Variational formulation for the stationary fractional advection dispersion equation

TLDR: In this paper, a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented, and appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs.

Abstract: In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Figures

Table 3: Experimental error results for Example 3.

Table 2: Experimental error results for Example 2.

Table 1: Experimental error results for Example 1.

Full text

Variational Formulation for the Stationary Fractional Advection
Dispersion Equation
Vincent J. Ervin
∗
John Paul Roop
†
Abstract
In this pap er a theoretical framework for the Galerkin finite element approximation to the steady
state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are
defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H
s
. Existence and
uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical
results are included which confirm the theoretical estimates.
Key words. Finite element method, fractional differential operator, fractional diffusion equation,
fractional advection dispersion equation.
AMS Mathematics subject classifications. 65N30, 35J99
1 Introduction
In this paper, we investigate the Galerkin approximation to the steady state Fractional Advection Dis-
p ersion Equation (FADE)
−D a (p
0
D
−β
x
+ q
x
D
−β
1
)Du + b(x)Du + c(x)u = f , (1)
where D represents a single spatial derivative, and
0
D
−β
x
,
x
D
−β
1
represent left and right fractional
integral operators, respectively, with 0 ≤ β < 1, and 0 ≤ p, q ≤ 1, satisfying p + q = 1.
Our interest in (1) arises from its application as a model for physical phenomena exhibiting anomalous
diffusion, i.e. diffusion not accurately modeled by the usual advection dispersion equation. Anomalous
diffusion has been used in modeling turbulent flow [4, 12], and chaotic dynamics of classical conservative
systems [14]. In viscoelasticity, fractional differential operators have been used to describe materials’
constitutive equations [7]. Recently articles involving fractional differential operators have appeared in
∗
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, email: vjervin@clemson.edu. Partially
supported by the National Science Foundation under Award Number DMS-0410792.
†
Corresponding author. Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123. email: jroop@vt.edu.
1

Physics Today [13], and Nature [6]. The application of central interest to us is that of contaminant trans-
p ort in groundwater flow. In [2] the authors state that solutes moving through aquifers do not generally
follow a Fickian, second-order, governing equation because of large deviations from the stochastic process
of Brownian motion. This give rise to superdiffusive motion.
Fractional differential operators have a long history, having been mentioned by Leibniz in a letter to
L’Hospital in 1695. Early mathematicians who contributed to the study of fractional differential oper-
ators include Liouville, Riemann and Holmgrem. (See [8] for a history of the development of fractional
differential operators). A number of definitions for the fractional derivative has emerged over the years:
Gr ¨uwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative, and the Caputo frac-
tional derivative [9]. In this article we restrict our attention to the use the Riemann-Liouville fractional
derivative.
To date most solution techniques for equations involving fractional differential operators have ex-
ploited the properties of the Fourier and Laplace transforms of the operators to determine a classical
solution. Finite difference have also been applied to construct numerical approximation [9]. Aside from
[5] we are not aware of any other papers in the literature which investigate the Galerkin approximation
and associated error analysis for the FADE.
There are two properties of fractional differential operators which make the analysis of the variational
solution to the FADE more complicated than than that for the usual advection dispersion equation.
These are
(i) fractional differential operators are not lo cal operators, and
(ii) the adjoint of a fractional differential operator is not the negative of itself.
Because of (i) and (ii) the correct function space setting for the variational solution is not obvious. Related
to the left fractional derivative we introduce the J
µ
L
space and corresponding to the right fractional
derivative the J
µ
R
space. We are able to relate these spaces to the fractional Sobolev space H
µ
through
an intermediate space J
µ
S
.
This paper is organized as follows. In Section 2 we develop the appropriate functional setting for
the variational solution of FADEs. In Section 3 we then prove existence and uniqueness of the varia-
tional solution. The Galerkin approximation is introduced in Section 4, and convergence results for the
Galerkin approximation are derived. Numerical results demonstrating the convergence of the Galerkin
approximation are presented in Section 5. Contained in the Appendix are the definitions of the Riemann-
Liouville fractional derivative and integral operators, together with some other useful properties of these
op erators.
2 Fractional Derivative Spaces
In this section we develop the abstract setting for the analysis of the approximation to FADEs. We
introduce associated left, right, and symmetric fractional derivative spaces. The equivalence of the
2

