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CHESTER RESEARCH ONLINE

Research in Mathematics and its Applications

ISSN 2050-0661

Series Editors: CTH Baker, NJ Ford

A finite element method for time fractional partial differential equations

Neville J. Ford, Jingyu Xiao and Yubin Yan

2012:RMA:7

© 2011 NJ Ford, J Xiao & Y Yan

Chester Research Online Research in Mathematics and its Applications ISSN 2050-0661

2012:RMA:7

Please reference this article as follows: Ford, N. J., Xiao, J., & Yan, Y. (2011). A finite element method for time fractional

partial differential equations. Fractional Calculus and Applied Analysis, 14(3), 454-474. doi: 10.2478/s13540-011-0028-2

http://dx.doi.org/10.2478/s13540-011-0028-2


2012:RMA:7

Author Biographies

Prof. Neville J. Ford

Neville Ford is Dean of Research at the University of Chester, where he has been employed since 1986. He holds degrees from Universities of Oxford (MA), Manchester (MSc) and Liverpool (PhD). He founded the Applied Mathematics Research Group in 1991 and became Professor of Computational Applied Mathematics in 2000. He has research interests in theory, numerical analysis and modelling using functional differential equations, including delay, mixed and fractional differential equations. Jingyu Xiao Jingyu Xiao is a PhD student at the Harbin Institute of Technology in China, and held a 2010/11 Chinese Scholarship Council Fellowship to study at the University of Chester. Dr Yubin Yan Yubin Yan graduated from China in 1985 with a BSc in Mathematics and gained his MSc in Mathematics in 1991 from the Harbin institute of Technology in China. In 1997, he visited the Chalmers University of Technology in Sweden for one year. Yubin completed his PhD in Mathematics at Chalmers in 2003. Before he joined Chester in 2007 Yubin worked as a research associate, first in in the Department of Mathematics at the University of Manchester and then in the Department of Automatic Control and System Engineering at the University of Sheffield.

A FINITE ELEMENT METHOD FOR TIME FRACTIONAL

PARTIAL DIFFERENTIAL EQUATIONS

Neville J Ford 1, Jingyu Xiao 2, Yubin Yan 3

Abstract

In this paper, we consider the finite element method for time fractionalpartial differential equations. The existence and uniqueness of the solutionsare proved by using the Lax-Milgram Lemma. A time stepping method isintroduced based on a quadrature formula approach. The fully discretescheme is considered by using a finite element method and optimal con-vergence error estimates are obtained. The numerical examples at the endof the paper show that the experimental results are consistent with ourtheoretical results.

MSC 2010 : Primary 65M12: Secondary 65M06; 65M60; 65M70;35S10

Key Words and Phrases: Fractional partial differential equations, finiteelement method, error estimates, numerical examples

1. Introduction

In this paper, we will consider the finite element method for the timefractional partial differential equation

R0D

αt u(t, x)−∆u(t, x) = f(t, x), t ∈ [0, T ], x ∈ Ω, (1.1)

u(0, x) = 0, x ∈ Ω, (1.2)

u(t, x) = 0, t ∈ [0, T ], x ∈ ∂Ω, (1.3)

where 0 < α < 1 and Ω is the bounded open domain in Rd, d = 1, 2, 3 and

∂Ω is the boundary of Ω. Here ∆ = ∂2

∂x21+ ∂2

∂x22+ ∂2

∂x23denotes the Lapla-

cian operator with respect to the x variable, R0D

αt u(t, x) denotes the left

c© Year Diogenes Co., Sofia

pp. xxx–xxx


2012:RMA:7

2 N.J. Ford, J. Xiao, Y. Yan

Riemann-Liouville fractional derivative with respect to the time variable tdefined by

R0D

αt u(t, x) =

1

Γ(1− α)

∂

∂t

∫ t

0

u(τ, x)

(t− τ)αdτ, 0 < α < 1, (1.4)

where Γ denotes the Gamma function.Time fractional partial differential equations have many applications in

areas such as diffusion processes, electromagnetics, electrochemistry, mate-rial science, turbulent flow, chaotic dynamics, etc. [3], [4], [14], [15], [24],[25], [28], [29]. Analytical solutions of time fractional partial differentialequations have been studied using Green’s functions or Fourier-Laplacetransforms [26], [23], [30], [31].

Numerical methods for fractional ordinary differential equations werestudied in, for example, [6], [7], [8], [9] [12], [13]. Numerical methods forfractional partial differential equations were also studied by some authors.Liu et al. [22] employed the finite difference method in both space and timeand analyzed the stability condition. Sun and Wu [32] proposed a finitedifference method for the fractional diffusion-wave equation. Langlands andHenry [18] considered an implicit numerical scheme for fractional diffusionequation. Lin and Xu [20] proposed a finite difference method in time andLegendre spectral method in space. Li and Xu [19] proposed a time-spacespectral method for time-space fractional partial differential equation basedon a weak formulation and a detailed error analysis was carried out.

