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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 171620, 20 pages
Research Article

A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation

1Department of Mathematics, Huaibei Normal University, Huaibei 235000, China
2Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China
3Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 13 May 2011; Revised 27 July 2011; Accepted 27 July 2011

Academic Editor: Delfim Soares Jr.

Copyright © 2011 Yunying Zheng and Zhengang 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 𝐽(𝑥,𝑡)=𝑎𝜕𝑐𝜕𝑥+𝑏𝑐,(1.1) where 𝐽 is the diffusive flow. Inserting the above equation into the equation of mass conservation 𝜕𝐽𝜕𝑥=𝜕𝑐𝜕𝑡,(1.2) 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, (𝑐𝑐)2𝑡.

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 (𝑐𝑐)2𝑡2𝛼 when 𝛼>1/2, or does the subdiffusion rate (𝑐𝑐)2𝑡2𝛼 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𝜕𝑡=𝐷𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢+𝑏(𝑢)𝐷𝑢+𝑓(𝑥,𝑡,𝑢),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 [110]. 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, 1120].

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𝜕𝑡=𝐷𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢],+𝑏(𝑢)𝐷𝑢+𝑓(𝑥,𝑡,𝑢),𝑥Ω,𝑡(0,𝑇𝑢𝑡=0=𝜑(𝑥),𝑥Ω,𝑢𝜕Ω],=𝑔,𝑡(0,𝑇(1.4) 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 of𝑥,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 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 𝑢(𝑥) are defined as follows 𝑎𝐼𝛼𝑥1𝑢(𝑥)=Γ(𝛼)𝑥𝑎(𝑥𝑠)𝛼1𝑢(𝑠)𝑑𝑠,𝑥𝐼𝛼𝑏1𝑢(𝑥)=Γ(𝛼)𝑏𝑥(𝑠𝑥)𝛼1𝑢(𝑠)𝑑𝑠,(2.1) where 𝛼>0, and Γ() is the Gamma function.

Definition 2.2. The 𝛼th order Caputo derivative of function 𝑢(𝑥) is defined as, 𝑎𝐷𝛼𝑥𝑢(𝑥)=𝑎𝐼𝑥𝑛𝛼𝑑𝑛𝑢(𝑥)𝑑𝑥𝑛,𝑛1<𝛼<𝑛𝑍+,𝑥𝐷𝛼𝑏𝑢(𝑥)=(1)𝑛𝑥𝐼𝑏𝑛𝛼𝑑𝑛𝑢(𝑥)𝑑𝑥𝑛,𝑛1<𝛼<𝑛𝑍+.(2.2) The 𝛼th order Riemann-Liouville derivative of function 𝑢(𝑥) is defined by changing the order of integration and differentiation.

Lemma 2.3 (see [8]). If 𝑢(0)=𝑢(0)==𝑢(𝑛1)(0)=0, then the Caputo fractional derivative is equal to the Riemann-Liouville derivative.

Definition 2.4. The fractional derivative space 𝐽𝛼(Ω) is defined as follows: 𝐽𝛼(Ω)=𝑢𝐿2(Ω)𝑎𝐷𝛼𝑥𝑢𝐿2(Ω),𝑛1𝛼<𝑛,(2.3) endowed with the seminorm |𝑢|𝐽𝛼=𝑎𝐷𝛼𝑥𝑢𝐿2(Ω),(2.4) and the norm 𝑢𝐽𝛼=|𝑢|2𝐽𝛼+[𝛼]𝑘𝐷𝑘𝑢21/2.(2.5)

Let 𝐽𝛼0(Ω) denote the closure of 𝐶0(Ω) with respect to the above norm and seminorm.

Definition 2.5. Define the seminorm |𝑢|𝐻𝛼=|𝑖𝑤|𝛼𝐹(𝑢)𝐿2(Ω),(2.6) and the norm 𝑢𝐻𝛼=|𝑢|2𝐻𝛼+[𝛼]𝑘𝐷𝑘𝑢21/2,(2.7) where 𝑖 is the imaginary unit, and 𝐹 is the Fourier transform, and which can define another fractional derivative space 𝐻𝛼(Ω).

Let 𝐻𝛼0(Ω) denote the closure of 𝐶0(Ω) with respect to the norm and seminorm.

Definition 2.6. The fractional space 𝐽𝛼𝑠(Ω) is defined below 𝐽𝛼𝑠(Ω)=𝑢𝐿2(Ω)𝑎𝐷𝛼𝑥𝑢𝐿2(Ω),𝑥𝐷𝛼𝑏𝑢𝐿2(Ω),𝑛1𝛼<𝑛,(2.8) endowed with the seminorm |𝑢|𝐽𝛼𝑠=|||𝑎𝐷𝛼𝑥𝑢,𝑥𝐷𝛼𝑏𝑢1/2|||𝐿2(Ω),(2.9) and the norm 𝑢𝐽𝛼𝑠=[𝛼]𝑘𝐷𝑘𝑢2+|𝑢|2𝐽𝛼𝑠1/2.(2.10)

Theorem 2.7 (see [3, 6]). 𝐽𝛼𝑠, 𝐽𝛼, and 𝐻𝛼 are equal with equivalent seminorm and norm.
The following are some useful results.