fractional derivative spaces with fractional order Hilbert spaces is then established.
In order to define the spaces, for G ⊂ IR an open interval (which may be unbounded) we let C
∞
0
(G)
denote the set of all functions u ∈ C
∞
(G) that vanish outside a compact subset K of G.
Definition 2.1 [Left Fractional Derivative Space] Let µ > 0. Define the semi-norm
|u|
J
µ
L
(IR)
:= kD
µ
uk
L
2
(IR)
,
and norm
kuk
J
µ
L
(IR)
:= (kuk
2
L
2
(IR)
+ |u|
2
J
µ
L
(IR)
)
1/2
, (2)
and let J
µ
L
(IR) denote the closure of C
∞
0
(IR) with respect to k · k
J
µ
L
(IR)
.
In the following analysis, we define a semi-norm for functions in H
µ
(IR) in terms of the Fourier
transform.
Definition 2.2 Let µ > 0. Define the semi-norm
|u|
H
µ
(IR)
:= k|ω|
µ
ˆuk
L
2
(IR)
, (3)
and norm
kuk
H
µ
(IR)
:= (kuk
2
L
2
(IR)
+ |u|
2
H
µ
(IR)
)
1/2
,
and let H
µ
(IR) denote the closure of C
∞
0
(IR) with respect to k · k
H
µ
(IR)
.
Theorem 2.1 The spaces J
µ
L
(IR) and H
µ
(IR) are equal with equivalent semi-norms and norms.
Proof. The proof follows immediately from the following lemma.
Lemma 2.2 Let µ > 0, be given. A function u ∈ L
2
(IR) belongs to J
µ
L
(IR) if and only if
|ω|
µ
ˆu ∈ L
2
(IR). (4)
Sp ecifically,
|u|
J
µ
L
(IR)
= k|ω|
µ
ˆuk
L
2
(IR)
. (5)
Proof. Let u ∈ J
µ
L
(IR) be given. Then D
µ
u ∈ L
2
(IR), and from (49) and (45)
F(D
µ
u) = (iω)
µ
ˆu. (6)
Using Plancherel’s theorem, we have
Z
IR
|ω|
2µ
|ˆu|
2
dω =
Z
IR
|D
µ
u|
2
dx.
Hence,
k|ω|
µ
ˆuk
L
2
(IR)
= |u|
J
µ
L
(IR)
.
Analogous to J
µ
L
(IR) we introduce J
µ
R
(IR), the right fractional derivative space, and establish their
equivalence.
3

Definition 2.3 [Right Fractional Derivative Space] Let µ > 0. Define the semi-norm
|u|
J
µ
R
(IR)
:= kD
µ∗
uk
L
2
(IR)
,
and norm
kuk
J
µ
R
(IR)
:= (kuk
2
L
2
(IR)
+ |u|
2
J
µ
R
(IR)
)
1/2
, (7)
and let J
µ
R
(IR) denote the closure of C
∞
0
(IR) with respect to k · k
J
µ
R
(IR)
.
Theorem 2.3 Let µ > 0. The spaces J
µ
L
(IR) and J
µ
R
(IR) are equal, with equivalent semi-norms and
norms.
Proof. We need only verify that the J
L
(IR) and J
R
(IR) semi-norms are equivalent. This is done using
the Fourier transform. Combining (44), the definitions of D
µ
, D
µ∗
, and Plancherel’s theorem yields
|u|
2
J
µ
L
(IR)
=
Z
IR
|(iω)
µ
ˆu(ω)|
2
dω
|u|
2
J
µ
R
(IR)
=
Z
IR
|(−iω)
µ
ˆu(ω)|
2
dω. (8)
Thus the semi-norms are equivalent, and, in fact, equal as |(iω)
µ
| = |(−iω)
µ
|.
In the finite element analysis of (1), we make use of the bilinear functional (D
µ
·, D
µ∗
·). For the case
of the entire real line, we can relate this mapping to | · |
J
µ
L
(IR)
.
Lemma 2.4 Let µ > 0, n be the smallest integer greater than µ (n − 1 ≤ µ < n), and σ = n − µ.
Then for u(x) a real valued function
(D
µ
u, D
µ∗
u) = cos(πµ)kD
µ
uk
2
L
2
(IR)
. (9)
Proof. Helpful in establishing this result is the Fourier transform property (
−−
denotes complex
conjugate)
Z
IR
uv dx =
Z
IR
ˆuˆv dω, (10)
and the observation that
(iω)
µ
=