Recently, Ervin and Roop [10], [11] used finite element methods to findthe variational solution of the fractional advection dispersion equation, inwhich the fractional derivative depends on the space, related to the nonlo-cal operator, but the time derivative term is of first order, related to thelocal operator. Adolfsson et al. [1], [2] considered an efficient numericalmethod to integrate the constitutive response of fractional order viscoelas-ticity based on the finite element method. Li et al. [16] considered a timefractional partial differential equation by using the finite element methodand obtained error estimates in both semidiscrete and fully discrete cases.Jiang et al. [17] considered a high-order finite element method for the timefractional partial differential equaions and proved the optimal order errorestimates.

In this paper, we will use the framework in Li and Xu [19] in which theauthors introduced suitable spaces and norms in which the time fractionaldifferential problem can be formulated into an elliptic problem. Using thesespaces, we introduce a finite element method for time fractional partialdifferential equation and obtain the optimal order error estimates both insemidiscrete and fully discrete cases.


2012:RMA:7

A FEM FOR TIME FRACTIONAL PDES . . . 3

The paper is organized as follows. In Section 2, we consider the exis-tence and uniqueness of the solution of the time fractional partial differ-ential equation. In Section 3, we introduce a time discretization schemeand prove the error estimate. In Section 4, we consider the finite elementmethod and obtain the optimal order error estimates in space discretiza-tion. Finally in Section 5, we give two numerical examples and show thatthe numerical results are consistent with the theoretical results.

2. Existence and uniqueness

Let C∞(0, T ) denote the space of infinitely differentiable functions on(0, T ) and C∞

0 (0, T ) denote the space of infinitely differentiable functionson (0, T ) with compact support in (0, T ). Let 0C

∞(0, T ) denote the space ofinfinitely differentiable functions on (0, T ) with compact support in (0, T ].Then we introduce the following Sobolev space 0H

α(0, T ), 0 < α < 1which is the closure of 0C

∞(0, T ) with respect to the norm ‖·‖Hα(0,T ), where‖·‖Hα(0,T ) denotes the norm in the usual fractional Sobolev space Hα(0, T )

[21]. Further, let L2(Ω),H1(Ω), H2(Ω) denote the usual Sobolev spaces

with corresponding norms ‖ · ‖L2(Ω), ‖ · ‖H1(Ω) and ‖ · ‖H2(Ω), respectively.

Denote H10 (Ω) = v ∈ H1(Ω), v|∂Ω = 0 with norm ‖ · ‖H1(Ω).

Define the space, with 0 < α < 1,

Bα((0, T )× Ω

)= 0H

α((0, T ), L2(Ω)

)∩ L2

((0, T ),H1

0 (Ω)).

Here Bα((0, T )× Ω

)is a Banach space with respect to the norm

‖v‖((0,T )×Ω) =(‖v‖Hα((0,T ),L2(Ω)) + ‖v‖L2((0,T ),H1

0 (Ω))

)1/2.

We have the following existence and uniqueness theorem.

Theorem 2.1. Assume that 0 < α < 1 and f ∈ L2((0, T )×Ω). Thenthe system (1.1) - (1.3) has a unique solution in Bα

((0, T ) × Ω

). Further

the following stability result holds:

‖u‖Bα/2((0,T )×Ω ≤ C‖f‖L2((0,T )×Ω). (2.1)

The proof of Theorem 2.1 can be found in [19]. For completeness, andbecause we use the approach later, we will give the ideas of the proof ofthis theorem here.

Recall that the right Riemann-Liouville fractional integral is defined as

Rt D

αT v(t) =

1

Γ(1− α)

∂

∂t

∫ T

t

v(τ)

(t− τ)αdτ, 0 < α < 1. (2.2)


2012:RMA:7


Definition 2.1. Define Hαr (0, T ) as the closure of C∞

0 (0, T ) withrespect to the norm ‖ · ‖Hα

r (0,T ), that is,

Hαr (0, T ) =

v ∈ L2(0, T )

∣∣∣ ∃ vn ∈ C∞0 (0, T ), such that ‖vn−v‖Hα

r (0,T ) → 0.

Here the norm ‖ · ‖Hαr (0,T ) is defined by

‖v‖Hαr (0,T ) =

(‖v‖2L2(0,T ) + |v|2Hα

r (0,T )

)1/2,

where the seminorm | · |Hαr (0,T ) is defined by

|v|Hαr (0,T ) = ‖Rt Dα

T v‖L2(0,T ).

Definition 2.2. Define Hαc (0, T ) as the closure of C∞

0 (0, T ) withrespect to the norm ‖ · ‖Hα

c (0,T ), that is,

Hαc (0, T ) =

v ∈ L2(0, T )

∣∣∣ ∃ vn ∈ C∞0 (0, T ), such that ‖vn−v‖Hα

c (0,T ) → 0.

Here the norm ‖ · ‖Hαc (0,T ) is defined by

‖v‖Hαc (0,T ) =

(‖v‖2L2(0,T ) + |v|2Hα

c (0,T )

)1/2,

where the seminorm | · |Hαc (0,T ) is defined by

|v|Hαc (0,T ) =

∣∣(0Dα

t v,Rt D

αT v

)L2(0,T )

∣∣1/2,where (·, ·)L2(0,T ) denotes the inner product in L2(0, T ).

