Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 171620, 20 pages
doi:10.1155/2011/171620
Research Article
A Fully Discrete Galerkin Method for a Nonlinear
Space-Fractional Diffusion Equation
Yunying Zheng1 and Zhengang Zhao2, 3
1
Department of Mathematics, Huaibei Normal University, Huaibei 235000, China
Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China
3
Department of Mathematics, Shanghai University, Shanghai 200444, China
2
Correspondence should be addressed to Zhengang Zhao, zgzhao999@yahoo.com.cn
Received 13 May 2011; Revised 27 July 2011; Accepted 27 July 2011
Academic Editor: Delfim Soares Jr.
Copyright q 2011 Y. Zheng and Z. Zhao. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The spatial transport process in fractal media is generally anomalous. The space-fractional
advection-diffusion equation can be used to characterize such a process. In this paper, a fully
discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion
equation. In the spatial direction, we use the finite element method, and in the temporal direction,
we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the
Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical
examples are also included which are in line with the theoretical analysis.
1. Introduction
The normal diffusive motion is modeled to describe the standard Brownian motion. The
relation between the flow and the divergence of the particle displacement represents
Jx, t −a
∂c
bc,
∂x
1.1
where J is the diffusive flow. Inserting the above equation into the equation of mass
conservation
∂c
∂J
− ,
∂x
∂t
1.2
2
Mathematical Problems in Engineering
we obtain the standard convection-diffusion equation. From the viewpoint of physics, it
means that during the method of time random walkers, the overall particle displacement
up to time t can be represented as a sum of independent random steps, in the case that both
the mean-squared displacement per step and the mean time needed to perform a step are
finite. The measured variance growth in the direction of flow of tracer plumes is typically at
a Fickian rate, c − c2 ∼ t.
The transport process in fractal media cannot be described with the normal diffusion.
The process is nonlocal and it does not follow the classical Fickian law. It depicts a particle
in spreading tracer cloud which has a standard deviation, and which grows like t2α for some
0 < α < 1, excluding the Fickian case α 1/2. The description of anomalous diffusion means
that the measure variance growth in the direction of flow has a deviation from the Fickian
case, it follows the super-Fickian rate c − c2 ∼ t2α when α > 1/2, or does the subdiffusion
rate c − c2 ∼ t2α if 0 < α < 1/2. With the help of the continuous time random walk and the
Fourier transform, the governing equation with space fractional derivative can be derived as
follows
∂u
β
D aua Dx u buDu fx, t, u,
∂t
0 < β < 1,
1.3
where D denotes integer derivative respect to x, and Dβ is fractional derivative. There are
some authors studying the spacial anomalous diffusion equation in theoretical analysis and
numerical simulations 1–10. Now the fractional anomalous diffusion becomes a hot topic
because of its widely applications in the evolution of various dynamical systems under
the influence of stochastic forces. For example, it is a well-suited tool for the description
of anomalous transport processes in both absence and presence of external velocities or
force fields. Since the groundwater velocities span many orders of magnitude and give rise
to diffusion-like dispersion a term that combines molecular diffusion and hydrodynamic
dispersion, the fractional diffusion is an important process in hydrogeology. It can be used
to describe the systems with reactions and diffusions across a wide range of applications
including nerve cell signaling, animal coat patterns, population dispersal, and chemical
waves. In general, fractional anomalous diffusions have numerous applications in statistical
physics, biophysics, chemistry, hydrogeology, and biology 4, 11–20.
In this paper, we mainly study one kind of typical nonlinear space-fractional partial
differential equations by using the finite element method, which reads in the following form:
∂u
β
D aua Dx u buDu fx, t, u,
∂t
u|t0 ϕx,
u|∂Ω g,
x ∈ Ω,
x ∈ Ω, t ∈ 0, T ,
1.4
t ∈ 0, T ,
where Ω is a spacial domain with boundary ∂Ω, Dβ is the βth 0 < β < 1 order fractional
derivative with respect to the space variable x in the Caputo sense which will be introduced
later on, a, b, f are functions of x, t, u, ϕ and g are known functions which satisfy the
conditions requested by the theorem of error estimations.
The rest of this paper is constructed as follows. In Section 2 the fractional integral,
fractional derivative, and the fractional derivative spaces are introduced. The error estimates
Mathematical Problems in Engineering
3
of the finite element approximation for 1.4 are studied in Section 3, and in Section 4,
numerical examples are taken to verify the theoretical results derived in Section 3.
2. Fractional Derivative Space
In this section, we firstly introduce the fractional integral or Riemann-Liouville integral, the
Caputo fractional derivative, and their corresponding fractional derivative space.
Definition 2.1. The αth order left and right Riemann-Liouville integrals of function ux are
defined as follows
α
a Ix ux 1
Γα
1
α
x Ib ux Γα
x
x − sα−1 usds,
a
b
2.1
s − xα−1 usds,
x
where α > 0, and Γ· is the Gamma function.