exp(−iπµ) (−iω)
µ
if ω ≥ 0
exp(iπµ) (−iω)
µ
if ω < 0
. (11)
Thus
(D
µ
u, D
µ∗
u) =
³
D
n
−∞
D
−σ
x
u, (−D)
n
x
D
−σ
∞
u
´
=
¡
(iω)
µ
ˆu, (−iω)
µ
ˆu
¢
=
Z
0
−∞
(iω)
µ
ˆu (−iω)
µ
ˆu dω
+
Z
∞
0
(iω)
µ
ˆu (−iω)
µ
ˆu dω.
Using (11) this becomes
(D
µ
u, D
µ∗
u) =
Z
0
−∞
(iω)
µ
ˆu exp(−iπµ) (iω)
µ
ˆu dω
4

+
Z
∞
0
(iω)
µ
ˆu exp(iπµ) (iω)
µ
ˆu dω
= cos(πµ)
Z
∞
−∞
(iω)
µ
ˆu (iω)
µ
ˆu dω
+i sin(πµ)
µ
Z
∞
0
(iω)
µ
ˆu (iω)
µ
ˆu dω
−
Z
0
−∞
(iω)
µ
ˆu (iω)
µ
ˆu dω
¶
. (12)
For f (x) real we have that F(f)(−ω) = F(f)(ω). Thus
Z
0
−∞
(iω)
µ
ˆu (iω)
µ
ˆu dω =
Z
∞
0
(iω)
µ
ˆu (iω)
µ
ˆu dω. (13)
Therefore, combining (12) and (13) we obtain
(D
µ
u, D
µ∗
u) = cos(πµ) (D
µ
u, D
µ
u) .
Remark. Note that for µ = n − 1/2, n ∈ IN, (D
µ
u, D
µ∗
u) = 0. For example, (D
1/2
u, D
1/2∗
u) =
(Du, u) = 0, provided that u vanishes on the boundary.
Definition 2.4 [Symmetric Fractional Derivative Space] Let µ > 0, µ 6= n −1/2, n ∈ IN. Define
the semi-norm
|u|
J
µ
S
(IR)
:=
¯
¯
(D
µ
u, D
µ∗
u)
L
2
(IR)
¯
¯
1/2
,
and norm
kuk
J
µ
S
(IR)
:= (kuk
2
L
2
(IR)
+ |u|
2
J
µ
S
(IR)
)
1/2
(14)
and let J
µ
S
(IR) denote the closure of C
∞
0
(IR) with respect to k · k
J
µ
S
(IR)
.
Theorem 2.5 For µ > 0, µ 6= n−1/2, n ∈ IN, the spaces J
µ
L
(IR) and J
µ
S
(IR) are equal, with equivalent
semi-norms and norms.
Proof. We must show that the J
µ
L
(IR) and J
µ
S
(IR) semi-norms are equivalent. We have from the
previous lemma that
|u|
2
J
µ
S
(IR)
= |cos(πµ)| |u|
2
J
µ
L
(IR)
. (15)
Let Ω = (l, r) be a bounded open subinterval of IR. We now restrict the fractional derivative spaces
to Ω.
Definition 2.5. Define the spaces J
µ
L,0
(Ω), J
µ
R,0
(Ω), J
µ
S,0
(Ω) as the closures of C
∞
0
(Ω) under their
resp ective norms.
Following are several useful intermediate results which we use in order to relate the spaces J
µ
L,0
(Ω), J
µ
R,0
(Ω),
and J
µ
S,0
(Ω) to the fractional Sobolev space H
µ
0
(Ω).
Lemma 2.6 Let µ > 0. The following mapping properties hold.
(i) D
−µ
: L
2
(Ω) → L
2
(Ω) is a bounded linear operator.
5

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Variational formulation for the stationary fractional advection dispersion equation" ?