Lemma 2.1. Let 0 < α < 1. We have

Hαr (0, T ) = Hα

c (0, T ) = Hα0 (0, T ),

and the norms ‖ · ‖Hαr (0,T ), ‖ · ‖Hα

c (0,T ) and ‖ · ‖Hα0 (0,T ) are equivalent.

P r o o f. We first prove Hαr (0, T ) = Hα

c (0, T ). In fact, ∀ v ∈ Hαr (0, T ),

there exists a sequence vn ∈ C∞0 (0, T ) such that

‖vn − v‖Hαr (0,T ) → 0, n→ ∞,

which implies that ‖vn − vm‖Hαr (0,T ) → 0, n → ∞, m → ∞, that is, vn

is a Cauchy sequence in Hαr (0, T ). We will show that the norm ‖ · ‖Hα

r (0,T )

is equivalent to the norm ‖ · ‖Hαc (0,T ) in C∞

0 (0, T ). Assuming this for themoment, we see that vn is also a Cauchy sequence in ‖ · ‖Hα

c (0,T ). Thusthere exists v′ ∈ Hα

c (0, T ) such that

‖vn − v′‖Hαc (0,T ) → 0, n→ ∞.


2012:RMA:7


Hence, we have

‖v − v′‖L2(0,T ) ≤ ‖v − vn‖L2(0,T )+ ‖vn − v′‖

L2(0,T )

≤ ‖v − vn‖Hαr (0,T ) + ‖vn − v′‖Hα

c (0,T ) → 0, n→ ∞,

(2.3)

which implies that v = v′ and therefore Hαr (0, T ) ⊂ Hα

c (0, T ). Similarly,we can prove Hα

c (0, T ) ⊂ Hαr (0, T ). Hence we get

Hαc (0, T ) = Hα

r (0, T ).

Denote the norm equivalence by ‖·‖Hαr (0,T )

∼= ‖·‖Hαc (0,T ). We now need

to prove that the norm ‖ · ‖Hαr (0,T ) is equivalent to the norm ‖ · ‖Hα

c (0,T )

in C∞0 (0, T ). In fact, ∀ v ∈ C∞

0 (0, T ), let v be the extension of v by zerooutside (0, T ). Then we have

|v|Hαr (0,T ) = |v|Hα

r (R)∼= |v|Hα

c (R) = |v|Hαc (0,T ),

where we use the fact that

|v|Hαc (0,T ) = |v|Hα

c (R),

which follows from

|v|Hαc (R) =

∣∣(−∞D

αt v, tD

α+∞v

)∣∣1/2 = ∣∣(0Dα

t v, tDαT v

)∣∣1/2 = |v|Hαc (0,T ).

Here we also use the fact that Hαr (R) = Hα

c (R) and the norm ‖ · ‖Hαr (R)

∼=‖ · ‖Hα

c (R) which can be found in [11].Next we prove that the norm ‖ · ‖Hα

r (0,T ) is equivalent to the norm‖ · ‖Hα

c (0,T ) in space Hαr (0, T ) = Hα

c (0, T ). In fact, following the ideas ofthe proof above, ∀ v ∈ Hα

r (0, T ), there exists a sequence vn ∈ C∞0 (0, T )

such that‖vn − v‖Hα

r (0,T ) → 0, n→ ∞,

and‖vn − v‖Hα

c (0,T ) → 0, n→ ∞.

Thus

‖v‖Hαr (0,T ) ≤ ‖v − vn‖Hα

r (0,T ) + ‖vn‖Hαr (0,T )

≤ ‖v − vn‖Hαr (0,T ) + C‖vn‖Hα

c (0,T )

≤ ‖v − vn‖Hαr (0,T ) + C‖vn − v‖Hα

c (0,T ) + C‖v‖Hαc (0,T ).

Let n → ∞, we get ‖v‖Hαr (0,T ) ≤ C‖v‖Hα

c (0,T ) for any v ∈ Hαc (0, T ). Sim-

ilarly we can prove ‖v‖Hαc (0,T ) ≤ C‖v‖Hα

r (0,T ) for any v ∈ Hαr (0, T ). Thus

the norm ‖ · ‖Hαr (0,T ) is equivalent to the norm ‖ · ‖Hα

c (0,T ) in Hαr (0, T ) =

Hαc (0, T )Finally we turn to the proof of Hα

c (0, T ) = Hα0 (0, T ) and the norm

‖·‖Hαc (0,T ) is equivalent to the norm ‖·‖Hα

0 (0,T ) inHαc (0, T ) = Hα

0 (0, T ). The


2012:RMA:7


arguments of the proof are the same as the proof for Hαr (0, T ) = Hα

c (0, T )above. Therefore it suffices to prove that the norm ‖ · ‖Hα

c (0,T ) is equivalentto the norm ‖ · ‖Hα

0 (0,T ) in C∞0 (0, T ) which follows from

|v|Hαc (0,T ) = |v|Hα

c (R)∼= |v|Hα

c (R) = ‖F(tDα

T v)‖L2(R)

= ‖(iw)αF(v)‖L2(R)∼= |v|Hα(R) = |v|Hα(0,T ).