Definition 2.2. The αth order Caputo derivative of function ux is defined as,
α
n−α d
a Dx uxa Ix
α
x Db ux
−1
n
n
ux
,
dxn
n−α d
x Ib
n
n − 1 < α < n ∈ Z ,
ux
,
dxn
2.2
n−1<α<n∈Z .
The αth order Riemann-Liouville derivative of function ux is defined by changing the order
of integration and differentiation.
Lemma 2.3 see 8. If u0 u 0 · · · un−1 0 0, then the Caputo fractional derivative is
equal to the Riemann-Liouville derivative.
Definition 2.4. The fractional derivative space J α Ω is defined as follows:
J α Ω u ∈ L2 Ω : a Dαx u ∈ L2 Ω, n − 1 ≤ α < n ,
2.3
endowed with the seminorm
|u|J α a Dαx uL2 Ω ,
2.4
and the norm
⎛
uJ α
⎞1/2
2
k ⎠
⎝|u|2J α .
D u
k≤α
2.5
4
Mathematical Problems in Engineering
Let J0α Ω denote the closure of C0∞ Ω with respect to the above norm and seminorm.
Definition 2.5. Define the seminorm
|u|H α |iw|α Fu
L2 Ω ,
2.6
and the norm
⎛
uH α ⎝|u|2H α
⎞1/2
2
k ⎠
,
D u
2.7
k≤α
where i is the imaginary unit, and F is the Fourier transform, and which can define another
fractional derivative space H α Ω.
Let H0α Ω denote the closure of C0∞ Ω with respect to the norm and seminorm.
Definition 2.6. The fractional space Jsα Ω is defined below
Jsα Ω u ∈ L2 Ω : a Dαx u ∈ L2 Ω, x Dbα u ∈ L2 Ω, n − 1 ≤ α < n ,
2.8
endowed with the seminorm
1/2 |u|Jsα a Dαx u, x Dαb u
L2 Ω
,
2.9
and the norm
⎞1/2
2
2
k
⎝
D u
|u|Jsα ⎠ .
⎛
uJsα
2.10
k≤α
Theorem 2.7 see 3, 6. Jsα , J α , and H α are equal with equivalent seminorm and norm.
The following are some useful results.
Lemma 2.8 see 3. For u ∈ J0α Ω, 0 < β < α, then
α
a D x ux
α−β
a Dx
β
a D x u.
2.11
Lemma 2.9 see 2. For u ∈ H0α Ω, one has
uL2 Ω ≤ c|u|H0α .
2.12
|u|H β Ω ≤ c|u|H0α .
2.13
For 0 < β < α,
0
Mathematical Problems in Engineering
5
Since Jsα , J α , and H α are equal with equivalent seminorm and norm, the norms with
each space which will be used following are without distinction, and the notations are used
seminorm | · |α and norm · α .
3. Finite Element Approximation
Let Ω a, b, and 0 ≤ β < 1. Define α 1 β/2. In this section, we will formulate a fully
discrete Galerkin finite element method for a type of nonlinear anomalous diffusion equation
as follows.
Problem 1 Nonlinear spacial anomalous diffusion equation. We consider equations of the
form
∂u
β
D aua Dx u buDu fx, t, u,
∂t
ux, t φx, t,
x, t ∈ Ω × 0, T ,
3.1
x ∈ ∂Ω × 0, T ,
x ∈ Ω.
ux, 0 gx,
We always assume that
0 < m < au < M,
0 < m < bu < M,
0 < m < fu < M.
3.2
The algorithm and analysis in this paper are applicable for a large class of linear and
nonlinear functions including polynomials and exponentials in the unknown variables.
Throughout the paper, we assume the following mild Lipschitz continuity conditions
on a, b, and f: there exist positive constants L and c such that for x ∈ Ω, t ∈ 0, T , and
s, t ∈ R,
|ax, t, s − ax, t, r| ≤ L|s − r|,
3.3
|bx, t, s − bx, t, r| ≤ L|s − r|,
3.4
fx, t, s − fx, t, r ≤ L|s − r|.
3.5
In order to derive a variational form of Problem 1, we suppose that u is a sufficiently
smooth solution of Problem 1. Multiplying an arbitrary v ∈ H0α Ω in both sides yields
∂u
v dx ∂t
Ω
Ω
β
D aua Dx u v dx buDuv dx fx, t, uv dx.
Ω
Ω
3.6
Rewriting the above expression yields
Ω
∂u
v dx ∂t
Ω
β
aua Dx uDv dx
−
Ω
buDuv dx Ω
fx, t, uv dx.
3.7
6
Mathematical Problems in Engineering
We define the associated bilinear form A : J0α Ω × J0α Ω → R as
β
Au, v aua Dx u, Dv − buDu, v,
3.8
where ·, · denotes the inner product on L2 Ω and J0α Ω.
For given f ∈ J −α Ω, we define the associated function F : J0α Ω → R as
Fv f, v .