Lemma 2.8 (see [3]). For 𝑢𝐽𝛼0(Ω), 0<𝛽<𝛼, then 𝑎𝐷𝛼𝑥𝑢(𝑥)=𝑎𝐷𝑥𝑎𝛼𝛽𝐷𝛽𝑥𝑢.(2.11)

Lemma 2.9 (see [2]). For 𝑢𝐻𝛼0(Ω), one has 𝑢𝐿2(Ω)𝑐|𝑢|𝐻𝛼0.(2.12) For 0<𝛽<𝛼, |𝑢|𝐻𝛽0(Ω)𝑐|𝑢|𝐻𝛼0.(2.13)

Since 𝐽𝛼𝑠, 𝐽𝛼, and 𝐻𝛼 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 0𝛽<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𝜕𝑡=𝐷𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢],],+𝑏(𝑢)𝐷𝑢+𝑓(𝑥,𝑡,𝑢),(𝑥,𝑡)Ω×(0,𝑇𝑢(𝑥,𝑡)=𝜙(𝑥,𝑡),𝑥𝜕Ω×(0,𝑇𝑢(𝑥,0)=𝑔(𝑥),𝑥Ω.(3.1) We always assume that 0<𝑚<𝑎(𝑢)<𝑀,0<𝑚<𝑏(𝑢)<𝑀,0<𝑚<𝑓(𝑢)<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 𝑥Ω,𝑡(0,𝑇], and 𝑠,𝑡𝑅, ||||𝑎(𝑥,𝑡,𝑠)𝑎(𝑥,𝑡,𝑟)𝐿|𝑠𝑟|,(3.3)||||𝑏(𝑥,𝑡,𝑠)𝑏(𝑥,𝑡,𝑟)𝐿|𝑠𝑟|,(3.4)||||𝑓(𝑥,𝑡,𝑠)𝑓(𝑥,𝑡,𝑟)𝐿|𝑠𝑟|.(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 𝑣𝐻𝛼0(Ω) in both sides yields Ω𝜕𝑢𝜕𝑡𝑣𝑑𝑥=Ω𝐷𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢𝑣𝑑𝑥+Ω𝑏(𝑢)𝐷𝑢𝑣𝑑𝑥+Ω𝑓(𝑥,𝑡,𝑢)𝑣𝑑𝑥.(3.6)

Rewriting the above expression yields Ω𝜕𝑢𝜕𝑡𝑣𝑑𝑥+Ω𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢𝐷𝑣𝑑𝑥Ω𝑏(𝑢)𝐷𝑢𝑣𝑑𝑥=Ω𝑓(𝑥,𝑡,𝑢)𝑣𝑑𝑥.(3.7)

We define the associated bilinear form 𝐴𝐽𝛼0(Ω)×𝐽𝛼0(Ω)𝑅 as 𝐴(𝑢,𝑣)=𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢,𝐷𝑣(𝑏(𝑢)𝐷𝑢,𝑣),(3.8) where (,) denotes the inner product on 𝐿2(Ω) and 𝐽𝛼0(Ω).

For given 𝑓𝐽𝛼(Ω), we define the associated function 𝐹𝐽𝛼0(Ω)𝑅 as 𝐹(𝑣)=𝑓,𝑣.(3.9)

Definition 3.1. A function 𝑢𝐽𝛼0(Ω) is a variational solution of Problem 1 provided that 𝜕𝑢𝜕𝑡,𝑣+𝐴(𝑢,𝑣)=𝐹(𝑣),𝑣𝐽𝛼0(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}𝑁𝑛=0 be a uniform partition of the time interval (0,𝑇] such that 𝑡𝑛=𝑛𝜏, where 𝜏=𝑇/𝑁, and let 𝑡𝑛1/2=𝑡𝑛𝜏/2. Throughout the paper, we use the following notation for a function 𝜙: 𝜙𝑛𝑡=𝜙𝑛,𝜕𝑡𝜙𝑛=𝜙𝑛𝜙𝑛1𝜏,𝜙𝑛=𝜙𝑛+𝜙𝑛12,𝜙𝑛=3𝜙𝑛1𝜙𝑛22.(3.11)

Let 𝒦={𝐾} be a partition of spatial domain Ω. Define 𝑘 as the diameter of the element 𝐾 and =max𝐾𝒦𝐾. And let 𝑆 be a finite element space 𝑆=𝑣𝐻𝛼0(Ω)𝑣𝐾𝑃𝑟1(𝐾),𝐾𝒦,(3.12) where 𝑃𝑟1(𝐾) is the set of polynomials of degree 𝑟1 on a given domain 𝐾. And the functions in 𝑆 are continuous on Ω. Our fully discrete quadrature scheme to solve Problem 1 is to find 𝑢: for 𝑣𝑆 such that 𝜕𝑡𝑢𝑛+𝑎,𝑣̃𝑢𝑛𝑎𝐷𝛽𝑥𝑢𝑛𝑏,𝐷𝑣̃𝑢𝑛𝐷𝑢𝑛=𝑓,𝑣̃𝑢𝑛,𝑣.(3.13)

The linear systems in the above equation requires selecting the value of 𝑢0 and 𝑢1. Given 𝑢0 depending on the initial data 𝑔(𝑥), we select 𝑢1 by solving the following predictor-corrector linear systems: 𝑢1,0𝑢0𝜏+𝑎𝑢,𝑣0𝑎𝐷𝛽𝑥𝑢1,0+𝑢02𝑏𝑢,𝐷𝑣0𝐷𝑢1,0+𝑢02=𝑓𝑢,𝑣0,𝑢,𝑣1𝑢0𝜏+𝑎𝑢,𝑣1,0+𝑢02𝑎𝐷𝛽𝑥𝑢1+𝑢02𝑏𝑢,𝐷𝑣1,0+𝑢02𝐷𝑢1+𝑢02=𝑓𝑢,𝑣1,0+𝑢02.,𝑣(3.14)

