2 Fractional derivative models for anomalous transport 2.1 Time-fractional derivative model The following time-fractional Feller diffusion equation can model subdiffusion with-
out drift:
𝜕
α c(x, t)
𝜕
t α =
K 𝜕
2
c(x, t)
𝜕
x 2
,
(2)
in which K is the effective diffusion coefficient, d α /
dt α is the Caputo fractional deriva-
tive, α ∈ (0, 1] is the order of the fractional derivative, and c(x, t) denotes the transition
probability (which represents the concentration of solutes in aquifers, rivers, or soil).
When c(x, 0) = δ(x), the MSD described by model (2) in an unbounded domain can be
written as
⟨
x 2
(
t)⟩ = 2Kt α Γ(1 + α)
,
(3)
which describes subdiffusion for α ∈ (0, 1).
2.2 Space-fractional derivative model Spatial non-local transport of solutes can be caused by preferential flow paths in het-
erogeneous aquifers and natural soil (or fractal systems), or the velocity field mixed
196 | H. G. Sun et al.
with turbulence [30]. In this case the solute concentration distribution in space is of-
ten found to be non-Gaussian and may follow Lévy distribution. Superdiffusion can
be characterized by the following space-fractional derivative model:
𝜕
c(x, t)
𝜕
t =
K 𝜕
β c(x, t)
𝜕|
x|
β ,
(4)
where 1 < β ≤ 2 is the order of the space-fractional derivative.