In this paper a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. 

Q2. what is the property of the left and right Riemann-Liouville fractional?

The left and right Riemann-Liouville fractional integral operators are adjoints in the L2 sense, i.e. for all σ > 0,( aD −σ x u, v ) L2(a,b) = ( u, xD −σ b v ) L2(a,b) , ∀ u, v ∈ L2(a, b). 

Q3. What is the simplest way to define the finite-dimensional subspace?

Assume that there exist positive constants c1, c2 such that c1h ≤ hK ≤ c2h, where hK is the width of the subinterval K, and h = maxK∈Sh hK .Let Pk(K) denote the space of polynomials of degree less than or equal to k on K ∈ Sh. Associated with Sh, define the finite-dimensional subspace Xh ⊂ Hα0 (Ω) asXh := {v ∈ Hα0 (Ω) ∩ C0(Ω̄) : v|K ∈ Pm−1(K), ∀K ∈ Sh} . 

Q4. What is the definition of the left fractional derivative space?

In order to define the spaces, for G ⊂ IR an open interval (which may be unbounded) the authors let C∞0 (G) denote the set of all functions u ∈ C∞(G) that vanish outside a compact subset K of G.Definition 2.1 [Left Fractional Derivative Space] Let µ > 

Q5. What is the equivalence of the functions of Lemma 2.4?

Note that as 1/2 < α ≤ 1, Lemma 2.4 implies (Dαu, Dα∗u) < 0. The semi-norm equivalence of JαS,0(Ω) and H α 0 (Ω), Theorem 2.12, implies thatB(u, u) ≥ C|u|2Hα(Ω). 

Q6. what is the right Riemann-Liouville fractional integral?

The left (right) Riemann-Liouville fractional derivative of order µ acts as a left inverseof the left (right) Riemann-Liouville fractional integral of order µ, i.e.aD µ x aD −µ x u(x) = u(x), (47) xD µ b xD −µ b u(x) = u(x), ∀ µ > 0. (48)Property A.5 [Fourier Transform Property] Let µ > 0, u ∈ C∞0 (Ω), Ω ⊂ IR. 

Q7. What is the proof for Lemma 2.7?

An elementary calculation shows that for φn ∈ C∞0 (Ω),Dµ−sφn(l) = 0.Hence, by Property A.6,‖D−s Ds Dµ−sφn −Dµ−sφn‖L2(Ω) = 0.The stated result then follows from the convergence of φn to u and the mapping properties in Lemma 2.6. 

Q8. what is the right Riemann-Liouville fractional derivative?

Then the left fractional derivative of order µ is defined to beDµu := −∞D µ xu = D n −∞D −σ x u(x) = 1Γ(σ)dndxn ∫ x −∞ (x− ξ)σ−1u(ξ)dξ. (45)Definition A.4 [Right Riemann-Liouville Fractional Derivative] 

Q9. What is the right Riemann-Liouville fractional integral?

Then the left Riemann-Liouville fractional integral of order σ is defined to beaD −σ x u(x) := 1Γ(σ) ∫ x a (x− s)σ−1u(s)ds. (40)Definition A.2 [Right Riemann-Liouville Fractional Integral] 

Q10. what is the r in the l2 norm?

u is the exact solution to the boundary value problem−2D(p 0D−1/2x + q xD−1/21 )Du + Du + u = fu(0) = 0u(1) = 0.As u ∈ H2(Ω), Theorem 4.4 predicts a rate of convergence of 2 in the L2 norm. 

Q11. What are the properties of the Fourier and Laplace transforms of the operators?

To date most solution techniques for equations involving fractional differential operators have exploited the properties of the Fourier and Laplace transforms of the operators to determine a classical solution.