Lemma 3.2. For 𝑢,𝑣,𝑤𝐽𝛼𝑠,0(Ω),0<𝑚𝑎(𝑢)𝑀,𝛼=(1+𝛽)/2, there exist constants 𝛾1,𝛾2 such that 𝑎(𝑢)𝑎𝐷𝛽𝑥𝑢,𝐷𝑣𝛾1𝑢𝛼𝑣𝛼,𝑎(𝑤)𝑎𝐷𝛽𝑥𝑣,𝐷𝑣𝛾2𝑣2𝛼.(3.15)

Proof. With the assumption of 𝑎(𝑢) in (3.3) and the property of dual space 𝑎(𝑤)𝑎𝐷𝛽𝑥𝑢,𝐷𝑣𝑎(𝑤)𝑎𝐷𝛽𝑥𝑢1𝛼𝐷𝑣(1𝛼)𝑀𝑐𝑢1𝛼+𝛽𝑣(1𝛼)+1𝛾1𝑢𝛼𝑣𝛼,𝑎(𝑤)𝑎𝐷𝛽𝑥𝑣,𝐷𝑣=𝐷𝑎(𝑤)𝑎𝐷𝛽𝑥𝑣,𝑣=𝑎𝐷𝑥(1𝛽)/2𝑎(𝑤)𝑎𝐷𝛽𝑥𝑣,𝑥𝐷𝑏(1+𝛽)/2𝑣𝑚|𝑣|2𝐽𝛼𝑠𝛾2𝑣2𝛼.(3.16)

Lemma 3.3 (see [2]). For Ω𝑅𝑛,𝛼>𝑛/4,𝑣,𝑤𝐻𝛼0(Ω),𝜀>0, one has (𝑣𝑏(𝑤),𝑣)𝑐0(𝑞𝜀)𝑝/𝑞𝑝𝑏(𝑤)𝑝𝑣2+𝜀𝑣2𝛼,(3.17) where 𝑝=4𝛼/(4𝛼𝑛),𝑞=4𝛼/𝑛.

Theorem 3.4. Let 𝑢𝑛 be bounded, then for a sufficiently small step 𝜏, there exists a unique solution 𝑢𝑛𝑆 satisfying scheme (3.13).

Proof. As scheme represents a finite system of problem, the continuity and coercivity of (𝑢𝑛,𝜔𝑛)/𝜏+𝐴(𝑢𝑛,𝜔𝑛) is the sufficient and essential condition for the existence and uniqueness of 𝑢𝑛. Let 𝑣=𝑢𝑛,𝑤=𝜔𝑛, then (𝑣,𝑣)𝜏+𝐴(𝑣,𝑣)=(𝑣,𝑣)𝜏+𝑎(𝑤)𝑎𝐷𝛽𝑥𝑣,𝐷𝑣(𝑏(𝑤)𝐷𝑣,𝑣)𝑣2𝜏+𝛾2𝑣𝛼𝑐0𝐷𝑏(𝑤)2𝑣2𝜀𝑣2𝛼=𝛾2𝜀𝑣2𝛼+𝜏1𝑐0𝐷𝑏(𝑤)2𝑣2𝑐𝑣2𝛼.(3.18) For the chosen sufficiently small 𝜏, the above inequality holds. (𝑣,𝑤)𝜏+𝐴(𝑣,𝑤)=(𝑣,𝑤)𝜏+𝑎(𝑢)𝑎𝐷𝛽𝑥𝑣,𝐷𝑤+(𝐷𝑏(𝑢)𝑣,𝐷𝑤)𝑢𝑤𝜏+𝛾1𝑣𝛼𝑤𝛼+𝑣𝐷(𝑏(𝑢)𝑤)𝑢𝑤𝜏+𝛾1𝑣𝛼𝑤𝛼+𝑀𝑣𝑤𝑐𝑣𝛼𝑤𝛼.(3.19) Hence, the scheme (3.13) is uniquely solvable for 𝑢𝑛.
Let 𝜌𝑛=𝑃𝑢𝑛𝑢𝑛, and 𝜃𝑛=𝑢𝑛𝑃𝑢𝑛, then 𝑢𝑛𝑢𝑛=𝑢𝑛𝑃𝑢𝑛+𝑃𝑢𝑛𝑢𝑛=𝜃𝑛+𝜌𝑛,(3.20) where 𝑃𝑢𝑛 is a Rits-Galerkin projection operator defined as follows: 𝑎(𝑤)𝑎𝐷𝛽𝑥𝑢𝑛𝑃𝑢𝑛𝑎𝑢,𝐷𝑣=0,0𝑎𝐷𝛽𝑥𝑢𝑛𝑃𝑢𝑛,𝐷𝑣=0.(3.21)

Lemma 3.5. Let 𝑎(𝑢),𝑏(𝑢) be smooth functions on Ω, 0<𝑚𝑎(𝑢),𝑏(𝑢)𝑀, and 𝑃𝑢𝑛 is defined as above, then 𝑎𝐷𝛼𝑥𝑢𝑛𝑃𝑢𝑛𝑐𝑘+1𝛼𝑢𝑘+1,𝑃𝑢𝑛𝑢𝑛𝑐𝑘+1𝑢𝑘+1.(3.22)

