Fractional diffusion equation and Green function approach: Exact solutions

• Published in: Physica A Vol. 360; no. 2; pp. 215 - 226
• Main Authors: Lenzi, E.K., Mendes, R.S., Gonçalves, G., Lenzi, M.K., da Silva, L.R.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-02-2006
• Subjects: Anomalous diffusion; Exact solutions; Fractional diffusion equation; Green function; Fractional diffusion equation; Green function; Anomalous diffusion; Exact solutions
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2005.06.073

We investigate the solutions of a fractional diffusion equation with radial symmetry by using the Green function approach and by taking the
N
-dimensional case into account. In our analysis, a spatial time-dependent diffusion coefficient is considered, i.e.,
D
(
r
,
t
)
=
D
t
δ
-
1
r
-
θ
/
Γ
(
α
)
. The presence of external forces
F
(
r
)
=
K
r
ε
with
ε
=
-
1
-
θ
and
F
(
r
)
=
-
kr
+
K
r
ε
is also taken into account. In particular, we discuss the results obtained by employing boundary conditions defined on a finite interval, and afterwards the analysis is extended to a semi-infinite interval. Finally, we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work.