Here we used the Plancherel Theorem [26] and remark that F(v) denotesthe Fourier transform of v. 2

Lemma 2.2. [26] We have

(1) If 0 < p < 1, 0 < q < 1, v(0) = 0, t > 0, then

0Dp+qt v(t) =

(0Dp

t

)(0Dq

t

)v(t) =

(0Dq

t

)(0Dp

t

)v(t), ∀w ∈ Hp+q(0, T )

(2) Let 0 < α < 1. Then we have(0Dα

t w, v)L2(0,T )

=(w, tD

αT v

)L2(0,T )

, ∀w ∈ Hα(0, T ), v ∈ C∞0 (0, T ).

Lemma 2.3. Let 0 < α < 1. Then for any w ∈ 0Hα(0, T ), v ∈

0Hα/2(0, T ), we have(

0Dα

t w, v)L2(0,T )

=(0D

α/2t w, tD

α/2T v

)L2(0,T )

.

P r o o f. Since 0 < α < 1, we have [21]

0Hα/2(0, T ) = H

α/20 (0, T ).

Thus, ∀ v ∈ 0Hα/2(0, T ), there exists a sequence vn ∈ C∞

0 (0, T ) such that

‖v − vn‖Hα/2(0,T ) → 0, n→ ∞.

By Lemma 2.2, we have, for any w ∈ 0Hα(0, T ) with w(0) = 0,(

0Dα

t w, vn)L2(0,T )

=((

0D

α/2t

)(0D

α/2t

)w, vn

)L2(0,T )

=(0D

α/2t w, tD

α/2T vn

)L2(0,T )

.

We now prove that(0Dα

t w, vn)L2(0,T )

→(0Dα

t w, v)L2(0,T )

, n→ ∞, (2.4)(0D

α/2t w, tD

α/2T vn

)L2(0,T )

→(0D

α/2t w, tD

α/2T v

)L2(0,T )

, n→ ∞, (2.5)


2012:RMA:7


It is easy to prove (2.4). For (2.5), we have∣∣∣(0Dα/2t w, tD

α/2T vn

)L2(0,T )

−(0D

α/2t w, tD

α/2T v

)L2(0,T )

∣∣∣≤ ‖0Dα/2

t w‖L2(0,T ) ·∥∥tD

α/2T vn − tD

α/2T v

∥∥L2(0,T )

≤ ‖0Dα/2t w‖L2(0,T ) · |vn − v|

Hα/2r (0,T )

≤ C‖0Dα/2t w‖L2(0,T ) · ‖vn − v‖Hα/2(0,T ) → 0, n→ ∞,

where we used Lemma 2.1 in the last inequality.Together these estimates complete the proof of the Lemma 2.3. 2

Proof of Theorem 2.1. The weak formulation of (1.1)-(1.3) is to find u ∈Bα/2

((0, T )× Ω

)such that

A(u, v) = F(v), ∀ v ∈ Bα/2((0, T )× Ω

), (2.6)

where the bilinear forms A(·, ·) and F(v) are defined by using Lemma 2.3,

A(u, v) =(0Dα

t u, v)L2((0,T )×Ω)

+ (∇u,∇v)L2((0,T )×Ω)

=(0D

α/2t u, tD

α/2T v

)L2((0,T )×Ω)

+ (∇u,∇v)L2((0,T )×Ω),

and

F(v) = (f, v)L2((0,T )×Ω).

It is easy to prove the continuities of the bilinear form A(·, ·) and the righthand functional F(v), that is, there exists a constant C > 0, such that

|A(u, v)| ≤ C‖u‖Bα/2((0,T )×Ω)‖v‖Bα/2((0,T )×Ω),

and

|F(v)| ≤ ‖f‖L2((0,T )×Ω)‖v‖Bα/2((0,T )×Ω). (2.7)

We next prove the coercivity of the bilinear operatorA(·, ·) on Bα/2((0, T )×

Ω). Note that [20](

0D

α/2t ϕ, tD

α/2T ϕ

)L2(0,T )

=(∞D

α/2t ϕ, tD

α/2∞ ϕ

)L2(R)

= cos(α2π)· ‖∞Dα/2

t ϕ‖L2(R), ∀ϕ ∈ C∞0 (0, T ),

where ϕ is the extension of ϕ by zero outside of (0, T ). Thus we find that(0D

α/2t v, tD

α/2T v

)L2(0,T )

is nonnegative for v ∈ Hα/2(0, T ), 0 < α < 1 since

cos(α2π

)is nonnegative for 0 < α < 1.


2012:RMA:7


Combining this with Lemma 2.1, we have

A(v, v) =(0D

α/2t v, tD

α/2T v

)L2((0,T )×Ω)

+ (∇v,∇v)L2((0,T )×Ω)

≥ C(0D

α/2t v, 0D

α/2t v

)L2((0,T )×Ω)

+ (∇v,∇v)L2((0,T )×Ω)

≥ C‖v‖Bα/2((0,T )×Ω), (2.8)

where we applied the Poincare inequalities in the last inequality, that is

‖∇ϕ‖L2(Ω)∼= ‖ϕ‖H1(Ω), ∀ϕ ∈ H1

0 (Ω),

and ∥∥0D

α/2t ψ

∥∥L2(0,T )

∼= ‖ψ‖Hα/2(0,T ), ∀ ψ ∈0 Hα/2(0, T ).