Proof. Using the definition of 𝑃𝑢𝑛, one gets 𝑎𝐷𝛼𝑥𝑃𝑢𝑛𝑢𝑛2=||𝑎𝐷𝛼𝑥𝑃𝑢𝑛𝑢𝑛,𝑎𝐷𝛼𝑥𝑃𝑢𝑛𝑢𝑛||𝑐𝑎𝐷𝛼𝑥𝑃𝑢𝑛𝑢𝑛𝑎𝐷𝛼𝑥(𝜒𝑢𝑛),(3.23) where 𝜒𝑆. Utilizing the interpolation of 𝐼𝑢𝑛 leads to 𝑎𝐷𝛼𝑥𝑃𝑢𝑛𝑢𝑛inf𝜒𝑆𝑐𝜒𝑢𝛼𝐼𝑐𝑢𝑛𝑢𝑛𝛼𝑐𝑘+1𝛼𝑢𝑘+1.(3.24) Next we estimate 𝑃𝑢𝑛𝑢𝑛. For all𝜙𝐿2(Ω), 𝑤 is the solution of the following equation: 𝑎𝐷𝑥2𝛼𝑤=𝜙,𝑤Ω,𝑤=0,𝑤𝜕Ω.(3.25)
So we have 𝑤2𝛼𝛾3𝜙.(3.26) For all𝜒𝑆, with the help of approximation properties of 𝑆 and the weak form, we can obtain 𝑃𝑢𝑛𝑢𝑛𝑃,𝜙=𝑢𝑛𝑢𝑛,𝑎𝐷𝑥2𝛼𝑤=𝑥𝐷𝛼𝑏𝑃𝑢𝑛𝑢𝑛,𝑎𝐷𝛼𝑥𝑤=𝑥𝐷𝛼𝑏𝑃𝑢𝑛𝑢𝑛,𝑎𝐷𝛼𝑥𝑃(𝑤𝜒)𝑢𝑛𝑢𝑛𝛼𝑤𝜒𝛼𝑃𝑢𝑛𝑢𝑛𝛼inf𝜒𝑆𝑤𝜒𝛼𝑐𝑟𝛼𝑢𝑟𝛼𝑤2𝛼=𝑐𝑟𝑢𝑟𝑃𝜙,𝑢𝑛𝑢𝑛=sup0𝜙𝐿2(Ω)𝑃𝑢𝑛𝑢𝑛,𝜙𝜙𝑐𝑟𝑢𝑟.(3.27)

Lemma 3.6 (see [21]). Let 𝑇,0<1, denote a quasiuniform family of subdivisions of a polyhedral domain Ω𝑅𝑑. Let (𝐾,𝑃,𝑁) be a reference finite element such that 𝑃𝑊𝑙,𝑝(𝐾)𝑊𝑚,𝑞(𝐾) is a finite-dimensional space of functions on 𝐾,𝑁 is a basis for 𝑃, where 1𝑝,1𝑝, and 0𝑚𝑙. For 𝐾𝑇, let (𝐾,𝑃𝐾,𝑁𝐾) be the affine equivalent element, and 𝑉=𝑣𝑣 is measurable and 𝑣|𝐾𝑃𝐾,forall𝐾𝑇. Then there exists a constant 𝐶=𝐶(𝑙,𝑝,𝑞) such that 𝑘𝑇𝑣2𝑊𝑙,𝑝(𝐾)1/𝑝𝐶𝑚𝑙+min(0,𝑑/𝑝𝑑/𝑞)𝑘𝑇𝑣𝑞𝑊𝑚,𝑞(𝐾)1/𝑞.(3.28) The following Gronwall’s lemma is useful for the error analysis later on.

Lemma 3.7 (see [2]). Let Δ𝑡,𝐻 and 𝑎𝑛,𝑏𝑛,𝑐𝑛,𝛾𝑛 (for integer 𝑛0) be nonnegative numbers such that 𝑎𝑁+Δ𝑡𝑁𝑛=0𝑏𝑛Δ𝑡𝑁𝑛=0𝛾𝑛𝑎𝑛+Δ𝑡𝑁𝑛=0𝑐𝑛+𝐻,(3.29) for 𝑁0. Suppose that Δ𝑡𝛾𝑛<1, for all 𝑛, and set 𝜎𝑛=(1Δ𝑡𝛾𝑛)1. Then 𝑎𝑁+Δ𝑡𝑁𝑛=0𝑏𝑛expΔ𝑡𝑁𝑛=0𝜎𝑛𝛾𝑛Δ𝑡𝑁𝑛=0𝑐𝑛+𝐻,(3.30) for 𝑁0.
The following norms are also used in the analysis: |𝑣|,𝑘=max0𝑛𝑁𝑣𝑛𝑘,|𝑣|0,𝑘=𝑁𝑛=0𝜏𝑣𝑛2𝑘1/2.(3.31)

Theorem 3.8. Assume that Problem 1 has a solution 𝑢 satisfying 𝑢𝑡𝑡,𝑢𝑡𝑡𝑡𝐿2(0,𝑇,𝐿2(Ω)) with 𝑢,𝑢𝑡𝐿2(0,𝑇,𝐻𝑘+1). If Δ𝑡𝑐, then the finite element approximation is convergent to the solution of Problem 1 on the interval (0,T], as Δ𝑡,0. The approximation 𝑢 also satisfies the following error estimates 𝑢𝑢0,𝛼𝐶𝑘+1𝑢𝑡0,𝑘+1+𝑘+1𝛼𝑢0,𝑘+1+𝜏2𝑢𝑡𝑡0,0+𝜏𝑘+1𝛼𝑢𝑡𝑡0,𝑘+1+𝜏2𝑢𝑡𝑡𝑡0,0,(3.32)𝑢𝑢,0𝐶𝑘+1𝑢𝑡0,𝑘+1+𝑘+1𝛼𝑢0,𝑘+1+𝜏2𝑢𝑡𝑡𝑡0,0+𝜏𝑘+1𝛼𝑢𝑡𝑡0,𝑘+1+𝜏2𝑢𝑡𝑡0,0+𝑘+1𝑢2,𝑘+1.(3.33)