3.9
Definition 3.1. A function u ∈ J0α Ω is a variational solution of Problem 1 provided that
∂u
,v
∂t
Au, v Fv,
∀v ∈ J0α Ω.
3.10
Now we are ready to describe a fully discrete Galerkin finite element method to solve
nonlinear Problem 1. In our new scheme, the finite element trial and test spaces for Problem
1 are chosen to be same.
For a positive integer N, let t {tn }N
n0 be a uniform partition of the time interval
0, T such that tn nτ, where τ T/N, and let tn−1/2 tn − τ/2. Throughout the paper, we
use the following notation for a function φ:
φn φtn , ∂t φn φn − φn−1
,
τ
n
φ φn φn−1
,
2
3φn−1 − φn−2
φn .
2
3.11
Let Kh {K} be a partition of spatial domain Ω. Define hk as the diameter of the
element K and h maxK∈Kh hK . And let Sh be a finite element space
Sh v ∈ H0α Ω : v|K ∈ Pr−1 K, K ∈ Kh ,
3.12
where Pr−1 K is the set of polynomials of degree r−1 on a given domain K. And the functions
in Sh are continuous on Ω. Our fully discrete quadrature scheme to solve Problem 1 is to find
uh : for v ∈ Sh such that
β
n ∂t unh , v a u
nh a Dx unh , Dv − b u
nh Dunh , v f u
h , v .
3.13
Mathematical Problems in Engineering
7
The linear systems in the above equation requires selecting the value of u0h and u1h .
Given u0h depending on the initial data gx, we select u1h by solving the following predictorcorrector linear systems:
u1,0
− u0h
h
τ
u1h − u0h
τ
,v
f
1,0
u0h
βu
a u0h a Dx h
, Dv
2
,v
a
u0h
u1,0
h
2
u0h
u1,0
h
2
1
β uh
a Dx
u0h
2
u1,0 u0
h
b u0h D h
,v
2
−
−
, Dv
b
u0h
u1,0
h
2
D
f u0h , v ,
u1h u0h
2
,v
,v .
3.14
α
Lemma 3.2. For u, v, w ∈ Js,0
Ω, 0 < m ≤ au ≤ M, α 1 β/2, there exist constants γ1 , γ2
such that
β
aua Dx u, Dv ≤ γ1 uα · vα ,
β
awa Dx v, Dv ≥ γ2 v2α .
3.15
Proof. With the assumption of au in 3.3 and the property of dual space
β
β awa Dx u, Dv ≤ awa Dx u
1−α
· Dv−1−α
≤ Mcu1−αβ · v−1−α1 ≤ γ1 uα · vα ,
β
β
awa Dx v, Dv − Dawa Dx v, v
−
1−β/2
β
1β/2
awa Dx v, x Db
v
a Dx
3.16
≥ m|v|2Jsα ≥ γ2 v2α .
Lemma 3.3 see 2. For Ω ⊂ Rn , α > n/4, v, w ∈ H0α Ω, ε > 0, one has
vbw, ∇v ≤ c0
−p/q
qε
∇bwp · v2 εv2α ,
p
3.17
where p 4α/4α − n, q 4α/n.
Theorem 3.4. Let unh be bounded, then for a sufficiently small step τ, there exists a unique solution
unh ∈ Sh satisfying scheme 3.13.
8
Mathematical Problems in Engineering
Proof. As scheme represents a finite system of problem, the continuity and coercivity
of unh , ωnh /τ Aunh , ωnh is the sufficient and essential condition for the existence and
uniqueness of unh . Let v unh , w ωnh , then
v, v
v, v β
Av, v awa Dx v, Dv − bwDv, v
τ
τ
v2
γ2 vα − c0 Dbw2 v2 − εv2α
τ
γ2 − ε v2α τ −1 − c0 Dbw2 v2
≥
3.18
≥ cv2α .
For the chosen sufficiently small τ, the above inequality holds.
v, w v, w
β
Av, w aua Dx v, Dw Dbuv, Dw
τ
τ
≤
u · w
γ1 vα wα v · Dbuw
τ
3.19
u · w
v · w
γ1 vα wα M
≤
τ
h
≤ cvα wα .
Hence, the scheme 3.13 is uniquely solvable for unh .
Let ρn Ph un − un , and θn unh − Ph un , then
unh − un unh − Ph un Ph un − un θn ρn ,
3.20
where Ph un is a Rits-Galerkin projection operator defined as follows:
β
awa Dx un − Ph un , Dv 0,
β
au0 a Dx un − Ph un , Dv 0.
3.21
Lemma 3.5. Let au, bu be smooth functions on Ω, 0 < m ≤ au, bu ≤ M, and Ph un is
defined as above, then
a Dαx un − Ph un ≤ chk1−α uk1 ,
Ph un − un ≤ chk1 uk1 .
