This may be the author’s version of a work that was submitted/accepted for publication in the following source: Zheng, Minling, Liu, Fawang, Liu, Qingxia, Burrage, Kevin,& Simpson, Matthew (2017) Numerical solution of the time fractional reaction-diffusion equation with a moving boundary. Journal of Computational Physics, 338, pp. 493-510. This file was downloaded from: https://eprints.qut.edu.au/104419/ c Consult author(s) regarding copyright matters This work is covered by copyright. Unless the document is being made available under a Creative Commons Licence, you must assume that re-use is limited to personal use and that permission from the copyright owner must be obtained for all other uses. If the docu- ment is available under a Creative Commons License (or other specified license) then refer to the Licence for details of permitted re-use. It is a condition of access that users recog- nise and abide by the legal requirements associated with these rights. If you believe that this work infringes copyright please provide details by email to [email protected]License: Creative Commons: Attribution-Noncommercial-No Derivative Works 2.5 Notice: Please note that this document may not be the Version of Record (i.e. published version) of the work. Author manuscript versions (as Sub- mitted for peer review or as Accepted for publication after peer review) can be identified by an absence of publisher branding and/or typeset appear- ance. If there is any doubt, please refer to the published source. https://doi.org/10.1016/j.jcp.2017.03.006
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This may be the author’s version of a work that was submitted/acceptedfor publication in the following source:
Zheng, Minling, Liu, Fawang, Liu, Qingxia, Burrage, Kevin, & Simpson,Matthew(2017)Numerical solution of the time fractional reaction-diffusion equation with amoving boundary.Journal of Computational Physics, 338, pp. 493-510.
This file was downloaded from: https://eprints.qut.edu.au/104419/
This work is covered by copyright. Unless the document is being made available under aCreative Commons Licence, you must assume that re-use is limited to personal use andthat permission from the copyright owner must be obtained for all other uses. If the docu-ment is available under a Creative Commons License (or other specified license) then referto the Licence for details of permitted re-use. It is a condition of access that users recog-nise and abide by the legal requirements associated with these rights. If you believe thatthis work infringes copyright please provide details by email to [email protected]
Notice: Please note that this document may not be the Version of Record(i.e. published version) of the work. Author manuscript versions (as Sub-mitted for peer review or as Accepted for publication after peer review) canbe identified by an absence of publisher branding and/or typeset appear-ance. If there is any doubt, please refer to the published source.
Received date: 11 July 2016Revised date: 3 March 2017Accepted date: 5 March 2017
Please cite this article in press as: M. Zheng et al., Numerical solution of the time fractional reaction-diffusion equation with a movingboundary, J. Comput. Phys. (2017), http://dx.doi.org/10.1016/j.jcp.2017.03.006
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Numerical solution of the Time FractionalReaction-Diffusion Equation with a Moving Boundary�
Minling Zhenga, Fawang Liub,∗, Qingxia Liuc, Kevin Burrageb,d, Matthew JSimpsonb
aSchool of Science, Huzhou University, Huzhou 313000, ChinabSchool of Mathematical Sciences, Queensland University of Technology, GPO Box 2434,
Brisbane, Qld. 4001, AustraliacSchool of Mathematical Sciences, Xiamen University, Xiamen 361005, China
dACEMS, ARC Centre of Excellence for Mathematical and Statistical Frontiers,Queensland University of Technology, Queensland 4001, Australia
Abstract
A fractional reaction-diffusion model with a moving boundary is presented inthis paper. An efficient numerical method is constructed to solve this movingboundary problem. Our method makes use of a finite difference approximationfor the temporal discretization, and spectral approximation for the spatial dis-cretization. The stability and convergence of the method is studied, and theerrors of both the semi-discrete and fully-discrete schemes are derived. Nu-merical examples, motivated by problems from developmental biology, show agood agreement with the theoretical analysis and illustrate the efficiency of ourmethod.
In this paper, we shall investigate time fractional reaction-diffusion equationon a uniformly growing domain. The immobilized form is a class of fractionalintegro-differential equations given by
C0D
γt u(t, x) = d(t)∂xxu(t, x) +Ku(t, x) + I1−γ
0+ [v(t, x)∂xu(t, x)] (1.1)
�This work is partly supported by the Natural Science Foundation of Zhejiang Province(LY16A010011), and partly supported by the Australian Research Council (FT130100148)and the ARC Centre of Excellence ACEMS.
I1−γ0+ the Riemann-Liouville fractional integral and K is a reaction parameter.It is worthwhile to point out that the diffusive coefficient d(t) depends on themoving boundary, and v(t, x) denotes the velocity of domain growth.
Here, we first review some recent results on moving boundary problems.Moving boundary problems are mainly concerned with fluid flow in porous me-dia and with diffusion and heat flow incorporating phase transformations orchemical reactions. Such problems are encountered in many industrial processes,for example, seepage through porous media, freezing or melting problems, andgas-solid reactions [1]. In reality, moving-boundary problems include both un-known boundary and prescribed-boundary problems. The former is often calleda Stefan problem. For the unknown boundary moving boundary problem, onehas to determine the motion of the interface together with the solution.
Exact solutions for moving boundary problems are only available under cer-tain, limited circumstances. A similarity solution has been constructed for aprescribed-boundary problem in [2]. Muntean et al. [3] studied a two phasecarbonation reaction model that has a moving unknown internal boundary, andpresented the global existence and uniqueness of the solution.