Proof. For 𝑡=𝑡𝑛𝜏/2=𝑡𝑛1/2,𝑛=0,1,,𝑁, find 𝑢𝑛1/2 such that 𝜕𝑡𝑢𝑛1/2+𝑎𝑢,𝑣𝑛1/2𝑎𝐷𝑥𝑢𝑛1/2𝑏𝑢,𝐷𝑣𝑛1/2𝐷𝑢𝑛1/2=𝑓𝑢,𝑣𝑛1/2,𝑣.(3.34)
Subtracting the above equation from the fully discrete scheme (3.13), and substituting 𝑢𝑛𝑢𝑛=(𝑢𝑛𝑃𝑢𝑛)+(𝑃𝑢𝑛𝑢𝑛)=𝜃𝑛+𝜌𝑛 into it, we obtain the following error formulation relating to 𝜃𝑛 and 𝜌𝑛: 𝜕𝑡𝜃𝑛+𝑎,𝑣̃𝑢𝑛𝑎𝐷𝛽𝑥𝜃𝑛𝑏,𝐷𝑣̃𝑢𝑛𝐷𝜃𝑛=𝑎,𝑣̃𝑢𝑛𝑎𝐷𝛽𝑥𝐼𝑢𝑛+𝑏,𝐷𝑣̃𝑢𝑛𝑎𝐷𝛽𝑥𝐼𝑢𝑛+𝜕,𝑣𝑡𝑢𝑛1/2,𝑣𝜕𝑡𝐼𝑢𝑛+𝑎𝑢,𝑣𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑏𝑢,𝐷𝑣𝑛1/2𝐷𝑢𝑛1/2+𝑓,𝑣̃𝑢𝑛𝑓𝑢,𝑣𝑛1/2𝑎,𝑣=̃𝑢𝑛𝑎𝐷𝛽𝑥𝜌𝑛+𝑎𝑢,𝐷𝑣𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑎̃𝑢𝑛𝑎𝐷𝛽𝑥𝑢𝑛+𝑏,𝐷𝑣̃𝑢𝑛𝐷𝜌𝑛+𝑏,𝑣̃𝑢𝑛𝐷𝑢𝑛𝑏𝑢𝑛1/2𝐷𝑢𝑛1/2+𝑓,𝐷𝑣̃𝑢𝑛𝑢𝑓𝑛1/2+𝜕,𝑣𝑡𝑢𝑛1/2𝜕𝑡𝐼𝑢𝑛,𝑣=𝑅1(𝑣)+𝑅2(𝑣)+𝑅3(𝑣)+𝑅4(𝑣)+𝑅5(𝑣)+𝑅6(𝑣).(3.35) Setting 𝑣=𝜃𝑛, we obtain 𝜕𝑡𝜃𝑛,𝜃𝑛+𝑎(̃𝑢𝑛)𝑎𝐷𝛽𝑥𝜃𝑛,𝐷𝜃𝑛𝑏(̃𝑢𝑛)𝐷𝜃𝑛,𝜃𝑛=𝑅1𝜃𝑛+𝑅2𝜃𝑛+𝑅3𝜃𝑛+𝑅4𝜃𝑛+𝑅5𝜃𝑛+𝑅6𝜃𝑛.(3.36) Note that 𝜕𝑡𝜃𝑛,𝜃𝑛=𝜃𝑛𝜃𝑛1𝜏,𝜃𝑛+𝜃𝑛12=12𝜏𝜃𝑛2𝜃𝑛12.(3.37) According to (3.2) and Lemma 3.2, we have 𝑎(̃𝑢𝑛)𝑎𝐷𝛽𝑥𝜃𝑛,𝐷𝜃𝑛|||𝑚𝜃𝑛|||2𝛼||𝜃𝑐𝑛||2𝛼+||𝜃𝑛1||2𝛼.(3.38) From Lemma 3.3, the following inequality can be derived: 𝑏(̃𝑢𝑛)𝜃𝑛,𝐷𝜃𝑛𝑐0𝜀𝑐12𝐷𝑏(̃𝑢𝑛)𝑐2𝜃𝑛2+𝜀3𝜃𝑛2𝛼=𝑐0𝜀𝑐12𝐷𝑏(̃𝑢𝑛)𝑐2𝜃𝑛+𝜃𝑛122+𝜀3𝜃𝑛+𝜃𝑛122𝛼𝑐3𝜀𝑐12𝐷𝑏(̃𝑢𝑛)𝑐2𝜃𝑛2+𝜃𝑛12+𝑐4𝜀3𝜃𝑛2𝛼+𝜃𝑛12𝛼.(3.39)
Substituting (3.37)–(3.39) into (3.36) then multiplying (3.36) by 2𝜏, summing from 𝑛=1 to 𝑁, we have 𝜃𝑛2𝜃22+𝜏𝑁𝑛=12𝑚𝑐2𝑐4𝜀3𝜃𝑛2𝛼+𝜃𝑛12𝛼2𝜏𝑁𝑛=1𝑐3𝜀𝑐12𝐷𝑏̃𝑢𝑛𝑐2𝜃𝑛2+𝜃𝑛12+2𝜏𝑁𝑛=3𝑅1𝜃𝑛+𝑅2𝜃𝑛+𝑅3𝜃𝑛+𝑅4𝜃𝑛+𝑅5𝜃𝑛+𝑅6𝜃𝑛.(3.40) We now estimate 𝑅1 to 𝑅6 in the right hand of (3.40), 𝑅1𝜃𝑛=𝑎𝐷𝑥1𝛼𝑎̃𝑢𝑛𝑎𝐷𝛽𝑥𝜌𝑛,𝑎𝐷𝛼𝑥𝜃𝑛𝑀𝑎𝐷𝛼𝑥𝜌𝑛,𝑎𝐷𝛼𝑥𝜃𝑛𝑀𝑎𝐷𝛼𝑥𝜌𝑛𝑎𝐷𝛼𝑥𝜃𝑛𝜀4𝜃𝑛2𝛼+𝑐254𝜀4𝜌𝑛2𝛼=𝜀42𝜌𝑛+𝜌𝑛12𝛼+𝑐2516𝜀4𝜃𝑛+𝜃𝑛12𝛼𝜀4𝑐6𝜃𝑛2𝛼+𝜃𝑛12𝛼+𝑐7𝜀4𝜌𝑛2𝛼+𝜌𝑛12𝛼.(3.41)