By using the well-known Lax-Milgram Lemma, there exists a uniquesolution u ∈ Bα/2((0, T )× Ω) such that (2.6) holds.

To prove the stability estimate (2.1), we take v = u in (2.6) to get, byusing (2.8) and (2.7),

C‖u‖Bα/2((0,T )×Ω) ≤ A(u, u) = F(u) ≤ C‖f‖L2((0,T )×Ω)‖u‖Bα/2((0,T )×Ω),

which implies that

‖u‖Bα/2((0,T )×Ω) ≤ C‖f‖L2((0,T )×Ω).

The proof is complete. 2

3. Time discretization

In this section, we will consider the time discretization of (1.1)- (1.3).Define A = −∆, D(A) = H1

0 (Ω) ∩H2(Ω). Then the system (1.1)-(1.3)can be written in the abstract form

R0D

αt u(t) +Au(t) = f(t), 0 < t < T, 0 < α < 1, (3.1)

u(0) = u0, (3.2)

or, equivalently,

R0D

αt [u− u0](t) +Au(t) = f(t), 0 < t < T, 0 < α < 1. (3.3)

Note that

R0D

αt u(t) =

1

Γ(−α)d

dt

∫ t

0

u(τ)

(t− τ)αdτ =

1

Γ(−α)

∫ t

0

u(τ)

(t− τ)α+1dτ,

where the integral must be interpreted as a Hadamand finite-part integral[6].

Let 0 = t0 < t1 < · · · < tn = T be a partition of [0, T ]. Then, for fixedtj , j = 1, 2, . . . n, we have


2012:RMA:7


R0D

αt [u− u0](tj) =

1

Γ(−α)

∫ tj

0

u(τ)− u(0)

(t− τ)1+αdτ

=t−αj

Γ(−α)

∫ 1

0

u(tj − tjw)− u(0)

w1+αdw =

t−αj

Γ(−α)

∫ 1

0g(w)w−1−α dw,

where g(w) = u(tj − tjw)− u(0).Now, for every j, we replace the integral by a first-degree compound

quadrature formula with the equispaced nodes 0, 1j ,2j , . . . ,

jj and obtain∫ 1

0g(w)w−1−α dw =

j∑k=0

αkjg(k/j) +Rj(g),

where the weights αkj satisfy that [6]

α(1−α)j−ααkj =

−1, for k = 0,

2k1−α − (k − 1)1−α − (k + 1)1−α, for k = 1, 2, . . . , j − 1,

(α− 1)k−α − (k − 1)1−α + k1−α, for k = j,

and the remainder term Rj(g) satisfies

‖Rj(g)‖ ≤ Cjα−2 sup0≤t≤T

‖g′′(t)‖.

Thus we have

R0D

αt [u− u0](tj) =

t−αj

Γ(−α)

( j∑k=0

αkj

(u(tj − tk)− u(0)

)+Rj(g)

)

= ∆t−αj∑

k=0

wkj

(u(tj − tk)− u(0)

)+

t−αj

Γ(−α)Rj(g),

where

Γ(2−α)wkj =

1, for k = 0,

−2k1−α + (k − 1)1−α + (k + 1)1−α, for k = 1, 2, . . . , j − 1,

−(α− 1)k−α + (k − 1)1−α − k1−α, for k = j.

Let t = tj . We can write (3.3) as

∆t−αj∑

k=0

wkj

(u(tj−tk)−u(0)

)+Au(tj) = f(tj)−

t−αj

Γ(−α)Rj(g), j = 1, 2, 3, . . . .

(3.4)


2012:RMA:7


Denote U j ≈ u(tj) as the approximation of u(tj). We can define thefollowing time stepping method

∆t−αj∑

k=0

(U j−k − U0

)+AU j = fj , fj = f(tj). (3.5)

Let ej = U j − u(tj) denote the error. Then we have the following errorestimate:

Theorem 3.1. Let Un and u(tn) be the solutions of (3.5) and (3.1),respectively. Then we have

‖Un − u(tn)‖ ≤ C∆t2−α.

In order to prove this Theorem, we need the following Lemma.

Lemma 3.1. [6] For 0 < α < 1, let the sequence dj j = 1, 2, . . . begiven by d1 = 1 and

dj = 1 + α(1− α)j−αj−1∑k=1

αkjdj−k, j = 2, 3, · · · ,

where αkj is as in (3.4). Then,

1 ≤ dj ≤sinπα

πα(1− α)jα, j = 1, 2, . . . .

Proof of Theorem 3.1. Subtracting (3.5) from (3.4), we get the error equa-tion

∆t−αj∑

k=0

wkj

(ej−k − e0

)+Aej = −

t−αj

Γ(−α)Rj(g),

or

ej =(∆t−αw0j +A

)−1(∆t−α

j∑k=1

wkjej−k −t−αj

Γ(−α)Rj(g)

)

= (α0j + Γ(−α)tαj A)−1( j∑

k=1

αkjej−k −Rj(g)).