Some efficient numerical methods have been presented for the classical Ste-fan problems, such as the spectral Petrov-Galerkin method [4, 5], the finiteelement method [6], and the spectral element method [7]. For a prescribed-boundary moving boundary problem, Baines et al. proposed the moving meshmethod [8], and Lee et al. studied the velocity-based moving mesh method [9].In [10] the boundary element method was studied. The finite element methodwas employed to solve a model of vibrating elastic membrane in [11]. Gawliket al. [12] reviewed some existing numerical method for prescribed-boundarymoving boundary problems and proposed a high-order finite element method.In [13], a spectral method has also been studied. Additionally, the finite differ-ence method was also applied to solve the prescribed-boundary moving bound-ary problems [14]. Recently, Yuan et al. [15] studied a three-dimensional movingboundary problem on the compressible miscible (oil and water) displacement bya second-order upwind difference fractional steps scheme applicable to parallelcomputing.
Over the past few decades, fractional differential equations have started toattract more and more attention. Fractional derivatives are extensively used asthe tools for dealing with complex systems, such as anomalous diffusion, turbu-lence and amorphous material [16, 17]. The non-local nature of fractional deriva-tives mean that these models are more suitable for studying history-related andtime-related problems.
Based on fractional derivatives, the standard moving boundary problemshave been extended in several areas of engineering and industry during thelast few years. In [18], the authors presented a fractional anomalous diffusionmodel of drug release that is obtained by replacing the time derivative of theclassical Stefan problem by the Caputo fractional derivative. In [19], a math-ematical model containing the space-time fractional derivative was applied tomodel the melting and solidification process. Rajeev et al. presented a time
2
fractional model of a generalized Stefan problem—a shoreline problem in [20].Atkinson [21] also considered time fractional diffusion with a moving boundaryand explicit results were obtained for the motion of planar, cylindrical, andspherical boundaries. In general, the numerical solution of fractional Stefanproblem has been obtained by the homotopy perturbation method (for examplesee [19, 20, 22]). Recently, the finite difference method is also employed to solvethe fractional Stefan problem [23].
In a biological system, domain expansion has been considered in classicalFickian reaction-diffusion models for biological pattern formation [24, 25]. In[26] the authors studied the phenomenon of cellular migration on an underlyingtissue, and examined the question of how long does it take for a wave of cellsto colonize the whole tissue, or whether it is possible, while the tissue itself isexpanding. More recently, Simpson et al. [27, 28] extended the work in [26] byfinding an exact solution of a linearized model of cell colonization in one, twoand three dimensions.
In this paper, we shall investigate a linear fractional reaction-diffusion pro-cess describing anomalous diffusion on a growing domain with a moving bound-ary. The model, which extends the standard reaction-diffusion models [29], willbe proposed in next section. In this model, the domain growth is determined bya local velocity v(x, t). We focus mainly on the numerical method for such typeof problem. Equation (1.1) can be derived by the use of a transformation con-verting the moving boundary into a fixed boundary problem. By the definitionand properties of fractional derivative we can transform (1.1) into a fractionaladvection-reaction-diffusion equation
∂tw(t, x) + a(t, x)∂xw(t, x) =RL0 D
1−γt [Kγb(t)∂xxw(t, x)] + f(w, t, x). (1.2)
In the simpler case where the model has constant coefficients, the solution of thefractional advection diffusion equation, a simple case of equation (1.2), has beeninvestigated using a finite difference method [30]. Also, the variable coefficientsspace-time fractional advection diffusion equation has been studied by using thedifference method in [31].
However, this is the first approach to solve equation (1.2) numerically. Weshall use the finite difference method for time discretization and the spectralmethod for spatial discretization. Here, the main difficulty lies in two aspects.The first one is caused by the fact that the coefficients depend on the boundaryfunction that leads to difficulties in the analysis of stability and convergenceof the numerical approximation. The other arises from the convective termthat is coupled with reactive term leads to more complexity in the temporaldiscretization.
The paper is arranged as follows. In next section, the fractional reaction-diffusion model with moving boundary is presented and the analytical solutionis also presented for this problem with non-growing domain. The temporaldiscretization is considered in the third section. The stability and convergenceof the semi-discrete scheme is derived in section 4. In section 5, the spectralapproximation is analyzed and the full-discrete error is derived. In section 6, we
3
consider the implementation of our method. Some examples are given in section7 to show the efficiency and high order accuracy of our method. Finally, someremarks are given in section 8.
2. Mathematical model
We consider a conservation statement for a density, C(t, x), and make thefollowing assumptions:
• the direction and speed of migration is determined by a gradient;
• the tissue growth is independent of the cell density, C(t, x);
• the system diffuses with a sub-diffusion and undergoes reaction at rateR(C, t, x).
Application of the Reynolds transport theorem [24] for mass conservation for Con a domain 0 < x < L(t), gives
where the domain velocity v(t, x) contributes an additional convective term.Here, 0 < γ < 1, D > 0 is the diffusivity coefficient, and C
0Dγt denotes the
Caputo fractional derivative defined as
C0D
γt C =
1
Γ(1− γ)
∫ t
0
∂sC(s, x)
(t− s)γds.