Secondly, we deduce the estimation of 𝑅2, 𝑅2𝜃𝑛=𝑎̃𝑢𝑛𝑎𝐷𝛽𝑥𝑢𝑛,𝐷𝜃𝑛+𝑎𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2,𝐷𝜃𝑛=𝑎𝑢𝑛1/2𝑎̃𝑢𝑛𝑎𝐷𝛽𝑥𝑢𝑛,𝐷𝜃𝑛+𝑎𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛,𝐷𝜃𝑛=𝑅21+𝑅22,(3.42) where 𝑅21=𝑎𝑢𝑛1/2𝑎(̃𝑢𝑛)𝑎𝐷𝛽𝑥̃𝑢𝑛,𝐷𝜃𝑛𝑐84𝜀5𝑎𝑢𝑛1/2𝑎(̃𝑢𝑛)𝑎𝐷𝛽𝑥̃𝑢𝑛21𝛼+𝜀5𝐷𝜃𝑛2𝛼1𝑐9𝑎𝑢𝑛1/2𝑎(̃𝑢𝑛)𝑎𝐷𝛽𝑥̃𝑢𝑛21𝛼+𝜀5𝜃𝑛2𝛼𝑐9𝐿𝑢𝑛1/2̃𝑢𝑛+𝜀5𝜃𝑛2𝛼,𝑅22=𝑎𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛,𝐷𝜃𝑛𝑐104𝜀6𝑎𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛21𝛼+𝜀6𝐷𝜃𝑛21+𝛼𝑐10𝑎𝑢𝑛1/22𝑎𝐷𝛽𝑥𝑢𝑛1/2𝑎𝐷𝛽𝑥𝑢𝑛21𝛼+𝜀6𝜃𝑛2𝛼𝑐10𝑀2𝑢𝑛1/2𝑢𝑛2𝛼+𝜀6𝑐𝜃𝑛2𝛼+𝜃𝑛12𝛼.(3.43) The estimations of ̃𝑢𝑛𝑢𝑛1/2 and 𝑢𝑛𝑢𝑛1/2𝛼 can be derived as follows: ̃𝑢𝑛𝑢𝑛1/2=32𝑢𝑛1/2𝜏2𝑢𝑡𝑛1/2+𝑢𝑛1/2𝑡𝑡𝜏2!22𝜏+𝑂312𝑢𝑛1/23𝜏2𝑢𝑡𝑛1/2+𝑢𝑛1/2𝑡𝑡2!3𝜏22𝜏+𝑂3𝑢𝑛1/2𝑐11𝜏2𝑢𝑡𝑡𝑡𝑛1/2𝑐11𝜏2𝑡𝑛𝑡𝑛1𝑢𝑡𝑡(,𝑠)𝑑𝑠,𝑢𝑛𝑢𝑛1/2𝛼=𝜏1𝑡𝑛𝑡𝑛1/2𝑠𝑡𝑛2𝑢𝑡𝑡(𝑠)𝑑𝑠+𝑡𝑛1/2𝑡𝑛1𝑠𝑡𝑛12𝑢𝑡𝑡(𝑠)𝑑𝑠𝛼𝑐12𝜏𝑡𝑛𝑡𝑛1𝑢𝑡𝑡(𝑠)𝑑𝑠𝛼𝑐12𝜏𝑡𝑛𝑡𝑛1𝑢𝑡𝑡(𝑠)𝛼𝑑𝑠𝑐12𝜏𝑘+1𝛼𝑡𝑛𝑡𝑛1𝑢𝑡𝑡(𝑠)𝑘+1𝑑𝑠.(3.44) Thirdly, it is turn to consider 𝑅3, 𝑅3𝜃𝑛=𝑏̃𝑢𝑛𝐷𝜌𝑛,𝜃𝑛𝑏̃𝑢𝑛𝐷𝜌𝑛𝛼𝜃𝑛𝛼𝑐134𝜀7𝑏̃𝑢𝑛2𝜌𝑛21𝛼+𝜀7𝜃𝑛2𝛼𝑐144𝜀7𝜌𝑛21𝛼+𝜌𝑛121𝛼+𝜀7𝑐15𝜃𝑛2𝛼+𝜃𝑛12𝛼.(3.45) Next, 𝑅4𝜃𝑛=𝑏̃𝑢𝑛𝐷𝑢𝑛,𝜃𝑛𝑏𝑢𝑛1/2𝐷𝑢𝑛1/2,𝜃𝑛=𝑏̃𝑢𝑛𝐷𝑢𝑛𝑢𝑏𝑛1/2𝐷𝑢𝑛,𝜃𝑛+𝑏𝑢𝑛1/2𝐷𝑢𝑛𝐷𝑢𝑛1/2,𝜃𝑛=𝑅41+𝑅42,(3.46) where 𝑅41𝑐164𝜀8𝑏̃𝑢𝑛𝑢𝑏𝑛1/2𝐷𝑢𝑛21𝛼+𝜀8𝜃𝑛2𝛼𝑐16𝐿4𝜀8̃𝑢𝑛𝑢𝑛1/22𝐷𝑢𝑛21𝛼+𝜀8𝜃𝑛2𝛼=𝑐16𝐿4𝜀8̃𝑢𝑛̃𝑢𝑛+̃𝑢𝑛𝑢𝑛1/22𝐷𝑢𝑛21𝛼+𝜀8𝜃𝑛2𝛼𝑐17̃𝑢𝑛̃𝑢𝑛2+𝑐17̃𝑢𝑛𝑢𝑛1/22+𝜀8𝜃𝑛2𝛼𝑐17̃𝜃𝑛+̃𝜌𝑛2+𝑐17̃𝑢𝑛𝑢𝑛1/22+𝜀8𝜃𝑛2𝛼𝑐18̃𝜃𝑛2+̃𝜌𝑛2+𝑐17̃𝑢𝑛𝑢𝑛1/22+𝜀8𝜃𝑛2𝛼.(3.47) Rewriting 𝑅42 by the aid of (3.20), we have 𝑅42𝑐194𝜀9𝑢𝑛𝑢𝑛1/22+𝜀9𝜃𝑛2𝛼.(3.48) The estimation of 𝑅5 is deduced as follows: 𝑅5𝜃𝑛𝑓̃𝑢𝑛𝑢𝑓𝑛1/2𝜃𝑛𝐿̃𝑢𝑛𝑢𝑛1/2𝜃𝑛𝐿𝑐204𝜀10̃𝑢𝑛𝑢𝑛1/22+𝜀10𝜃𝑛2𝐿𝑐21̃𝜃𝑛+̃𝜌𝑛2+̃𝑢𝑛𝑢𝑛1/22+𝜀10𝜃𝑛2𝑐22̃𝜃𝑛2+̃𝜌𝑛2+𝐿𝑐21̃𝑢𝑛𝑢𝑛1/22+𝜀10𝜃𝑛2.(3.49) Last, we estimate 𝑅6, 𝑅6𝜃𝑛=𝜕𝑡𝑢𝑛1/2,𝜃𝑛𝜕𝑡𝑃𝑢𝑛,𝜃𝑛=𝜕𝑡𝑢𝑛1/2𝜕𝑢𝑛,𝜃𝑛+𝜕𝑡𝑢𝑛𝜕𝑡𝑃𝑢𝑛,𝜃𝑛=𝜕𝑡𝑢𝑛1/2𝜕𝑡𝑢𝑛,𝜃𝑛+𝜕𝑡𝜌𝑛,𝜃𝑛𝜕𝑡𝑢𝑛1/2𝜕𝑡𝑢𝑛𝜃𝑛+𝜕𝑡𝜌𝑛𝜃𝑛,(3.50) where 𝜕𝑡𝑢𝑛1/2𝜕𝑢𝑛=(2𝜏)1𝑐23𝑡𝑛𝑡𝑛1/2𝑠𝑡𝑛2𝑢𝑡𝑡𝑡(𝑠)𝑑𝑠+𝑡𝑛1/2𝑡𝑛1𝑠𝑡𝑛12𝑢𝑡𝑡𝑡(𝑠)𝑑𝑠𝑐23𝜏𝑡𝑛𝑡𝑛1𝑢𝑡𝑡𝑡(𝑠)𝑑𝑠𝑐23𝜏𝑡𝑛𝑡𝑛1𝑢𝑡𝑡𝑡(𝑠)𝑑𝑠,𝜕𝑡𝜌𝑛=𝜌𝑛𝜌𝑛1𝜏𝜏1𝑡𝑛𝑡𝑛1𝜌𝑛𝑡(𝑠)𝑑𝑠𝜏1𝑡𝑛𝑡𝑛1𝑢𝑡(𝑠)𝑑𝑠𝜏1𝑡𝑛𝑡𝑛11𝑑𝑠𝑡𝑛𝑡𝑛1𝑢𝑡=(𝑠)𝑑𝑠𝑡𝑛𝑡𝑛1𝑢𝑡(𝑠)𝑘+1𝑡𝑛𝑡𝑛1𝑢𝑡(𝑠)𝑘+1𝑑𝑠.(3.51)
