Abstract and Applied Analysis

Volume 2013 (2013), Article ID 868035, 15 pages

http://dx.doi.org/10.1155/2013/868035

Research Article

A Finite Element Method for the Multiterm Time-Space Riesz Fractional Advection-Diffusion Equations in Finite Domain

Jingjun Zhao, Jingyu Xiao, and Yang Xu

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

Received 13 December 2012; Revised 28 February 2013; Accepted 28 February 2013

Academic Editor: Xinan Hao

Copyright © 2013 Jingjun Zhao et al. 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.

Abstract.

We present an effective finite element method (FEM) for the multiterm time-space Riesz fractional advection-diffusion equations (MT-TS-RFADEs). We obtain the weak formulation of MT-TS-RFADEs and prove the existence and uniqueness of weak solution by the Lax-Milgram theorem. For multiterm time discretization, we use the Diethelm fractional backward finite difference method based on quadrature. For spatial discretization, we show the details of an FEM for such MT-TS-RFADEs. Then, stability and convergence of such numerical method are proved, and some numerical examples are given to match well with the main conclusions.

1. Introduction

Fractional differential equations are different from integer ones, in which the nature of the fractional derivative introduces the memory effect, thus increasing its modeling ability. Recently, many mathematical models with fractional derivatives have been successfully applied in biology, physics, chemistry, and biochemistry, hydrology, and finance [1–3]. The multiterm fractional differential equations have been widely studied in rheology, and, in many cases, the exact solutions are known [4, 5]. Summary of the fractional differential equations can be found in monographs [6–9]. As one of the main branch, fractional partial differential equations have attracted great attention. Therefore, the numerical treatment and supporting analysis of fractional order partial differential equations have become an important research topic that offers great potential.

The FEM is one of the effective numerical methods for solving traditional partial differential equations. For fractional partial differential equations, FEM also can be a useful and effective numerical method. In recent years, some valuable papers are concerned with the FEM for fractional differential equations. Adolfsson et al. [10, 11] considered an efficient numerical method to integrate the constitutive response of fractional order, viscoelasticity based on the FEM. Roop and Ervin [12–15] investigated the theoretical framework for the Galerkin finite element approximation to some kinds of fractional partial differential equations. Li et al. [16] considered numerical approximation of fractional differential equations with subdiffusion and superdiffusion by using difference method and finite element method. Li and Xu [17, 18] proposed a time-space spectral method for time and time-space fractional partial differential equation based on a weak formulation, and a detailed error analysis was carried out. Jiang and Ma [19] considered a high-order FEM for time fractional partial differential equations and proved the optimal order error estimates. Ford et al. [20] studied an FEM for time fractional partial differential equations.

Fractional advection-diffusion equations especially are important in describing and understanding the dispersion phenomena. Analytical solutions of such equations in finite domain have been obtained by Park in [21]. Also, the Riesz fractional advection-diffusion equations (RFADEs) with a symmetric fractional derivative (the Riesz fractional derivative) were derived from the kinetics of chaotic dynamics by [22] and summarized by [23]. Ciesielski and Leszczynski [24] presented a numerical solution for such equations based on the finite difference method.

One often sees RFADEs defined in terms of the fractional Laplacian as follows:

for example, where is a solute concentration, and represent the dispersion coefficient and the average fluid velocity. Here, the fractional Laplacian operator uses the Fourier transformation on an infinite domain, with a natural extension to include finite domains when the function is subject to the zero Dirichlet boundary conditions (see [9]). Due to Lemma  1 in [25], the fractional Laplacian operator on an infinite domain is equivalent to the Riesz fractional derivative operator . In particular, the Riesz fractional derivatives include both the left and the right Riemann-Liouville derivatives that allow the modeling of flow regime impacts from either side of the domain. Yang et al. [25] investigated the numerical treatment for the RFADE with the Riesz space fractional derivative as

where is the Riesz space fractional operator defined in Section 2.

To increase the modeling ability, some authors considered the equations with the fractional order in both time and spatial variables in RFADEs, which include more information and hence are more interesting. For the time-space fractional advection-dispersion equations, Shen et al. [26] presented the fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation with initial and boundary conditions on a bounded domain, and derived the stability and convergence of their proposed numerical methods. Then, for fractional advection-diffusion equations, Shen et al. [27] presented an explicit difference approximation and an implicit difference approximation for the time-space Riesz-Caputo fractional advection-diffusion equations with initial and boundary conditions on a finite domain.