3.22
Mathematical Problems in Engineering
9
Proof. Using the definition of Ph un , one gets
a Dαx Ph un − un 2 | a Dαx Ph un − un , a Dαx Ph un − un |
≤ c a Dαx Ph un − un · a Dαx χ − un ,
3.23
where χ ∈ Sh . Utilizing the interpolation of Ih un leads to
a Dαx Ph un − un ≤ inf c
χ − u
α ≤ cIh un − un α ≤ chk1−α uk1 .
χ∈Sh
3.24
Next we estimate Ph un − un . For all φ ∈ L2 Ω, w is the solution of the following equation:
−a Dx2α w φ,
w 0,
w ∈ Ω,
3.25
w ∈ ∂Ω.
So we have
w2α ≤ γ3 φ
.
3.26
For all χ ∈ Sh , with the help of approximation properties of Sh and the weak form, we can
obtain
α
n
n
α
Ph un − un , φ − Ph un − un , a D2α
x w − x D b Ph u − u , a D x w
− x Dαb Ph un − un , a Dαx w − χ ≤ Ph un − un α w − χ
α
≤ Ph un − un α inf w − χ
α
χ∈Sh
≤ chr−α ur hα w2α chr ur φ
,
P h u n − un , φ
n
n
≤ chr ur .
Ph u − u sup
φ
0 φ∈L2 Ω
3.27
/
Lemma 3.6 see 21. Let Th , 0 < h ≤ 1, denote a quasiuniform family of subdivisions of a
polyhedral domain Ω ⊂ Rd . Let K , P, N be a reference finite element such that P ⊂ W l,p K ∩
W m,q K is a finite-dimensional space of functions on K , N is a basis for P , where 1 ≤ p ≤ ∞, 1 ≤
p ≤ ∞, and 0 ≤ m ≤ l. For K ∈ Th , let K, PK , NK be the affine equivalent element, and Vh v : v
is measurable and v|K ∈ PK , for all K ∈ Th . Then there exists a constant C Cl, p, q such that
k∈Th
1/p
v2W l,p K
≤ Ch
m−lmin0,d/p−d/q
·
k∈Th
1/q
q
vW m,q K
The following Gronwall’s lemma is useful for the error analysis later on.
.
3.28
10
Mathematical Problems in Engineering
Lemma 3.7 see 2. Let Δt, H and an , bn , cn , γn (for integer n ≥ 0 be nonnegative numbers
such that
aN Δt
N
N
N
bn ≤ Δt γn an Δt cn H,
n0
n0
3.29
n0
for N ≥ 0. Suppose that Δtγn < 1, for all n, and set σn 1 − Δtγn −1 . Then
aN Δt
N
N
bn ≤ exp Δt σn γn
n0
Δt
N
n0
!
cn H ,
3.30
n0
for N ≥ 0.
The following norms are also used in the analysis:
|v|∞,k max vn k ,
0≤n≤N
|v|0,k N
1/2
τvn 2k
3.31
.
n0
Theorem 3.8. Assume that Problem 1 has a solution u satisfying utt , uttt ∈ L2 0, T, L2 Ω with
u, ut ∈ L2 0, T, H k1 . If Δt ≤ ch, then the finite element approximation is convergent to the solution
of Problem 1 on the interval (0,T], as Δt, h → 0. The approximation uh also satisfies the following
error estimates
u − uh 0,α ≤ C hk1 ut 0,k1 hk1−α u0,k1 τ 2 utt 0,0
τhk1−α utt 0,k1 τ 2 uttt 0,0 ,
u − uh ∞,0 ≤ C hk1 ut 0,k1 hk1−α u0,k1 τ 2 uttt 0,0
τhk1−α utt 0,k1 τ 2 utt 0,0 hk1 u2∞,k1 .
3.32
3.33
Proof. For t tn − τ/2 tn−1/2 , n 0, 1, . . . , N, find un−1/2 such that
∂t un−1/2 , v a un−1/2 a Dx un−1/2 , Dv − b un−1/2 Dun−1/2 , v f un−1/2 , v .
3.34
Mathematical Problems in Engineering
11
Subtracting the above equation from the fully discrete scheme 3.13, and substituting unh −
un unh − Ph un Ph un − un θn ρn into it, we obtain the following error formulation
relating to θn and ρn :
β n
∂t θn , v a u
nh a Dx θ , Dv − b u
nh Dθn , v
β
β
n
n
nh a Dx Ih u , v ∂t un−1/2 , v − ∂t Ih un , v
a u
nh a Dx Ih u , Dv b u
β
nh , v − f un−1/2 , v
a un−1/2 a Dx un−1/2 , Dv − b un−1/2 Dun−1/2 , v f u
β
β
n β n
− a u
nh a Dx ρn , Dv a un−1/2 a Dx un−1/2 − a u
h a Dx u , Dv
n n n n n−1/2 n−1/2
Du
h Du − b u
, Dv
b u
h Dρ , v b u
f u
nh − f un−1/2 , v ∂t un−1/2 − ∂t Ih un , v
R1 v R2 v R3 v R4 v R5 v R6 v.
