Appl Math Optim 49:241–264 (2004)
2004 Springer-Verlag New York, LLC
The Riesz–Bessel Fractional Diffusion Equation
V. V. Anh and R. McVinish
Program in Statistics and Operations Research,
Queensland University of Technology, GPO Box 2434,
Brisbane, Queensland 4001, Australia
Communicated by B. Øksendal
Abstract. This paper examines the properties of a fractional diffusion equation
deﬁned by the composition of the inverses of the Riesz potential and the Bessel
potential. The ﬁrst part determines the conditions under which the Green function
of this equation is the transition probability density function of a L´evy motion.
This L´evy motion is obtained by the subordination of Brownian motion, and the
L´evy representation of the subordinator is determined. The second part studies the
semigroup formed by the Green function of the fractional diffusion equation. Ap-
plications of these results to certain evolution equations is considered. Some results
on the numerical solution of the fractional diffusion equation are also provided.
Key Words. Fractional diffusion equation, Anomalous diffusion, Stochastic evo-
lution equation, L´evy motion.
AMS Classiﬁcation. Primary 60G10, Secondary 60M20.
In the Eulerian theory of turbulence, the concentration ﬁeld c(t , x) is commonly assumed
to satisfy the advection–diffusion equation
+∇·(uc) = κc(t, x), t ∈ R
, x ∈ D ⊂ R
This research was partially supported by Australian Research Council Grant A10024117.