For a uniformly growing domain, the domain growth is associated with thevelocity v(t, x) by a relationship (see [26, 29] for details)
dL
dt=
∫ L(t)
0
∂xvdx =
∫ L(t)
0
σ(t)dx. (2.1)
From (2.1), it follows that σ(t) = L′(t)L(t) and v = xσ(t). In particular, for non-
growing domain, σ(t) ≡ 0; for an exponentially growing domain with L(t) =L(0) exp(αt)(α > 0), σ(t) = α; and for a linearly growing domain with L(t) =L(0) + bt(b > 0), σ(t) = b
L(t) .
Now, we consider the linear fractional reaction-diffusion equation with mov-ing boundary
Further, for convenience, we shall convert problem (2.4)–(2.5) into one withhomogeneous boundaries. To this goal, let u(t, x) = v(t, x) + ul(t) + (ur(t) −ul(t))x, then (2.4) is clearly rewritten into one with homogeneous boundaries:
To end this section, we consider the analytical solution for the case of anon-growing domain. On a non-growing domain, the analytical solutions oftime fractional partial differential equations [32] and multi-term time fractionalpartial differential equations [33] have been investigated by several authors. Inthis case, (2.4) is converted into
C0D
γt u(t, x) = D∂xxu(t, x) +Ku(t, x) (2.7)
with initial condition u(0, x) = u0(x) and boundary boundary condition u(t, 0) =ul(t), u(t, 1) = ur(t).
Thus, by using the method of separation of variables (see [34] for example),the analytical solution of (2.7) can be derived. Let
f(t, x) = ul(t) + [ur(t)− ul(t)]x− C0D
γt ul(t)− x · C0Dγ
t [ur(t)− ul(t)],
and expand f(t, x) in the Fourier series
f(t, x) =
∞∑n=1
fn(t) sin (nπx).
Then, one can obtain the analytical solution of (2.7)
u(t, x) =
∞∑n=1
Bn(t) sin (nπx) + ul(t) + [ur(t)− ul(t)]x,
where
Bn(t) =2Eγ,1
[−(n2π2D −K)tγ] ∫ 1
0
v0(x) sin (nπx)dx
+
∫ t
0
sγ−1Eγ,γ
[−(n2π2D −K)sγ]fn(t− s)ds, (2.8)
in which v0(x) = u0(x)− ul(0)− [ur(0)− ul(0)]x and Eα,β(t) =∑∞
n=0tn
Γ(nα+β)
is the Mittag-Leffler function based on the gamma function.
3. Semi-discrete approximation
Hereafter, we study the numerical solution of (2.6). For simplicity of analysisand without loss of generality, we consider the following equation
C0D
γt u(t, x) = d(t)∂xxu(t, x)−Υu(t, x) + I1−γ
t+ [x∂xu(t, x)σ(t)] + f(t, x), (3.1)
where d(t) = 1L2(t) , σ(t) =
L′(t)L(t) and Υ > 0.
Let tk = kτ, k = 0, 1, 2, · · · ,K, where τ = TK is the time step. By the
definition of the Caputo derivative, equation (3.1) is recast as
Set Λ = (0, 1), Ω = (0, T ]× (0, 1). Denote by (·, ·) the inner product on theHilbert space L2(Λ), by ‖ · ‖0 the norm of L2(Λ).
Let uk(x), vk(x)(k = 1, 2, · · · ) be the solutions of the semi-discrete approx-imation equations (3.10)-(3.11) associated to the initial values u0, v0, respec-tively. Let ek(x) = uk(x)− vk(x) for k = 0, 1, 2, · · · .Theorem 4.1 (stability). Let u0, v0 ∈ L2(Λ), d(t) = 1/L2(t), σ(t) = L′(t)/L(t).For the growing moving domain, where σ(t) > 0, if τ satisfies
K−1∑j=0
(bj − bj+1)
[L(T )
L((K − j)τ)
]2≤ 1, (4.1)
then the semi-discrete scheme (3.10) is stable. Moreover, for any k = 1, 2, · · · ,K,
‖ek‖0 ≤ ‖e0‖0. (4.2)
Proof. By (3.10) and (3.11), we have
e1(x) = e0(x) + τxσ1∂xe1(x) + rd1∂xxe
1(x)− rΥe1(x), (4.3)
ek+1(x) = ek(x) + τxσk+1∂xek+1(x) + rdk+1∂xxe
k+1(x)− rΥek+1(x)
+ rk−1∑j=0
(bj+1 − bj)[dk−j∂xxe
k−j(x)−Υek−j(x)], for k ≥ 1. (4.4)
Multiplying the both sides of (4.2) by ek+1 and integrating over Λ, we get
and making use of the Cauchy-Schwartz inequality, we have
‖ek+1‖20 ≤ 1
2‖ek‖20 +
1
2‖ek+1‖20 −
1
2τσk+1‖ek+1‖20 − rdk+1‖∂xek+1‖20
+r
2
k−1∑j=0
(bj − bj+1)dk−j
(‖∂xek−j‖20 + ‖∂xek+1‖20)
+rΥ
2
k−1∑j=0
(bj − bj+1)(‖ek−j‖20 + ‖ek+1‖20
)− rΥ‖ek+1‖20. (4.6)
By rearrangement, we have
‖ek+1‖20 + r
k∑j=0
bjdk+1−j‖∂xek+1−j‖20 + rΥ
k∑j=0
bj‖ek+1−j‖20
≤ ‖ek‖20 + rk−1∑j=0
bjdk−j‖∂xek−j‖20 + rΥk−1∑j=0
bj‖ek−j‖20 − τσk+1‖ek+1‖20
− rdk+1‖∂xek+1‖20 + rk−1∑j=0
(bj − bj+1)dk−j‖∂xek+1‖20
+ rΥ
k−1∑j=0
(bj − bj+1)‖ek+1‖20 − rΥ‖ek+1‖20. (4.7)
Let
En = ‖en‖20 + rn−1∑j=0
bjdn−j‖∂xen−j‖20 + rΥn−1∑j=0
bj‖en−j‖20.