Thus,

‖ej‖ ≤∥∥(α0j + Γ(−α)tαj A)−1

∥∥(( j∑k=1

αkj‖ej−k‖+ ‖Rj(g)‖).


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Note that A is a positive definite elliptic operator, we have, since α0j < 0and Γ(−α) < 0,∥∥(α0j + Γ(−α)tαj A)−1

∥∥ = supλ>0

∣∣∣(α0j + Γ(−α)tαj λ)−1∣∣∣ < −α−1

0j .

Hence

‖ej‖ ≤ −α−10j

( j∑k=1

αkj‖ej−k‖+ Cjα−2n−2 sup0≤t≤T

‖u′′(t)‖)

= α(1− α)j−αj∑

k=1

αkj‖ej−k‖+ α(1− α)Cn−2 sup0≤t≤T

‖u′′(t)‖.

Note that e0 = u(0)− U0 = 0. Denote

d1 = 1,

dj = 1 + α(1− α)j−αj−1∑k=1

αkjdj−k, j = 2, 3, . . . , n.

Then we have by induction,

‖ej‖ ≤ Cα(1− α)n−2 sup0≤t≤T

‖u′′(t)‖ · dj .

By Lemma 3.1, we get

‖ej‖ ≤ Csinπα

πsup

0≤t≤T‖u′′(t)‖j−αn−2 ≤ C∆t2−α.

The proof is complete.2

4. Space discretization

In this section, we will consider the space discretization of (1.1) - (1.3).The variational form of (1.1) - (1.3) is to find u(t) ∈ H1

0 (Ω), such that,(0Dα

t u(t), v)L2(Ω) + (∇u(t),∇v)L2(Ω) = (f(t), v)L2(Ω), ∀ v ∈ H10 (Ω).

(4.1)Let T denote a partition of Ω into disjoint triangles such that no vertex

of any triangle lies on the interior of a side of another triangle and suchthat the union of the triangles determines a polygonal domain Ωh ⊂ Ω withboundary vertices on ∂Ω. Let h denote the maximal length of the sides ofthe triangulation Th. We assume that the triangulations are quasiuniformin the sense that the triangles of Th are of essentially the same size.

Let r be any nonnegative integer. We denote by ‖ · ‖Hr(Ω) the norm in

Hr(Ω). Let Sh ⊂ H10 be a family of finite element space with the accuracy


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of order r ≥ 2, i.e., Sh consists of continuous functions on the closure Ω ofΩ which are polynomials of degree at most r− 1 in each triangle of Th andwhich vanish outside Ωh, such that, for small h, with v ∈ Hs(Ω) ∩H1

0 (Ω)[33]

infχ∈Sh‖v − χ‖L2(Ω) + h‖∇(v − χ)‖L2(Ω) ≤ Chs‖v‖L2(Ω), for 1 ≤ s ≤ r.

(4.2)The semidiscrete problem of (4.1) is to find the approximate solution uh(t) =uh(·, t) ∈ Sh for each t such that(

0Dα

t uh(t), χ)L2(Ω)+(∇uh(t),∇χ)L2(Ω) = (f(t), χ)L2(Ω), ∀ χ ∈ Sh. (4.3)

Let Rh : H10 (Ω) → Sh be the elliptic projection, or Ritz projection,

defined by

(∇(Rhu),∇χ) = (∇u,∇χ), ∀ χ ∈ Sh. (4.4)

We then have,

Lemma 4.1. [33] Assume that (4.2) holds. Then, with Rh defined by(4.4) we have, with v ∈ Hs(Ω) ∩H1

0 (Ω),

‖Rhv − v‖L2(Ω) + h‖∇(Rhv − v)‖L2(Ω) ≤ Chs‖v‖Hs(Ω), for 1 ≤ s ≤ r.

Lemma 4.2. Let 0 < α < 1 and assume that w ∈ Hα((0, T ), L2(Ω)).We have∫ T

0

(0Dα

t w(t), w(t))L2(Ω)

dt =

∫ T

0

(0D

α/2t w(t), tD

α/2T w(t)

)L2(Ω)

dt

=

∫ T

0

(0D

α/2t w(t), 0D

α/2t w(t)

)L2(Ω)

dt

P r o o f. We have, by Lemmas 2.3, 2.1,∫ T

0

(0Dα

t w(t), w(t))L2(Ω)

dt (4.5)

=

∫ T

0

∫Ω

(0Dα

t w(x, t))w(x, t) dxdt

=

∫Ω

(0Dα

t w(x, ·), w(x, ·))L2(0,T )

dx (4.6)

=

∫Ω

(0D

α/2t w(x, ·), tDα/2

T w(x, ·))L2(0,T )

dx

=

∫ T

0

(0D

α/2t w(t), tD

α/2T w(t)

)L2(Ω)

dt.


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Similarly, we can prove

∫ T

0

(0Dα

t w(t), w(t))L2(Ω)

dt =

∫ T

0

(0D

α/2t w(t), 0D

α/2t w(t)

)L2(Ω)

dt.