3.35
Setting v θn , we obtain
n
n n
β n
un a Dx θ , Dθ − b
un Dθ , θ
∂t θn , θn a
R1 θ
n
R2 θ
n
R3 θ
n
R4 θ
n
R5 θ
n
R6 θ
3.36
n
.
Note that
∂t θn , θn θn − θn−1 θn θn−1
,
τ
2
2 1 n 2 θ − θn−1 .
2τ
3.37
According to 3.2 and Lemma 3.2, we have
n
a
u
n
β n
a Dx θ , Dθ
n 2
n−1 2
n 2
≥ mθ ≥ c |θ |α θ .
α
3.38
α
From Lemma 3.3, the following inequality can be derived:
n 2
n
n
2 n 2
b
un θ , Dθ ≤ c0 ε2−c1 Db
un c θ ε3 θ α
2
n
θn θn−1 2
n−1 −c1
n c2 θ θ
c0 ε2 Db
u ε3 2
2
≤
2
c3 ε2−c1 Db
un c
3.39
α
n−1 2
n−1 2
n 2
θ θ c4 ε3 θ α θ .
n 2
α
12
Mathematical Problems in Engineering
Substituting 3.37–3.39 into 3.36 then multiplying 3.36 by 2τ, summing from
n 1 to N, we have
N
2
2
n−1 2
n 2
θ − θ τ 2mc − 2c4 ε3 θ α θ n 2
α
n1
≤ 2τ
N
2 n c2
h θn 2 θn−1 c3 ε2−c1 Db u
3.40
n1
2τ
N "
n
n
n
n
n
n #
R1 θ R2 θ R3 θ R4 θ R5 θ R6 θ
.
n3
We now estimate R1 to R6 in the right hand of 3.40,
n β n
a u
nh a Dx ρn , a Dαx θ
R1 θ a D1−α
x
n
n
≤ M a Dαx ρn , a Dαx θ ≤ M
a Dαx ρn a Dαx θ n 2
c2 2
≤ ε4 θ 5 ρn α
α
4ε4
3.41
2
2
c2 ε4 n
ρ ρn−1 5 θn θn−1 α
α
2
16ε4
2 c 2 7 n 2
ρ α ρn−1 .
≤ ε4 c6 θn 2α θn−1 α
α
ε4
Secondly, we deduce the estimation of R2 ,
n β
β
n
n
nh a Dx un , Dθ a un−1/2 a Dx un−1/2 , Dθ
R2 θ −a u
β
n β n
n
n
β
h a Dx u , Dθ a un−1/2 a Dx un−1/2 −a Dx un , Dθ
a un−1/2 − a u
R21 R22 ,
3.42
where
R21 #
" n
β n
un a D x u
a un−1/2 − a
, Dθ
n 2
" # β 2
c8 un a D x u
n ε5 Dθ a un−1/2 − a
α−1
1−α
4ε5
β n 2
n 2
un a Dx u
≤ c9 a un−1/2 − a
ε5 θ ≤
n 2
≤ c9 L
un−1/2 − u
n ε5 θ ,
α
1−α
α
Mathematical Problems in Engineering
β
n
β
R22 a un−1/2 a Dx un−1/2 −a Dx un , Dθ
13
n 2
" β
#
c10 β n 2
a un−1/2 a Dx un−1/2 −a Dx u ε6 Dθ −1α
1−α
4ε6
2
n 2
2 β
β
≤ c10 a un−1/2 a Dx un−1/2 −a Dx un ε6 θ ≤
1−α
2
2 ≤ c10 M2 un−1/2 − un ε6 c θn 2α θn−1 .
α
α
α
3.43
The estimations of un − un−1/2 and un − un−1/2 α can be derived as follows:
n−1/2 2
3
τ
τ n−1/2 utt
n
n−1/2 n−1/2
3
u −u
− ut
O τ
u
2
2
2!
2
n−1/2 1 n−1/2 3τ n−1/2 utt
3τ 2
3
n−1/2 − u
ut
−u
−
O τ
2
2
2!
2
≤ c11 τ 2 utt tn−1/2 ≤ c11 τ 2
tn
utt ·, sds,
tn−1
!
tn
tn−1/2
−1
n
2
2
n−1/2 s − tn utt sds s − tn−1 utt sds τ
u − u
α
tn−1/2
tn−1
tn
≤ c12 τ utt sds
tn−1
≤ c12 τ
tn
tn−1
3.44
α
α
utt sα ds
≤ c12 τhk1−α
tn
tn−1
utt sk1 ds.
Thirdly, it is turn to consider R3 ,
n n n
n
n
h Dρ −α θ nh Dρn , θ ≤ b u
R3 θ b u
α
n 2
n 2 n 2
c13 b u
h ρ 1−α ε7 θ ≤
α
4ε7
c14 n−1 2
n−1 2
n 2
n 2
ρ 1−α ρ ≤
ε7 c15 θ α θ .