For the growing domain moving boundary problem, where σ(t) > 0, then
−τσk+1‖ek+1‖20 + rΥk−1∑j=0
(bj − bj+1)‖ek+1‖20 − rΥ‖ek+1‖20 ≤ 0.
On the other hand, noting that dj = 1/L2(jτ), if
k−1∑j=0
(bj − bj+1)
[L((k + 1)τ)
L((k − j)τ)
]2≤ 1,
thenk−1∑j=0
(bj − bj+1)dk−j ≤ dk+1,
and it follows from (4.7) that
Ek+1 ≤ Ek, for k ≥ 1. (4.8)
10
For k = 0, we have from (4.3)
E1 ≤ E0 = ‖e0‖20.
Making use of (4.8), we have for all k ∈ N, Ek ≤ ‖e0‖20. By the definition ofEk, the stability is derived. This completes the theorem. �
Remarks 4.1. i) The semi-discrete approximation (3.10) is unconditionallystable for the non-growing boundary problem.
ii) For the uniform growing domain, the stability condition (4.1) can be solvedto obtain the condition that the step size satisfies. For example, in the caseof the exponentially growing domain with L(t) = eαt,
K−1∑j=0
(bj − bj+1)e2αT
e2α(k−j)τ=
K−1∑j=0
(bj − bj+1)e2α(j+1)τ ≤ 1,
here, one can obtain τ by some numerical methods. Actually, due to
(bj − bj+1)e2α(j+1)τ ≤ e2αT (bj − bj+1),
it follows that∑∞
j=0(bj − bj+1)e2α(j+1)τ is convergent. Therefore, there
exists τ0 such that
K−1∑j=0
(bj − bj+1)e2α(j+1)τ ≤
∞∑j=0
(bj − bj+1)e2ατ0(j+1) ≤ 1
holds for any τ ≤ τ0. The same result can be obtained for a linearly growingboundary.
Theorem 4.2 (convergence). Let σ(t) ≥ θ ≥ 0. Assume that d(t), ∂xxu(·, x) ∈C2[0, T ]. Then, when the condition (4.1) holds the semi-discrete approxima-tion (3.10) is convergent. Further, for any k = 1, 2, · · · , there exists a positiveconstant C independent of τ and k, such that
‖u(tk, x)− uk‖0 ≤ Cτ.
Proof. Let ηk(x) = u(tk, x)− uk(x). From (3.8) and (3.10), we have
ηk+1(x) = ηk(x) + τxσk+1∂xηk+1(x) + rdk+1∂xxη
k+1(x)− rΥηk+1(x)
+ rk−1∑j=0
(bj+1 − bj)[dk−j∂xxη
k−j(x)−Υηk−j(x)]+ τRk+1(x). (4.9)
Note that ηk(0) = ηk(1) = 0. Hence, the following equation still holds
(x∂xη
k+1, ηk+1)= −1
2
(ηk+1, ηk+1
).
11
Performing a similar proof to Theorem 4.1, we have
Y k+1 ≤Y k − rbkdk+1‖∂xηk+1‖20 − rΥbk‖ηk+1‖20− τσk+1‖ηk+1‖20 + 2τ |(Rk+1, ηk+1)|, (4.10)
where
Y k = ‖ηk‖20 + r
k−1∑j=0
bjdk−j‖∂xηk−j‖20 + rΥ
k−1∑j=0
bj‖ηk−j‖20.
Therefore, by Young’s inequality we obtain
Y k+1 ≤ Y k − rΥbk‖ηk+1‖20 + 2τ |(Rk+1, ηk+1)|≤ Y k − rΥbk‖ηk+1‖20 +
τ
ε‖Rk+1‖20 + ετ‖ηk+1‖20. (4.11)
Taking ε = Υbkτγ−1
Γ(1+γ) , then it follows that
Y k+1 ≤ Y k +τΓ(1 + γ)
Υbkτγ−1‖Rk+1‖20 ≤ Y k +
c2γΓ(1 + γ)τ2
Υbkτ
γ , (4.12)
by using (3.9).Noting that Y 0 = ‖η0‖20 = 0, hence we obtain
Y k ≤ c2γΓ(1 + γ)τ2
Υ
k−1∑j=0
bjτγ ≤ c2γΓ(1 + γ)τ2
Υ(kτ)γ ≤ c2γΓ(1 + γ)T γ
Υτ2. (4.13)
This finishes the proof of the theorem. �
5. Spectral approximation
In the following, we consider the full discretization scheme of (3.1) by thespectral method. For convenience, we introduce the following Sobolev norm
‖u‖21,γ = ‖u‖20 + τγ‖∂xu‖20 for u ∈ H10 (Λ).
Additionally, we denote by ‖ · ‖m the standard Sobolev norm on space Hm(Λ).