The proof is complete.2

We now come to the main theorem in this section.

Theorem 4.1. Let uh and u be the solutions of (4.1) and (4.3). Then

∫ T

0‖0Dα/2

t (uh(t)− u(t))‖2L2(Ω) dt ≤ Ch2r∫ T

0‖0Dα/2

t u(t)‖2Hr(Ω) dt.

P r o o f. We write

uh − u = θ + ρ, where θ = uh −Rhu, ρ = Rhu− u.

The second term is easily bounded by Lemma 3.1 and has the obviousestimates∫ T

0‖0Dα/2

t ρ(t)‖2L2(Ω) dt ≤ Ch2r∫ T

0‖0Dα/2

t u(t)‖2Hr(Ω) dt. (4.7)

In order to estimate θ, we note that by our definitions,

(0Dαt θ(t), χ)L2(Ω) + (∇θ(t),∇χ)L2(Ω)

= (0Dαt (uh(t)−Rhu(t)), χ)L2(Ω) + (∇(uh(t)−Rhu(t)),∇χ)L2(Ω)

= (0Dαt uh(t), χ)L2(Ω) + (∇uh(t),∇χ)L2(Ω) − (0D

αt Rhu(t), χ)L2(Ω) + (∇Rhu(t),∇χ)L2(Ω)

= (f(t), χ)L2(Ω) − (0Dαt Rhu(t), χ)L2(Ω) + (∇Rhu(t),∇χ)L2(Ω)

= (0Dαt u(t), χ)L2(Ω) − (0D

αt Rhu(t), χ)L2(Ω)

= ((I −Rh)0Dαt Rhu(t), χ)L2(Ω) = (0D

αt ρ(t), χ)L2(Ω), ∀ χ ∈ Sh.

Choose χ = θ(t) and integrating on both sides with respect to t on[0, T ], we obtain

∫ T

0

(0Dα

t θ(t), θ(t))L2(Ω)

dt+

∫ T

0

(∇θ(t),∇χ

)L2(Ω)

dt =

∫ T

0

(0Dα

t ρ(t), θ(t))L2(Ω)

dt.


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By Lemma 4.2, we have, for any small ε > 0,∫ T

0‖0Dα/2

t θ(t)‖2L2(Ω) dt+

∫ T

0‖∇θ(t)‖2L2(Ω) dt

=

∫ T

0(0D

αt ρ(t), θ(t))L2(Ω) dt =

∫ T

0(0D

α/2t ρ(t), 0D

α/2t θ(t))L2(Ω) dt

≤∫ T

0

(Cε‖0Dα/2

t ρ(t)‖2L2(Ω) dt+ ε‖0Dα/2t θ(t)‖2L2(Ω)

)dt.

For sufficiently small ε > 0, we get, by (4.7)∫ T

0‖0Dα/2

t θ(t)‖2L2(Ω) dt+

∫ T

0‖∇θ(t)‖2L2(Ω) dt

≤ C

∫ T

0‖0Dα/2

t ρ(t)‖2L2(Ω) dt ≤ Ch2r∫ T

0‖0Dα/2

t u(t)‖2Hr(Ω) dt. (4.8)

Combining (4.7) with (4.8), we complete the proof of the theorem.2

Corollary 4.1. Let uh and u be the solutions of (4.1) and (4.3).Then ∫ T

0‖uh(t)− u(t)‖2L2(Ω) dt ≤ Ch2r

∫ T

0‖0Dα/2

t u(t)‖2Hr(Ω) dt. (4.9)

P r o o f. Note that, by Theorem 4.1,We have∫ T

0‖uh(t)− u(t)‖2L2(Ω) dt =

∫ T

0

∫Ω|uh(x, t)− u(x, t)|2 dxdt

=

∫Ω

∫ T

0|uh(x, t)− u(x, t)|2 dtdx ≤

∫Ω

∫ T

0|0Dα

t (uh(x, t)− u(x, t))|2 dtdx

≤∫ T

0‖0Dα

t (uh(t)− u(t)‖2L2(Ω) dt ≤ Ch2r∫ T

0‖0Dα/2

t u(t)‖2Hr(Ω) dt,

which is (4.9). The proof of the Lemma is complete. 2

5. Numerical simulations

In this section, we present some numerical results by using the finiteelement method for solving the time-fractional partial differential equation(1.1) – (1.3). The numerical results are consistent with our theoreticalresults. We can see that convergence rate of numerical solutions is of order


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2 − α as the time stepsize tends to zero and of order 2 as the space stepsize tends to zero, on condition that the exact solution is smooth.

Example 5.1. Consider

R0D

αt u(x, t)−

d2

dx2u(x, t) = f(x, t), t ∈ [0, T ], 0 < x < 1, (5.1)

u(x, 0) = 0, 0 < x < 1, (5.2)

u(0, t) = u(1, t) = 0, t ∈ [0, T ], (5.3)

where

f(x, t) =2

Γ(3− α)t2−α sin(2πx) + 4π2 sin(2πx)t2.

The exact solution is u(x, t) = t2 sin 2πx.