1−α
α
4ε7
3.45
14
Mathematical Problems in Engineering
Next,
n n
n
nh Dun , θ − b un−1/2 Dun−1/2 , θ
R4 θ b u
n n
b u
nh Dun − b un−1/2 Dun , θ b un−1/2 Dun − Dun−1/2 , θ
R41 R42 ,
3.46
where
2
n 2
c16 n
h − b un−1/2 Dun ε8 θ b u
α
1−α
4ε8
2
c16 L n
2 n 2
uh − un−1/2 Dun 1−α ε8 θ ≤
α
4ε8
n 2
2
c16 L n
2 uh − u
n u
n − un−1/2 Dun 1−α ε8 θ α
4ε8
2
n 2
n
2
n
h − u
u − un−1/2 ε8 θ ≤ c17 u
n c17 R41 ≤
3.47
α
2
2
n 2
n
u − un−1/2 ε8 θ ≤ c17 θn ρn c17 α
2
n 2
n
2 2
u − un−1/2 ε8 θ .
≤ c18 θn ρn c17 α
Rewriting R42 by the aid of 3.20, we have
R42 ≤
2
n 2
c19 n
u − un−1/2 ε9 θ .
α
4ε9
3.48
The estimation of R5 is deduced as follows:
n nh − f un−1/2 θ R5 θn ≤ f u
n n
≤ L
uh − un−1/2 θ 2
n 2
Lc20 n
uh − un−1/2 ε10 θ 4ε10
2 2 n 2
n
n
n
n−1/2 ≤ Lc21 θ ρ u −u
ε10 θ ≤
2
n 2
n
n 2 n 2
u − un−1/2 ε10 θ .
≤ c22 θ ρ
Lc21 3.49
Mathematical Problems in Engineering
15
Last, we estimate R6 ,
n n
n
R6 θ ∂t un−1/2 , θ − ∂t Ph un , θ
n
n
∂t un−1/2 − ∂un , θ ∂t un − ∂t Ph un , θ
∂t u
n−1/2
− ∂t u , θ
n
n
∂t ρ , θ
n
n
3.50
≤ ∂t un−1/2 − ∂t un θn ∂t ρn θn ,
where
tn−1/2
tn
n−1/2
−1
2
2
n
− ∂u 2τ c23 s − tn uttt sds s − tn−1 uttt sds
∂t u
tn−1/2
tn−1
tn
≤ c23 τ uttt sds
tn−1
≤ c23 τ
tn
uttt sds,
tn−1
3.51
tn
n
n−1 n
ρ − ρ −1 n
ρt sds
≤τ ∂t ρ tn−1
τ
≤τ
−1
tn
ut sds ≤ τ
−1
tn−1
tn
tn−1
ut s ≤ hk1
tn
tn
1ds
tn−1
tn
tn−1
ut sds
tn−1
ut sk1 ds.
The θ2 should be estimated with 3.14. Let n 1 then subtracting 3.34 from the
two equations of 3.14, respectively, one gets
β 1,0
1,0
1,0
0
0
∂t θ , v a uh a Dx θ , Dv − b uh Dθ , v
β
β
β
− a u0h a Dx ρ1,0 , Dv − a u1/2 a Dx u1/2 − a u0h a Dx u1,0 , Dv
b u0h Dρ1,0 , v b u0h Du1,0 − b u1/2 Du1/2 , Dv
f u0h − f u1/2 , v ∂t u1/2 − ∂t Ih u1,0 , v
R1 v R2 v R3 v R4 v R5 v R6 v,
16
Mathematical Problems in Engineering
β 1
1
∂t θ1 , v a u0h Dx θ , Dv − b u0h Dθ , v
a
− a
u0h u1,0
h
b
u0h u1,0
h
f
−f u
2
Dρ , v
u0h u1,0
h
−
1
2
0
!
β
uh u1,0
β 1
h
1/2
1/2
a u
−a
a Dx u
a D x u , Dv
2
β 1
a D x ρ , Dv
2
b
1/2
u0h u1,0
h
2
!
,v
Du − b u1/2 Du1/2 , Dv
!
1
∂t u1/2 − ∂t Ih u1 , v
R1 v R2 v R3 v R4 v R5 v R6 v.
3.52
Setting v θ1,0 , and using the similar estimation see 3.40, one has
t1
t1
2
1,0 utt s2α ds h2k1−α u2k1 τ 2
uttt s2 ds
θ ≤ c τ 2
t0
t0
ch2k1
t1
t0
!
ut s2k1 ds
3.53
.
Letting v θ1 , applying the above result of θ1,0 , and using the similar estimation see
3.53, we get
t1
t1
2
1
utt s2α ds h2k1−α u2k1 τ 2
uttt s2 ds
θ ≤ c τ 2
t0
ch
t0
2k1
t1
t0
!
ut s2k1 ds
3.54
.
Using T Nτ and Gronwall’s lemma, we get
|θ|20,α N
τθ2α .