5.1.Variational formulation.Multiplying (3.10) by a test function v ∈ H10 (Λ)
and integrating, we obtain the variational formulation of problem (3.1): to finduk+1(k = 0, 1, · · · ) ∈ H1
where g is defined as (3.4). Clearly, the bilinear operator B has the followingproperties:
Lemma 5.1. For any uk, v ∈ H10 (Λ), there exist positive constant κ1, κ2 inde-
pendent of uk, v and τ , such that
B(uk, uk) ≥ κ1‖uk‖21,γ ,∣∣B(uk, v)
∣∣ ≤ κ2‖uk‖1,γ‖v‖1,γ .
5.2. Legendre spectral method. Let PN (Λ) denote the set of polynomials ofdegree N . Denote by PN (Λ) = H1
0 (Λ) ∩ PN (Λ).The spatial discretization of the semi-discrete approximation (3.10) is to find
uk+1N ∈ PN (Λ)(k = 0, 1, · · · ), such that
B(uk+1N , vN ) = Fk
N (vN ), ∀vN ∈ PN (Λ), (5.2)
where
FkN (vN ) =(uk
N , vN ) + r
k−1∑j=0
(bj − bj+1)dk−j(∂xuk−jN , ∂xvN )
+ rΥk−1∑j=0
(bj − bj+1)(uk−jN , vN ) + τ(gk+1, vN ).
In light of the coercivity and continuity of B, the existence and uniquenessof the solution of (5.2) is assured by Lax-Milgram theorem. In what follows, weshall study the error estimate of the spectral approximation solution.
Let ΠN : H10 (Λ) → PN (Λ) be the orthogonal projection on PN (Λ) in H1
0 (Λ)such that
B(ΠNu, v) = B(u, v) for all v ∈ PN (Λ) (5.3)
for any u ∈ H10 (Λ). We have the error estimate (see Section 5.4 in [37])
‖u−ΠNu‖1,γ ≤ ‖u−ΠNu‖1 ≤ CN1−m‖u‖m (5.4)
for all u ∈ Hm0 (Λ), with m ≥ 1.
Theorem 5.2 (error estimate). Let 0 ≤ d(t) ≤ d, σ(t) ≥ θ > 0. Assume thatu0 ∈ L2(Λ) and {uk}Kk=1 be the solutions of problem (5.1), {uk
N}Kk=1 the solutionsof the full discrete problem (5.2). Suppose that uk ∈ Hm(Λ) ∩ H1
0 (Λ),m > 1and τ satisfies (4.1). Then, there exists a positive constant C independent ofk, τ and N , such that
‖uk − ukN‖1,γ ≤ Cτ−1N1−m max
0≤s≤K‖us‖m,
and‖uk − uk
N‖0 ≤ Cτ−1N−m max0≤s≤K
‖us‖m,
for all k = 1, 2, · · · ,K.
13
Proof. Let
ηj = uj −ΠNuj , ηj = ΠNuj − ujN , εj = uj − uj
N = ηj + ηj .
Subtracting (5.2) from (5.1), we have
B(uk+1 − uk+1N , vN ) =(εk, vN ) + r
k−1∑j=0
(bj − bj+1)dk−j(∂xεk−j , ∂xvN )
+ rΥ
k−1∑j=0
(bj − bj+1)(εk−j , vN ) (5.5)
for any vN ∈ PN (Λ). In light of (5.3), B(uk+1 − uk+1N , vN ) = B(ΠNuk+1 −
Ek+1 ≤ Ek + ‖ηk+1‖20 + rdk+1‖∂xηk+1‖20 + rΥ‖ηk+1‖20+
1
τθ‖ηk+1‖20 +
rdk+1
bk‖∂xηk+1‖20 +
rΥ
bk‖ηk+1‖20
≤ Ek + Cτ−1 max1≤s≤K
‖ηs‖21,γ (5.9)
since 1bk
≤ τγ−1
λ by Lemma 3.1. Here, C is a positive constant that depends
only on θ, λ, γ,Υ and d.Making use of (5.9) and E0 = 0, we have
Ek ≤ Ckτ−1 max1≤s≤K
‖ηs‖21,γ ≤ CTτ−2 max1≤s≤K
‖ηs‖21,γ .
By the definition of Ek, we obtain
‖uk − ukN‖1,γ ≤ Cτ−1 max
1≤s≤K‖us −ΠNus‖1,γ .
The first estimate of Theorem 5.2 is derived.Now, we prove the second error estimate in L2(Λ) sense by using the duality
argument. By the basic theory of elliptic equation and properties of B, it followsthat for any ψ ∈ L2(Λ), the following equation
−rdk+1∂xxu− τσk+1x∂xu+ (1 + rΥ)u = ψ (5.10)
has a unique solution u ∈ H2(Λ) ∩ H10 (Λ) and ‖u‖2 ≤ C‖ψ‖0. Let v be the
solution of the dual problem of (5.10), then ‖v‖2 ≤ C‖ψ‖0, and the followingholds
B(z, v) = (ψ, z) for any z ∈ H10 (Λ).
Taking z = uk+1 − uk+1N , then by Lemma 5.1
(ψ, uk+1 − uk+1N ) = B(uk+1 − uk+1
N , v −ΠNv)
≤ κ2‖uk+1 − uk+1N ‖1,γ‖v −ΠNv‖1,γ
≤ CN−1‖uk+1 − uk+1N ‖v‖2
≤ CN−1‖uk+1 − uk+1N ‖1,γ‖ψ‖0.