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 𝜕𝑡𝜃1,0+𝑎𝑢,𝑣0𝑎𝐷𝛽𝑥𝜃1,0𝑢,𝐷𝑣𝑏0𝐷𝜃1,0𝑎𝑢,𝑣=0𝑎𝐷𝛽𝑥𝜌1,0𝑎𝑢,𝐷𝑣1/2𝑎𝐷𝛽𝑥𝑢1/2𝑢𝑎0𝑎𝐷𝛽𝑥𝑢1,0+𝑏𝑢,𝐷𝑣0𝐷𝜌1,0+𝑏𝑢,𝑣0𝐷𝑢1,0𝑢𝑏1/2𝐷𝑢1/2+𝑓𝑢,𝐷𝑣0𝑢𝑓1/2+𝜕,𝑣𝑡𝑢1/2𝜕𝑡𝐼𝑢1,0,𝑣=𝑅1(𝑣)+𝑅2(𝑣)+𝑅3(𝑣)+𝑅4(𝑣)+𝑅5(𝑣)+𝑅6(𝑣),𝜕𝑡𝜃1+𝑎𝑢,𝑣0𝑎𝐷𝛽𝑥𝜃1𝑢,𝐷𝑣𝑏0𝐷𝜃1𝑎𝑢,𝑣=0+𝑢1,02𝑎𝐷𝛽𝑥𝜌1𝑎𝑢,𝐷𝑣1/2𝑎𝐷𝛽𝑥𝑢1/2𝑢𝑎0+𝑢1,02𝑎𝐷𝛽𝑥𝑢1+𝑏𝑢,𝐷𝑣0+𝑢1,02𝐷𝜌1+𝑏𝑢,𝑣0+𝑢1,02𝐷𝑢1𝑏𝑢1/2𝐷𝑢1/2+𝑓𝑢,𝐷𝑣0+𝑢1,02𝑢𝑓1/2+𝜕,𝑣𝑡𝑢1/2𝜕𝑡𝐼𝑢1,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 𝜃1,02𝜏𝑐2𝑡1𝑡0𝑢𝑡𝑡(𝑠)2𝛼𝑑𝑠+2(𝑘+1𝛼)𝑢2𝑘+1+𝜏2𝑡1𝑡0𝑢𝑡𝑡𝑡(𝑠)2𝑑𝑠+𝑐2(𝑘+1)𝑡1𝑡0𝑢𝑡(𝑠)2𝑘+1.𝑑𝑠(3.53)
Letting 𝑣=𝜃1, applying the above result of 𝜃1,0, and using the similar estimation (see (3.53)), we get 𝜃12𝜏𝑐2𝑡1𝑡0𝑢𝑡𝑡(𝑠)2𝛼𝑑𝑠+2(𝑘+1𝛼)𝑢2𝑘+1+𝜏2𝑡1𝑡0𝑢𝑡𝑡𝑡(𝑠)2𝑑𝑠+𝑐2(𝑘+1)𝑡1𝑡0𝑢𝑡(𝑠)2𝑘+1.𝑑𝑠(3.54)
Using 𝑇=𝑁𝜏 and Gronwall’s lemma, we get ||𝜃||20,𝛼=𝑁𝑛=0𝜏𝜃2𝛼.(3.55)
Hence, using the interpolation property and ||𝑢𝑢||0,𝛼||𝜃||0,𝛼+||𝜌||0,𝛼,(3.56) the estimate (3.32) holds.
Also using the interpolation property, Gronwall’s lemma, and the approximation properties, we get ||𝑢𝑢||,0||𝜃||,0+||𝜌||,0max0𝑛𝑁||𝜃𝑛||2+2𝑘+1|𝑢|2,𝑘+1,(3.57) 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 𝑆 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 𝑋 with the standard basis of hat functions on the uniform grid of size =1/𝑛. We have the predicted rates of convergence if the condition Δ𝑡=𝑐 of 𝑢𝑢0,𝛼𝑂2𝛼,𝑢𝑢,0𝑂2𝛼,(4.1) provided that the initial value 𝜑(𝑥) is smooth enough.