The main purpose of these experiments is to check the convergencerate of the numerical solutions with respect to the fractional order α. Weuse the linear finite element method and therefore the convergence order isO(∆t2−α +∆x2).

In the first test, we fix T = 1, α = 0.5 and ∆x = 0.001 which is smallenough such that the space discretization errors are negligible as comparedwith the time errors. We choose stepsize ∆t = 1/2i (i = 1, · · · , 5), then weobtain Table 1 with the estimated convergence rate when α = 0.5, tendingto a limit close to 1.5. In the same way, we can plot the errors in thelogscale as functions of the log(∆t−1) for α = 0.2, 0.5, 0.9 in Figures 1, 2,3 and obtain the convergence rates. Here we investigate both the L2-normand the H1-norm in space.

∆x ∆t H1-norm 2-norm estimated cvgce. rates0.001 0.5000 0.01822017 0.002500600.001 0.2500 0.00675406 0.00092695 1.43170.001 0.1250 0.00245740 0.00033726 1.45860.001 0.0625 0.00087822 0.00012053 1.48450.001 0.03125 0.00030493 4.18492564e-05 1.5261

Table 1. Example 1, Fix α = 0.5 ∆x = 0.001

In Figures 1-3 , one can observe that the error curves are all nearlystraight lines. The convergence orders are the slopes of the lines respec-tively. By example 1, we can observe that the convergence order of themethod with respect to the time step is 2 − α, which have been shown inTable 2.


2012:RMA:7


α 0.1 0.2 0.3 0.4 0.5est. cvgce. rate 1.8805 1.8343 1.6786 1.5712 1.4752

α 0.6 0.7 0.8 0.9est. cvgce. rate 1.3818 1.2882 1.1935 1.0976

Table 2. Convergence rates in Example 1

0.5 1 1.5 2 2.5 3 3.5−13

−12

−11

−10

−9

−8

−7

−6

−5

∆ t in logscale

erro

rs in

logs

cale

H1−norm2−norm

Figure 1. Example 1, H1 norm and L2 norm of errors withα = 0.2 ∆x = 0.001

0.5 1 1.5 2 2.5 3 3.5−11

−10

−9

−8

−7

−6

−5

−4

∆ t in logscale

erro

rs in

logs

cale

H1−norm2−norm



2012:RMA:7


0.5 1 1.5 2 2.5 3 3.5−8

−7

−6

−5

−4

−3

−2

∆ t in logscale

erro

rs in

logs

cale

H1−norm2−norm


On the other hand, if we fix ∆t small enough, then the convergencerate of space discretization errors can be shown clearly (see Table 3 andfigure 4). In this case α = 0.5 , the limiting vale of the convergence rate isconsistent with the value of 2 that is expected from the theory.

∆x ∆t H1-norm 2-norm est. cvge. rates0.2500 0.001 0.79733222 0.108857420.1250 0.001 0.23998057 0.03202152 1.76530.0625 0.001 0.06266713 0.00842864 1.92570.03125 0.001 0.01584965 0.00215046 1.97070.015625 0.001 0.00397465 0.00054228 1.9875

Table 3. Example 1, Fix α = 0.5 ∆t = 0.001

Example 5.2. Consider

R0D

αt u(x, t)−

d2

dx2u(x, t) = f(x, t), t ∈ [0, T ], 0 < x < 1, (5.4)

u(x, 0) = 0, x ∈ Ω, (5.5)

u(0, t) = u(1, t) = 0, t ∈ [0, T ]. (5.6)

The exact solution is u(x, t) = sinπt sinπx.

In this example, we fix T = 1, α = 0.5 and ∆x = 0.0002. We obtainthe convergence rate in Table 4. In Figure 5, we plot the errors in logscale


2012:RMA:7


1 1.5 2 2.5 3 3.5 4 4.5−8

−7

−6

−5

−4

−3

−2

−1

0

∆ x in logscale

erro

rs in

logs

cale

H1−norm2−norm

Figure 4. Example 1, H1 norm and L2 norm of errors withα = 0.5 ∆t = 0.001

as functions of time stepsize log(∆t−1) for α = 0.5. Here we can observemuch better convergence than predicted by the theory, and this is worthyof further investigation.

∆x ∆t H1-norm 2-norm est. cvge. rates0.0002 0.5000 0.10734868 0.025917700.0002 0.2500 0.02757097 0.00665659 1.96110.0002 0.1250 0.00541156 0.00130654 2.34900.0002 0.0625 0.00057297 0.00013833 3.2395

Table 4. Example 2, Fix α = 0.5 ∆x = 0.002.

6. Acknowledgements

The work of the second author was carried out during her stay at theUniversity of Chester, which is supported financially by China ScholarshipCouncil (CSC[2010]3006, No. 2010612215), P. R. China.

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2012:RMA:7


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1 Department of MathematicsUniversity of ChesterParkgate RoadChester, CH1 4BJ, UK

e-mail: [email protected] Received: May 8, 2011


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2 Department of MathematicsHarbin Institute of TechnologyHarbin, 150001, P.R. China

e-mail: [email protected]

3 Department of MathematicsUniversity of ChesterParkgate RoadChester, CH1 4BJ, UK

e-mail: [email protected]


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