15
Therefore, we deduce that
‖uk+1 − uk+1N ‖0 = sup
ψ∈L2(Λ)
|(uk+1 − uk+1N , ψ)|
‖ψ‖0≤ CN−1‖uk+1 − uk+1
N ‖1,γ .This finishes the proof of the theorem. �
Remark 5.1. This theorem still holds for θ = 0. It needs some modificationsby Young’s inequality to get an estimate.
Finally, we have the following error estimate by combining Theorem 4.2 withTheorem 5.2.
Theorem 5.3. Let d(t) ≥ 0 be increasing in t, σ(t) ≥ θ ≥ 0. Assume that u0 ∈L2(Λ) and d(t) ∈ C2[0, T ]. If u ∈ C2(0, T ;H2(Λ) ∩H1
0 (Λ)) ∩ L∞(0, T ;Hm(Λ))(m > 1) is the solution of (3.4), and τ satisfies (4.1), then
‖u(tk, x)− ukN‖0 ≤ C(τ + τ−1N−m),
where C is a constant independent of k, τ and N . �
5.3.An improved estimate.The coefficient τ−1 ahead of the spectral approx-imation estimate may be dropped by following the line in [38]. However, onlyan estimate of order N2−m is obtained.
Theorem 5.4 (improved error bound). Let d(t) ≥ 0, σ(t) ≥ θ > 0. Assumethat u0 ∈ L2(Λ) and {uk
N}Kk=1 the solutions of the full discrete problem (5.2).Suppose that u ∈ Hm(Λ) ∩ H1
0 (Λ),m > 1 and τ satisfies (4.1). Then, thereexists a positive constant C independent of k, τ and N , such that
‖u(tk, x)− ukN‖0 ≤ C(τ +N2−m),
for all k = 1, 2, · · · ,K.
Proof. Denote
∂tu(tk+1, x) =1
τ[ΠNu(tk+1, x)−ΠNu(tk, x)] +Rk+1
21 (τ, x),
where
Rk+121 (τ, x) = (I −ΠN )∂tu(tk+1, x) +ΠN
[∂tu(tk+1, x)− u(tk+1, x)− u(tk, x)
τ
].
Let
Rk+122 (τ, x) = ∂xu(tk+1, x)− ∂xΠNu(tk+1, x),
Rk+123 (τ, x) = ∂xxu(tk+1, x)− ∂xxΠNu(tk+1, x).
16
Then,
‖Rk+121 ‖0 = O(τ +N−m), ‖Rk+1
22 ‖0 = O(N1−m), ‖Rk+123 ‖0 = O(N2−m).
Similar to (3.8), we have
ΠNu(tk+1, x)−ΠNu(tk, x)
τ= xσ(tk+1)∂xΠNu(tk+1, x)+
r
τ
⎡⎣d(tk+1)∂xxΠNu(tk+1, x) +
k−1∑j=0
(bj+1 − bj)d(tk−j)∂xxΠNu(tk−j , x)
⎤⎦
− rΥ
τ
⎡⎣ΠNu(tk+1, x) +
k−1∑j=0
(bj+1 − bj)ΠNu(tk−j , x)
⎤⎦+ g(tk+1, x) +Rk+1
2 (τ, x),
(5.11)
where Rk+12 (τ, x) = Rk+1
21 (τ, x) +Rk+122 (τ, x) +Rk+1
23 (τ, x) +Rk+1(τ, x). In lightof (3.9), one has
|Rk+12 (τ, x)| ≤ cγbkτ
γ + C(τ +N2−m), ∀x ∈ Λ (5.12)
where C is a constant independent of τ, k andN . Therefore, for any vN ∈ PN (Λ)
Let ηk = ukN − ΠNu(tk, x). Then, by using the full discretization (5.2)and
(5.13), we obtain that
(ηk+1, vN ) = (ηk, vN ) + τ(xσk+1∂xη
k+1, vN)
− r
⎡⎣dk+1
(∂xη
k+1, ∂xvN)+
k−1∑j=0
(bj+1 − bj)dk−j
(∂xη
k−j , ∂xvN)⎤⎦
− rΥ
⎡⎣(ηk+1, vN
)+
k−1∑j=0
(bj+1 − bj)(ηk−j , vN
)⎤⎦+ (τRk+1
2 , vN ), (5.14)
for any vN ∈ PN (Λ). Taking vN = ηk+1, and performing a similar process toTheorem 4.2, we have
Y k+1 ≤Y k − rbkdk+1‖∂xηk+1‖20 − rΥbk‖ηk+1‖20− τσk+1‖ηk+1‖20 + 2|(τRk+1
2 , ηk+1)|, (5.15)
17
where
Y k = ‖ηk‖20 + rk−1∑j=0
bjdk−j‖∂xηk−j‖20 + rΥk−1∑j=0
bj‖ηk−j‖20.
Now, |τRk+12 | ≤ cγbkτ
1+γ + Cτ(τ +N2−m), thus
Y k+1 ≤ Y k − rΥbk‖ηk+1‖20 − τθ‖ηk+1‖20 + 2|(cγbkτ1+γ , ηk+1)|+ 2| (Cτ(τ +N2−m), ηk+1
) |≤ Y k − rΥbk‖ηk+1‖20 − τθ‖ηk+1‖20 + ε1‖ηk+1‖20 +
1
ε1c2γb
2kτ
2(1+γ)
+ ε2‖ηk+1‖20 +1
ε2C2τ2(τ +N2−m)2. (5.16)
Taking ε1 = rΥbk, ε2 = τθ, it yields that
Y k+1 ≤ Y k +bkτ
γ
ΥΓ(1 + γ)c2γτ
2 +C2(τ +N2−m)2
θτ.