All of the above papers only considered single-term fractional equations in time variable, where only one fractional differential operator appeared. In this paper, we consider the multiterm fractional differential equation, which includes more than one fractional derivative. For example, the so-called Bagley-Torvik equation [28] is

where , and are certain constants and is a given function. The Basset equation is, in [6],

where , , and are positive real numbers. This equation describes the forces that occur when a spherical object sinks in an incompressible viscous fluid.

Recently, some authors considered the applications of multiterm fractional differential equations [29] and the numerical methods for such equations [30–32]. At the same time, the multiterm fractional partial differential equations have been proposed in [33, 34]. The analytical solution and the numerical methods for multiterm time fractional wave-diffusion equations have been investigated in [35, 36]. This motivates us to consider the effective numerical solution for such multiterm fractional partial differential equations.

In this paper, we consider MT-TS-RFADEs in finite domain with the zero Dirichlet boundary conditions. The analytical solution of such MT-TS-RFADEs has been investigated by Jiang et al. in [37]. Here, we present an FEM for a simplified MT-TS-RFADEs and obtain the optimal order error estimates both in semidiscrete and fully discrete cases and derive the stability of such FEM. As far as we are aware, there are few research papers in the published literature written on this topic.

This paper is organized as follows. In Section 2, the preliminaries of the fractional calculation are shown. Then, we give the weak formulation of MT-TS-RFADEs and prove the existence and uniqueness of this problems by the well-known Lax-Milgram theorem. In Section 3, we present the convergence rate of Diethelm’s fractional backward difference method (see [38, 39]) for time discretization. In Section 4, we propose a finite element method based on the weak formulations and carry out the error analysis. In Section 5, we prove the stability of such FEM for MT-TS-RFADEs. Finally, some numerical examples are considered in Section 6.

2. Existence and Uniqueness

We consider the MT-TS-RFADEs with multiterm time fractional derivatives and the Riesz space fractional derivatives in the following form:

where , , , and are respectively the space and time variables and , are positive constants. We consider this problem with the zero Dirichlet boundary value conditions and the initial value condition defined as follows:

For nonzero boundary value conditions, we need to transform the problem into one with zero boundary value conditions before using the method in this paper.

Here, we consider the multiterm time fractional differential operator which has the subdiffusion process (see [16]). It is different from (2), which only has one integer order differential operator in time.

Note that the analytical solutions for MT-TS-RFADEs have been studied in [37], in which this problem is well defined. For this problem, some new techniques have been used, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between the fractional Laplacian operator and the Riesz fractional derivative.

For convenience, we introduce the following definitions and properties. The space derivatives and are the Riesz space fractional derivatives of order and , respectively. The definitions of them can be found in [40].

Let denote the gamma function. For any positive integer and real number (), there are different definitions of fractional derivatives with order in [8]. During this paper, we consider the left, (right) Caputo derivative and left (right) Riemann-Liouville derivative defined as follows:

(i)the left Caputo derivative:

(ii)the right Caputo derivative:

(iii) the left Riemann-Liouville derivative:

(iv) the right Riemann-Liouville derivative:

The Riesz fractional operators and in (5) can be defined by the left and the right Riemann-Liouville fractional derivatives.

Definition 1 (see [40]). The Riesz fractional derivatives of order for , on a finite interval is defined as

where .

The multiterm fractional operator in (5) is

where or or , and , . Here, and denote the left Caputo fractional derivatives with respect to the time variable of order and . There are three cases for multiterm time-space fractional derivatives.

Case 1. If , (5) is a generalized MT-TS-RFADE with multiterm time fractional diffusion terms with initial conditions given as (6). And especially, if , and , then (5) becomes a space fractional advection-diffusion equation with the Riesz space fractional derivatives, which was discussed by Yang et al. [25].

Case 2. If , (5) is a generalized MT-TS-RFADE with multiterm time fractional wave terms. In this case, the initial conditions are given as follows:

Case 3. If , (5) becomes a generalized MT-TS-RFADE, which we refer to as a multiterm time-space fractional mixed wave-diffusion equation. In this case, the initial conditions are also given by (13).

In this paper, we just consider Case 1 of (5) with . Another two cases with and will be studied in our following work.

In order to establish the weak formulation of the problem (5), we need some preparatory work. We use definitions of functional spaces and derive some properties related to these spaces. Let denote the space of infinitely differentiable functions on , and let denote the space of infinitely differentiable functions with compact support in . Let be the space of measurable functions whose square is the Lebesgue integrable in , which may denote a domain or or , where denotes the time domain and denotes the space domain. The inner product and norm of are defined by

For any real , we define the spaces and to be the closure of with respect to the norms and , respectively, where

In the usual Sobolev space , we also have the definition

From [18], for , the spaces , , and are equal, and their seminorms are all equivalent to .

