General solution of the diffusion equation with a nonlocal diffusive term and a linear force term
L. C. Malacarne,
1
R. S. Mendes,
1
E. K. Lenzi,
1
and M. K. Lenzi
2
1
Universidade Estadual de Maringá, Departamento de Física, 87020-900 Maringá, Paraná, Brazil
2
Departamento de Engenharia Química, Universidade Federal do Paraná, Setor de Tecnologia, Jardim das Américas,
Caixa Postal 19011, 81531-990 Curitiba, Paraná, Brazil
Received 19 June 2006; published 17 October 2006
We obtain a formal solution for a large class of diffusion equations with a spatial kernel dependence in the
diffusive term. The presence of this kernel represents a nonlocal dependence of the diffusive process and, by
a suitable choice, it has the spatial fractional diffusion equations as a particular case. We also consider the
presence of a linear external force and source terms. In addition, we show that a rich class of anomalous
diffusion, e.g., the Lévy superdiffusion, can be obtained by an appropriated choice of kernel.
DOI: 10.1103/PhysRevE.74.042101 PACS numbers: 05.40.Fb, 66.10.Cb, 05.20.-y, 05.60.-k
I. INTRODUCTION
The diffusive process is one of the most usual processes
in nature and, since the Brown study and Einstein’s first ex-
planation 1, it has attracted attention in all science fields. In
the last decades, diffusive processes that do not present the
usual asymptotic time dependence on the second moment,
i.e., x
2
t, have also been related to a large class of physi-
cal systems. Illustrative examples are fluid transport in po-
rous media 2, diffusion in plasmas 3, substance trans-
ported in a solvent from one vessel to another across a thin
membrane 4, asymmetry of DNA translocation 5, relative
diffusion in turbulent media 6, cetyltrimethylammonium
bromide CTAB micelles dissolved in salted water 7, sur-
face growth and transport of fluid in porous media 8, two
dimensional rotating flow 9, subrecoil laser cooling 10,
diffusion on fractals 11, anomalous diffusion at liquid sur-
faces 12, enhanced diffusion in active intracellular trans-
port 13, particle diffusion in a quasi-two-dimensional bac-
terial bath 14, and spatiotemporal scaling of solar surface
flows 15. Thus the existence of the anomalous diffusion
and its ubiquity has motivated the study of several ap-
proaches, in particular, the ones based on fractional diffusion
equations 16–19 that have intensively been investigated. In
fact, in Ref. 20 the fractional diffusion and wave equations
are discussed, in 21 the boundary value problems for frac-
tional diffusion equations are studied, in Ref. 22 a frac-
tional Fokker-Planck equation is derived from a generalized
master equation, in Ref. 23 the behavior of fractional dif-
fusion at the origin is analyzed, in Ref. 24 a harmonic
analysis of random fractional diffusion-wave equations is
done, in Ref. 25 a fractional Kramers equation is intro-
duced, and in Refs. 26–34 the solutions of the time-
fractional diffusion equations are obtained.
In this Brief Report, we consider the formal solution of a
large class of anomalous diffusion processes described by the
equation
t
P
ˆ
x, t =
D
2
-
Kx - x'
2
x'
2
P
ˆ
x', tdx'
-
x
Fx, t P
ˆ
x, t + t P
ˆ
x, t , 1
where Kx is the kernel which contains a nonlocal depen-
dence, D is the diffusion coefficient, Fx , t is the external
force, and t is a time-dependent source. Note that, due to
the broadness of Eq. 1, it encompasses several scenarios of
physical interest such as the distributed fractional diffusion
equations 35, truncated Lévy flights 36, and advection-
dispersion equations with a fractional Laplacian operator tak-
ing a general directional mixing measure into account 37.
In this direction, Eq. 1 may be used to investigate turbu-
lence 38, anomalous diffusion in disordered media 39,
and transport in the direction of flow in an aquifer with
heavy tailed distribution 40.
In this paper, we work out Eq. 1 by taking a general
kernel into account with Fx , t =-Cx. In particular, we show
how to obtain the generalized solution from the usual one for
the case characterized by the absence of external force.
We also discuss particular cases which emerge from
choices Kx 1/ |x|
1+
and Kx e
-a|x|
. These developments
are presented in Sec. II and in Sec. III, we present our
conclusions.
II. DIFFUSION EQUATION
Let us start our analysis by considering Eq. 1 without
external force and subject to the initial condition P
ˆ
x ,0
=
2x and the boundary condition P
ˆ
± , t =0. In order
to eliminate the source term of Eq. 1, we use the change
P
ˆ
x , t = exp
0
t
tdt Px , t, which leads us to the equation
t
Px, t =
D
2
-
Kx - x'
2
x'
2
Px', tdx' . 2
Note that the above equation reduces to the usual diffusion
equation for the kernel Kx =
2x and other kernels im-
ply a spatially nonlocal correlation. In addition, direct inves-
tigation shows that this diffusion equation can be correlated
with a continuous time random walk for a Poissonian
waiting time probability density function and a jump length
probability density function given by
˜
k =1- Dk
2
K
˜
k,
where is the characteristic waiting time and K
˜
k is the
Fourier transform F¯ =
1
2
-
dxe
ikx
¯ and F
-1
¯
=
1
2
-
dke
-ikx
¯ of Kx. Thus the presence of this kernel
PHYSICAL REVIEW E 74, 042101 2006
1539-3755/2006/744/0421014 ©2006 The American Physical Society 042101-1