Hence,
Y k ≤ Γ(1 + γ)c2γτ2
Υ
k−1∑j=0
bjτγ +
C2(τ +N2−m)2
θ
k−1∑j=0
τ,
that is, there exists a constant C independent of τ, k and N
‖ηk‖0 ≤ C(τ +N2−m).
Noting that
‖ukN − u(tk, x)‖0 ≤ ‖uk
N −ΠNu(tk, x)‖0 + ‖u(tk, x)−ΠNu(tk, x)‖0,
then, the theorem holds. �
6. Implementation
6.1.Galerkin spectral approximation.We firstly consider the solution ofthe full discrete approximation equation (5.2). Let us take ϕn(x) = Ln+2(2x−1)−Ln(2x−1), n = 0, 1, · · · , N−2, here Ln(x) denote the Legendre polynomialsof degree n. Hence, PN (Λ) = Span{ϕ0(x), ϕ1(x), · · · , ϕN−2(x)}. Let
ukN (x) =
N−2∑j=0
Ukj ϕj(x), for k = 0, 1, 2, · · · ,K,
18
and take vN = ϕl(x), l = 0, 1, · · · , N − 2, so that from (5.2)
(∂xuk+1N , ∂xϕl) =
N−2∑j=0
(∂xϕj , ∂xϕl)NUk+1j = AUk+1,
(uk+1N , ϕl) =
N−2∑j=0
(ϕj , ϕl)NUk+1j = BUk+1,
(x∂xuk+1N , ϕl) =
N−2∑j=0
(x∂xϕj , ϕl)NUk+1j = EUk+1,
where (·, ·)N denotes the Legendre-Gauss-Lobatto type discrete inner, and
A = (Alj)(N−1)×(N−1) , B = (Blj)(N−1)×(N−1) , E = (Elj)(N−1)×(N−1) ,
Uk+1 = (Uk+10 , Uk+1
1 , · · · , Uk+1N−2)
T .
Therefore, the full discretization scheme (5.2) can be rewritten in the matrixform as
[rdk+1A− τσk+1E + (1 + rΥ)B]Uk+1
= BUk + rk−1∑j=0
(bj − bj+1)dk−jAUk−j + rΥ
k−1∑j=0
(bj − bj+1)BUk−j + τF k+1N ,
(6.1)
where
F k+1N =
⎡⎢⎢⎢⎢⎢⎣
(gk+1, ϕ0)N
(gk+1, ϕ1)N
...
(gk+1, ϕN−2)N
⎤⎥⎥⎥⎥⎥⎦.
At k = 0, we have
[rd1A− τσ1E + (1 + rΥ)B]U1 = BU0 + τF 1N .
In order to evaluate U0, we need to solve the following equation
BU0 = cN =
⎡⎢⎢⎢⎢⎣
(u0, ϕ0)N
(u0, ϕ1)N
...
(u0, ϕN−2)N
⎤⎥⎥⎥⎥⎦ .
Since B is symmetric, we may make use of the conjugate gradient (CG) methodto solve U0. For k ≥ 1, the BiCGSTAB algorithm is adopted to obtain Uk. Thealgorithm for solving the problem (3.4) is as follows (see Algorithm 1).
19
Algorithm 1 Solving the moving boundary problem with homogeneous bound-ary conditions.
Input: r, σi, di, τ, bi,Υ.
1. Compute matrices A,B and E;
2. Compute U0 by solving BU0 = cN ;
3. Compute F 1N . To solve U1 by
[rd1A− τσ1E + (1 + rΥ)B]U1 = BU0 + τF 1N .
4. For k = 1 : K do
• Evaluate F k+1N ;
• Solve the equation (6.1).
Output: U0, U1, · · · , UK . To obtain u(tk, x).
6.2. Petrov-Galerkin approximation. In the Galerkin spectral method, theboundary conditions are reduced to homogeneous boundary conditions, the sub-stitutional work is evaluating f(t, x). Now, we consider the Legendre Petrov-Galerkin approximation for Dirichlet boundary conditions. For the moment, weneed to consider the boundary conditions, but it is not necessary to computef(t, x) since f(t, x) ≡ 0 here. For this case, the different basis functions shallbe considered. Taking ϕn(x) = Ln(2x− 1), n = 0, 1, 2, · · · , N . Let
ukN (x) =
N∑j=0
Ukj ϕj(x), for k = 0, 1, 2, · · · ,K,
and take the test functions as vj(x) = Lj+2(2x−1)−Lj(2x−1), j = 0, 1, · · · , N−2 such that vj(x) satisfy the homogeneous boundary conditions. Substitutingthese into (5.2), it gives
(∂xuk+1N , ∂xvl)N =
N∑j=0
(∂xϕj , ∂xvl)NUk+1j = AUk+1,
(uk+1N , vl)N =
N∑j=0
(ϕj , vl)NUk+1j = BUk+1,
(x∂xuk+1N , vl)N =
N∑j=0
(x∂xϕj , vl)NUk+1j = EUk+1,
where A, B, E are matrices of order (N − 1) × (N + 1). Thus, equations (6.1)is changed into ones in which A,B,E are replaced by A, B, E and F k
N vanishes,
20
that is
[rdk+1A− τσk+1E + (1 + rΥ)B]Uk+1
= BUk + r
k−1∑j=0
(bj − bj+1)dk−jAUk−j + rΥ
k−1∑j=0
(bj − bj+1)BUk−j . (6.2)
The advantage of the above choice of test functions is that the computation ofthe first derivative in E can be avoided, and that the computation in A canbecome much simpler, by using the properties of Legendre polynomials. Inaddition, consider the boundary conditions
N∑j=0
ϕj(0)Uk+1j = uk+1
l , andN∑j=0
ϕj(1)Uk+1j = uk+1
r , (6.3)
where uk+1l = ul(tk+1), u
k+1r = ur(tk+1). Adding the above two equations to
(6.2), a closed system is obtained. The algorithm is shown in Algorithm 2.