We now give some results for fractional operators on these spaces.

Lemma 2 (see [8, 18]). For real , if , then

Let . Then, one has

Lemma 3 (see [18]). Let . Then, for any , then

Following from the definitions and lemmas above, we define the space

when and .

Then, one obtains

Here, is a Banach space with respect to the following norm:

where

endowed with the norm

For obtaining a suitable weak solution for problem (5) with the Caputo time fractional derivation, we consider the connection between the Caputo and the Riemann-Liouville fractional definition. Based on the definitions of the Caputo and the Riemann-Liouville fractional differential operators, we have an immediate consequence, for any real order , in [38],

where denotes the Taylor polynomial for of order , centered at ,

where .

Based on (25), we make

Therefore, that led to the following weak formulation of (5). Let be the dual space of . For , we find such that

where the bilinear form is, based on Lemmas 2 and 3,

and the functional is given by

Lemma 4 (see [17]). For real , , then

Based on Lemma 4, we can prove the following existence and uniqueness theorem. During this paper, we use the expression to mean that there exists a positive real number such that (). At the same time, we denote to mean that , which means there exist positive real numbers such that and (i.e., ).

Theorem 5. Assume that , and . Then, system (28) has a unique solution in . Furthermore,

Proof. The existence and uniqueness of the solution is guaranteed by the well-known Lax-Milgram theorem. First, from the equivalence of , , and , for all , it follows that

This implies the continuity of the bilinear form and the right-hand side function .

We next prove the coercivity of the bilinear operator . Note that

for all , where is the extension of by zero outside of . Thus, we find that , is nonnegative for since is nonnegative for . That is the same for fractional operators with .

From the above analysis, we have

By using the well-known Lax-Milgram theorem, there exists a unique solution such that (28) holds.

To prove the stability estimate (32), by using (34) and (36), we take in (28) and obtain

which implies that .

3. Time Discretization

In this section, we consider the Diethelm fractional backward difference method based on quadrature, which was independently introduced by [39], for ordinary fractional differential equations. Here, we consider this method for the time discretization of (5) and derive the convergence rate for the time-discretization of MT-TS-RFADEs.

Lemma 6 (see [25]). For and a function defined on an unbounded domain , the following equality holds:

where is the Laplacian.

Definition 7 (see [26]). Suppose the Laplacian has a complete set of orthonormal eigenfunctions corresponding to eigenvalues on a bounded region , that is, on a bounded region ; on is one of the standard three homogeneous boundary conditions. Let

then for any , is defined by

Here, for any , we make a continuous prolongation in space for function as follows:

Let denote the one-variable function of , and let and denote the one-variable functions as and , respectively. By the definition (12) of and Lemma 6, (5) is equivalent to

Let , . Then, the system (5) and the initial condition (6) can be rewritten in the abstract form, for ,

Based on (25), (43) can be rewritten as

where .

Let be a partition of . Without loss of generality, for and fix ,, we have

where . Here, the integral is a Hadamard finite-part integral, in [38, 39].

Now, for every , we replace the integral by a first-degree compound quadrature formula with equispaced nodes and obtain

where the weights are

and the remainder term satisfies

where is a constant.

Thus, we have

where .

Let , we can write (45) as

for , where .

Denote as the approximation of . We can define the following time stepping method:

where .

Lemma 8 (see [38]). For , let the sequence be given by and . Then, .

Let denote the error in . Let be any given norm. Then, we have the following error estimate.

Theorem 9. Let and be the solutions of (43) and (52), respectively, and is the time step size. Then, one has

Proof. Subtracting (52) from (51), we obtain the error equation

Note that and denote

Thus, we have

From Definition 7, note that is a positive definite elliptic operator with all of eigenvalues . Since and when , we have

Hence,

Note that . Denote and

Then, by induction and Lemma 8, we have

4. Space Discretization

In this section, we consider the space discretization of (5) with homogeneous boundary condition. Using the FEM, we obtain the numerical approximation solution in a finite domain. Then, we prove the convergence rate of this method. Let be an interval in one-dimensional space. All of the results in this section can be generalized into the cases of high dimension.

Based on the time discretization in Section 3 and (25), we need to find such that

Let denote the maximal length of intervals in , and let be any nonnegative integer. We denote the norm in by . Let be a family of finite element space with the accuracy of order consists of continuous functions on the closure of which are polynomials of degree at most in each interval and which vanish outside such that for small , ,

The semidiscrete problem of (5) is to find the approximate solution for each such that

Let be the elliptic projection defined by

Lemma 10. Assume that (62) holds. Then, for defined by (64) and any , one has

Proof. Let be the projection operator from to . From the definition of -norm, we obtain

Let . From (64), we obtain

Note that

Thus, combining (67) and (68), we obtain

By (62), we see that

By interpolation properties, we obtain

Similarly, we have

Combining (71) and (72), we can obtain (65).

