic Press, NY) 1974.

\bibitem{callen} H. B. Callen, Thermodynamics and an introduction to
thermostatistics (second edition), 1985, John Wiley \& Sons.

\bibitem{bologna} M. Bologna, C. Tsallis and P. Grigolini, Phys. Rev. E 62,
2213 (2000).

\bibitem{klafter} J. Klafter, A. Blumen and M. F. Shlesinger 35, 3081 (1987).

\bibitem{sokolov} M. C. Jullien, J. Paret and P Tabeling, Phys. Rev. Lett.
82, 2872 (1999); G. Boffetta and A. Celani, Phys. Rev. E 60, 6734 (1999); G.
Boffetta and I. M. Sokolov, Phys. Rev. Lett. 88, 094501 (2002).
\end{thebibliography}

\end{document}

cond-mat0404352.extracted_bib
\begin{thebibliography}{99}
\bibitem{scher} H. Scher and E. Montroll, Phys. Rev. B 12, 2455 (1975); U.
Even et al., Phys. Rev. Lett. 52, 2164 (1984).

\bibitem{cates} M. Cates, Phys. Rev. Lett. 53, 926 (1984); H. Schiessel and
A. Blumen, Fractals 3, 483 (1995).

\bibitem{nikla} G. Niklasson and C. Granqvist, Phys. Rev. Lett. 56, 256
(1986).

\bibitem{berko} B. Berkowitz and H. Scher, Phys. Rev. Lett. 79, 4038 (1997);
A. Klemm, H. M\"{u}ller and R. Kimmih, Phys. Rev. E 55, 4413 (1997).

\bibitem{mainardi} F. Mainardi, Fractinal calculus: Some basic problems in
continuum and statistical mechanics, in : A. Carpinteri, F. Mainardi (Eds.),
Fractals and Fractional Calculus in Continnum Mechanics, Springer-Verlag,
NY, 1997, pp. 291-348; T.M. Atanackovic and B. Stankovic, Journal of sound
and vibration (2004), in press.

\bibitem{laskin} N. Laskin, Physica A 287, 482 (2000).

\bibitem{hillel} D. Hillel, Soil and Water: Physical Principles and
Processes, 1971, Academic Press, NY.

\bibitem{gardner} W. Gardner and J.A. Widtsoe, Soil Sci. 11, 215 (1921);
Nielsen et al., Soil Sci. Soc. Am. Proc. 26, 107 (1962); S.I. Rawlins and
W.H. Gardner, Soil Sci. Soc. Am. Proc. 27, 507 (1963); H. Ferguson and W.R.
Gardner, Soil Sci. Soc. Am. Proc. \ 27, 243 (1963).

\bibitem{pachep} Y. Pachepsky and D. Timlin, J. Hydrology 204, 98 (1998).

\bibitem{guerrini} I.A. Guerrini and D. Swartzendruber, Soil Sci. Soc. Am.
J. 56, 335 (1992).

\bibitem{metzler} R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).

\bibitem{oldham} K. B. Oldham and J. Spannier, The fractional calculus
(Academic Press, NY) 1974.

\bibitem{callen} H. B. Callen, Thermodynamics and an introduction to
thermostatistics (second edition), 1985, John Wiley \& Sons.

\bibitem{bologna} M. Bologna, C. Tsallis and P. Grigolini, Phys. Rev. E 62,
2213 (2000).

\bibitem{klafter} J. Klafter, A. Blumen and M. F. Shlesinger 35, 3081 (1987).

\bibitem{sokolov} M. C. Jullien, J. Paret and P Tabeling, Phys. Rev. Lett.
82, 2872 (1999); G. Boffetta and A. Celani, Phys. Rev. E 60, 6734 (1999); G.
Boffetta and I. M. Sokolov, Phys. Rev. Lett. 88, 094501 (2002).
\end{thebibliography}