Algorithm 2 Solving the moving boundary problem with non-homogeneousboundary conditions.
Input: r, σi, di, τ, bi,Υ.
1. Compute matrices A, B, E and the corresponding cN ;
2. Compute U0 by solving the system
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
BU0 = cNN∑j=0
ϕj(0)U0j = u0
l
N∑j=0
ϕj(1)U0j = u0
r.
3. Solve U1 by
[rd1A− τσ1E + (1 + rΥ)B]U1 = BU0,
N∑j=0
ϕj(0)U1j = u1
l , and
N∑j=0
ϕj(1)U1j = u1
r.
4. For k = 1 : K, solving (6.2) and (6.3).
Output: U0, U1, · · · , UK . To obtain u(tk, x).
7. Numerical examples
In this section, we shall study two examples to illustrate the efficiency of ourmethod, and to illustrate our theoretical analysis of the error estimates.
21
Example 1. Consider the following problem on a non-growing domain:⎧⎪⎪⎨⎪⎪⎩
C0D
γt u(t, x) = ∂xxu(t, x)− 1
2u(t, x), 0 < x < 1, t > 0,
u(t, 0) = u(t, 1) = 0,
u(0, x) = sinπx.
(7.1)
The simple example serves mainly as testing the theoretical analysis on theerror estimate of our method. It is easy to derive the analytical solution of (7.1)as
u(t, x) = Eγ,1
[−(π2 +
1
2
)tγ]sinπx.
The numerical results are given in Table 1. Here, we take N = 13 whichis large enough such that the spatial discretization error is negligible comparedwith the temporal discretization error. For different fractional derivative ordersγ = 0.1, 0.5 and 0.9, the results in Table 1 show an accuracy O(τ) is attained.We remark that the analytical solution is not in C2 near the singular pointt = 0. As for this type of singularity, McLean and Mustapha proved that thesame accuracy can be obtained when nonuniform meshes are used (see [39] fordetails).
Table 1: L2−errors of problem 7.1 versus τ and convergence order, with N = 13.
In order to show the spatial discretization error and convergence order, wetake τ = 5.0 × 10−6. The results are plotted in Figure 1. In the figure, weplot the log-linear error of the numerical solution as a function of polynomialdegree N . It can be seen that the errors start decaying exponentially as theerror variations are essentially linear versus the degree of polynomial, then stallwhen they are dominated by temporal discretization errors of order O(τ). Thisis also expected by the theoretical results.
Example 2. Consider the moving boundary problem with an exponentiallygrowing domain
C0D
γt C(t, x) = ∂xxC(t, x)− C(t, x), 0 < x < L(t), t > 0, (7.2)
where L(t) = L(0) exp(0.2t). Given the initial condition
C(0, x) = sinπx
22
3 5 7 9 11−6
−5
−4
−3
−2
−1
polynomial degree N
erro
rs in
logs
cale
gamma=0.1gamma=0.5gamma=0.9
Figure 1: L2−errors of the numerical solution of (7.1) in spatial discretization, with τ =5.0e− 06.
and the Dirichlet boundary conditions C(t, 0) = 0, C(t, L(t)) = 0.Let L(0) = 1. We check the efficiency of our method for solving some
problems on an exponentially growing domain. Taking γ = 0.35, 0.55, 0.75, 0.95,we see that our method is convergent only if τ ≤ 0.001. The results are shownin Figure 2. These plots show the evolution of C(t, x) and it is clear that thedomain is increasing in time. Our results show how the evolution of the movingboundary problem is affected by altering γ.
8. Conclusion
Moving boundary problems are important in many science and engineeringapplications. However, few studies that examine a fractional diffusion problemwith a moving boundary have been presented. In this paper, we first present afractional reaction-diffusion model with prescribed-boundary moving boundarycondition that arises from developmental biology. An efficient numerical methodis proposed to solve such a class of fractional moving boundary problem in thepresent paper. This method utilizes the finite difference scheme to discretize thetime variable and a spectral scheme for the space variable. The convergence rateof our method in the temporal discretization is O(τ) and is spectrally accuratein the spatial discretization. The restriction Υ > 0 should be omitted by sometechnological management, and we shall study it in the future work.
Acknowledgment
We would like to thank the referees for their careful reading of the paperand many constructive comments and suggestions.
Figure 2: The solution of moving boundary problem (7.2) with γ = 0.35, 0.55, 0.75, 0.95 atdifferent times.
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