Example 4.1. The following equation 𝜕𝑢𝑢𝜕𝑡=𝐷20𝐷𝑥0.5𝑢(𝑥,𝑡)2𝑥(𝑥1)2𝑥1.5𝑥Γ(2.5)0.5𝑒Γ(1.5)2𝑡𝐷𝑢𝑢(𝑥,𝑡)𝑢2𝑒𝑡2𝑥0.5𝑥Γ(1.5)0.5Γ(0.5),0𝑥1,0𝑡1,𝑢(𝑥,0)=𝑥(𝑥1),0𝑥1,𝑢(0,𝑡)=𝑢(1,𝑡)=0,0𝑡1,(4.2) has a unique solution 𝑢(𝑥,𝑡)=𝑒𝑡𝑥(𝑥1).
If we select Δ𝑡=𝑐 and note that the initial value 𝑢0 is smooth enough, then we have 𝑢𝑢0,0.75𝑂1.25,𝑢𝑢,0𝑂1.25.(4.3)

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.

Table 1: Numerical error result for Example 4.1.

Example 4.2. The function 𝑢(𝑥,𝑡)=cos(𝑡)𝑥2(2𝑥)2 solves the equation in the following form: 𝜕𝑢=𝜕𝑡0𝐷𝑥1.7[][𝑢(𝑥,𝑡)+𝑏(𝑢)𝐷𝑢𝑢4(1𝑥)+tan𝑡+𝑓(𝑥,𝑡),𝑥(0,2),𝑡0,1),𝑢(𝑥,0)=𝑥2(2𝑥)2𝑢,0𝑥2,(0,𝑡)=0,𝑢(2,𝑡)=0,0𝑡1,(4.4) where 𝑏(𝑢)=𝑢,cos𝑡𝑓(𝑥,𝑡)=cos𝑡𝑥cos(0.85𝜋)242.3+(2𝑥)2.3𝑥Γ(3.3)241.3+(2𝑥)1.38𝑥Γ(2.3)0.3+(2𝑥)0.3.Γ(1.3)(4.5)
If we select Δ𝑡=𝑐, then 𝑢𝑢0,0.85𝑂1.15,𝑢𝑢,0𝑂1.15.(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.

Table 2: Numerical error result for Example 4.2.

Example 4.3. Consider the following space-fractional differential equation with the nonhomogeneous boundary conditions, 𝜕𝑢=𝜕𝑡0𝐷𝑥1.73𝑢(𝑥,𝑡)𝑥𝑥0𝑢𝑑𝑥2𝑥0.3𝑒𝑡Γ(1.3),0𝑥1,0𝑡1,𝑢(𝑥,0)=𝑥2,0𝑥1,𝑢(0,𝑡)=0,𝑢(1,𝑡)=𝑒𝑡,0𝑡1,(4.7) whose exact solution is 𝑢(𝑥,𝑡)=𝑒𝑡𝑥2.
We still choose Δ𝑡=𝑐, then get the convergence rates 𝑢𝑢0,0.85𝑂1.15,𝑢𝑢,0𝑂1.15.(4.8)

The numerical results are presented in Table 3 which are in line with the theoretical analysis.

Table 3: Numerical error result for Example 4.3.

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.


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.


  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. V. J. Ervin, N. Heuer, and J. P. Roop, “Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 45, no. 2, pp. 572–591, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  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. View at Publisher · View at Google Scholar
  21. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15, Springer, Berlin, Germany, 1994.