3.55
n0
Hence, using the interpolation property and
|u − uh |0,α ≤ |θ|0,α ρ
0,α ,
the estimate 3.32 holds.
3.56
Mathematical Problems in Engineering
17
Also using the interpolation property, Gronwall’s lemma, and the approximation
properties, we get
|u − uh |∞,0 ≤ |θ|∞,0 ρ
∞,0
≤ max |θn |2 h2k1 |u|2∞,k1 ,
3.57
0≤n≤N
which is just the estimate 3.33.
4. Numerical Examples
In this section, we present the numerical results which confirm the theoretical analysis in
Section 3.
Let K denote a uniform partition on 0, a, and Sh the space of continuous piecewise
linear functions on K, that is, k 1.. In order to implement the Galerkin finite element
approximation, we adapt finite element discrete along the space axis, and finite difference
scheme along the time axis. We associate shape function of space Xh with the standard basis of
hat functions on the uniform grid of size h 1/n. We have the predicted rates of convergence
if the condition Δt ch of
u − uh 0,α ∼ O h2−α ,
u − uh ∞,0 ∼ O h2−α ,
4.1
provided that the initial value ϕx is smooth enough.
Example 4.1. The following equation
2x1.5
∂u
x0.5
2
0.5
D u 0 Dx ux, t − 2xx − 1
−
e−2t Du − ux, t
∂t
Γ2.5 Γ1.5
2x0.5
x−0.5
2 −t
−
, 0 ≤ x ≤ 1, 0 ≤ t ≤ 1,
−u e
Γ1.5 Γ0.5
ux, 0 xx − 1,
0 ≤ x ≤ 1,
u0, t u1, t 0,
0 ≤ t ≤ 1,
4.2
has a unique solution ux, t e−t xx − 1.
If we select Δt ch and note that the initial value u0 is smooth enough, then we have
u − uh 0,0.75 ∼ O h1.25 ,
u − uh ∞,0 ∼ O h1.25 .
4.3
18
Mathematical Problems in Engineering
Table 1: Numerical error result for Example 4.1.
h
1/5
1/10
1/20
1/40
1/80
1/160
u − uh ∞,0
2.2216E−003
1.3551E−003
5.5865E−004
3.0515E−004
1.2423E−004
5.1033E−005
u − uh 0,0.75
1.0213E−003
6.0779E−004
2.3188E−004
1.0545E−004
3.9883E−005
2.1310E−005
cvge. rate
—
0.7132
1.2784
0.8724
1.2964
1.2835
cvge. rate
—
0.74875
1.3901
1.1367
1.4027
0.9042
Table 1 includes numerical calculations over a regular partition of 0, 1. We can
observe the experimental rates of convergence agree with the theoretical rates for the
numerical solution.
Example 4.2. The function ux, t costx2 2 − x2 solves the equation in the following form:
∂u
0 Dx1.7 ux, t buDu − u41 − x tan t fx, t,
∂t
ux, 0 x2 2 − x2 ,
u0, t 0,
0 ≤ x ≤ 2,
u2, t 0,
x ∈ 0, 2, t ∈ 0, 1,
4.4
0 ≤ t ≤ 1,
where
√
u
bu √
,
cos t
⎡ ⎤
24 x1.3 2 − x1.3
8 x0.3 2 − x0.3
24 x2.3 2 − x2.3
cos t
⎢
⎥
−
−
fx, t ⎣
⎦.
cos0.85π
Γ3.3
Γ2.3
Γ1.3
4.5
If we select Δt ch, then
u − uh 0,0.85 ∼ O h1.15 ,
u − uh ∞,0 ∼ O h
1.15
4.6
.
Table 2 shows the error results at different size of space grid. We can observe that the
experimental rates of convergence still support the theoretical rates.
Mathematical Problems in Engineering
19
Table 2: Numerical error result for Example 4.2.
h
1/5
1/10
1/20
1/40
1/80
1/160
u − uh ∞,0
1.3010E−001
4.6402E−002
1.6843E−002
6.6019E−003
2.7979E−003
1.2665E−003
u − uh 0,0.85
3.2223E−002
1.4133E−002
6.2946E−003
2.8571E−003
1.3137E−003
6.0848E−004
cvge. rate
—
1.4878
1.4620
1.6843
1.2386
1.1434
cvge. rate
—
1.1890
1.1669
1.1395
1.1209
1.1103
Table 3: Numerical error result for Example 4.3.
h
1/5
1/10
1/20
1/40
1/80
1/160
u − uh ∞,0
8.3052E−002
3.6038E−002
1.3839E−002
5.0631E−003
1.8920E−003
9.9899E−004
u − uh 0,0.85
2.8009E−002
1.0086E−002
3.2327E−003
1.1789E−003
5.9555E−004
3.0034E−004
cvge. rate
—
1.2045
1.3807
1.4507
1.4201
0.9214
cvge. rate
—
1.4735
1.6414
1.4554
0.9851
0.9876
Example 4.3. Consider the following space-fractional differential equation with the nonhomogeneous boundary conditions,
∂u
3
0 D1.7
x ux, t −
∂t
x
x
u dx −
0
ux, 0 x2 ,
u0, t 0,
2x0.3 e−t
,
Γ1.3
0 ≤ x ≤ 1, 0 ≤ t ≤ 1,
0 ≤ x ≤ 1,
u1, t e−t ,
4.7
0 ≤ t ≤ 1,
whose exact solution is ux, t e−t x2 .
We still choose Δt ch, then get the convergence rates
u − uh 0,0.85 ∼ O h1.15 ,
u − uh ∞,0 ∼ O h1.15 .
4.8
The numerical results are presented in Table 3 which are in line with the theoretical analysis.
5. Conclusion
In this paper, we propose a fully discrete Galerkin finite element method to solve a type
of fractional advection-diffusion equation numerically. In the temporal direction we use the
modified Crank-Nicolson method, and in the spatial direction we use the finite element
method. The error analysis is derived on the basis of fractional derivative space. The
numerical results agree with the theoretical error estimates, demonstrating that our algorithm
is feasible.
20
Mathematical Problems in Engineering
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China
under grant no. 10872119, the Key Disciplines of Shanghai Municipality under grant no.
S30104, the Key Program of Shanghai Municipal Education Commission under grant no.
12ZZ084, and the Natural Science Foundation of Anhui province KJ2010B442.
References
1 S. Chen and F. Liu, “ADI-Euler and extrapolation methods for the two-dimensional fractional
advection-dispersion equation,” Journal of Applied Mathematics and Computing, vol. 26, no. 1, pp. 295–
311, 2008.
2 V. J. Ervin, N. Heuer, and J. P. Roop, “Numerical approximation of a time dependent, nonlinear, spacefractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 45, no. 2, pp. 572–591, 2008.
3 V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion
equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2006.
4 B. I. Henry, T. A. M. Langlands, and S. L. Wearne, “Anomalous diffusion with linear reaction
dynamics: from continuous time random walks to fractional reaction-diffusion equations,” Physical
Review E, vol. 74, no. 3, article 031116, 2006.
5 C. P. Li, A. Chen, and J. J. Ye, “Numerical approaches to fractional calculus and fractional ordinary
differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.
6 C. P. Li, Z. G. Zhao, and Y. Q. Chen, “Numerical approximation of nonlinear fractional differential
equations with subdiffusion and superdiffusion,” Computers & Mathematics with Applications, vol. 62,
pp. 855–875, 2011.
7 F. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,”
Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004.
8 I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
9 R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Fractional reactiondiffusion equations,” Astrophysics
and Space Science, vol. 305, no. 3, pp. 289–296, 2006.
10 Y. Y. Zheng, C. P. Li, and Z. G. Zhao, “A note on the finite element method for the space-fractional
advection diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1718–
1726, 2010.
11 B. Baeumer, M. Kovács, and M. M. Meerschaert, “Fractional reproduction-dispersal equations and
heavy tail dispersal kernels,” Bulletin of Mathematical Biology, vol. 69, no. 7, pp. 2281–2297, 2007.
12 S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,”
Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
13 A. V. Chechkin, V. Y. Gonchar, R. Gorenflo, N. Korabel, and I. M. Sokolov, “Generalized fractional
diffusion equations for accelerating subdiffusion and truncated Levy flights,” Physical Review E, vol.
78, no. 2, article 021111, 2008.
14 P. D. Demontis and G. B. Suffritti, “Fractional diffusion interpretation of simulated single-file systems
in microporous materials,” Physical Review E, vol. 74, no. 5, article 051112, 2006.
15 S. A. Elwakil, M. A. Zahran, and E. M. Abulwafa, “Fractional space-time diffusion equation on
comb-like model,” Chaos, Solitons & Fractals, vol. 20, no. 5, pp. 1113–1120, 2004.
16 A. Kadem, Y. Luchko, and D. Baleanu, “Spectral method for solution of the fractional transport
equation,” Reports on Mathematical Physics, vol. 66, no. 1, pp. 103–115, 2010.
17 R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through
fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance,
vol. 190, no. 2, pp. 255–270, 2008.
18 M. M. Meerschaert, D. A. Benson, and B. Baeumer, “Operator Levy motion and multiscaling
anomalous diffusion,” Physical Review E, vol. 63, no. 2 I, article 021112, 2001.
19 W. L. Vargas, J. C. Murcia, L. E. Palacio, and D. M. Dominguez, “Fractional diffusion model for force
distribution in static granular media,” Physical Review E, vol. 68, no. 2, article 021302, 2003.
20 V. V. Yanovsky, A. V. Chechkin, D. Schertzer, and A. V. Tur, “Levy anomalous diffusion and fractional
Fokker-Planck equation,” Physica A: Statistical Mechanics and its Applications, vol. 282, no. 1-2, pp. 13–
34, 2000.
21 S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer, Berlin,
Germany, 1994.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014