Theorem 11. For , let and be the solutions of (63) and (61), respectively. Then, it holds

Proof. We write

where , . The second term is easily bounded by Lemma 10 and has the obvious estimate

In order to estimate , for all , we get

Choosing and integrating on both sides with respect to on , we obtain

By Lemmas 2–4, for any small , we have

where is a constant respect to , and .

For sufficiently small , by (75), we obtain

Combining (75) with (79), we obtain that

For the reason that the time and space fractional derivatives, we introduce the complete form of this FEM. In view of space discretization, we first pose the finite-dimensional problem to find such that (63) holds.

In terms of the basis , write , and insert it into (63). After the time discretization on , in Section 3, we get

To obtain the value of , Let . From (81), we obtain a vector equation

where is the mass matrix; and are stiffness matrices with , as follows:

and is a vector valued function. Then, from (82), we can obtain the solution .

5. Stability of the Numerical Method

In this section, we analyze the stability of the FEM for MT-TS-RFADEs (5). Now, we do some preparation before proof. Based on the definition of coefficients in Section 3, we can obtain the following lemma easily.

Lemma 12. For , the coefficients , satisfy the following properties:

(i) and for , (ii).

Now, we report the stability theorem of this FEM for MT-TS-RFADEs as follows.

Theorem 13. The FEM defined in (81) is unconditionally stable.

Proof. Let denote the approximation to at and and the right-hand side . From (63), we have

Using the Cauchy-Schwarz inequality, for , by Definition 1 and Lemma 12, we obtain

We prove the stability of (63) by induction. From the beginning, we have

The induction basis is presupposed. For the induction step, we have . Then, using this result, by Lemma 12, we obtain

Here, and for . After squaring at both sides of the above inequality, we obtain .

6. Numerical Tests

Based on the above analysis, we present three numerical examples for MT-TS-RFADEs to demonstrate the efficiency of our theoretical analysis. The main purpose is to check the convergence behavior of numerical solutions with respect to time step size and space step size , which have been shown in Theorems 9 and 11.

Example 1. Consider MT-TS-RFADE, in ,

The exact solution of (88) is . From the definition of the Riemann-Liouville differential operator, it holds

where is a positive constant. From (89), we can choose right-hand side function to satisfy (88).

Choosing and in time fractional operators and in the space Riesz fractional operators, we can obtain the numerical approximation to the exact solution of (88) on finite domain , with space step size and time step size . In Figure 1, one can see that the numerical solution matches well with the exact solution.

Figure 1: Numerical solution and exact solution for Example 1.

Example 2. Consider MT-TS-RFADE (88) with conditions as follows:

Let the exact solution is . We choose and obtain the numerical solution and exact solution when . The results have been shown in Figure 2, where the exact solution is noted by lines and numerical solution is noted by squares. Here, space step size is ; time step size is .

Figure 2: Numerical solution and exact solution for Example 2.

Example 3. Consider MT-TS-RFADE (88) with the zero Dirichlet boundary conditions, for , . We require that the exact solution is .

For this example, in the first test, we obtain the numerical solution and exact solution when , , , in Figure 3, where we choose time step size and space step size with , , and , .

Figure 3: Numerical solution and exact solution for Example 3.

In the second test, we check the convergence rates of numerical solutions with respect to the fractional orders ,  and . We fix , and  and choose which is small enough such that the space discretization errors are negligible as compared with the time errors. Choosing step size , we present Table 1 with the convergence rate which is equal to 1.2, as Theorem 9 predicted. Table 2 shows the spatial approximate convergence rate, by fixing and choosing . From Theorem 11, the convergence rate should be equal to or less than 1.1 (i.e., for and ). In Table 2, the numerical results match well with such conclusion. Here, we also report both the -norm and -norm of errors in Figure 4.

Table 1: Convergence rate in time for Example 3.

-norm
-norm
cvge. rate

0.001
1/2

0.001
1/4

1.1893

0.001
1/8

1.1957

0.001
1/16

1.2012

0.001
1/32

1.2090

Table 2: Convergence rate in space for Example 3.

-norm
-norm
cvge. rate

0.001
1/2

0.001
1/4

1.0006

0.001
1/8

1.0014

0.001
1/16

1.0029

0.001
1/32

1.0057

(a)

(b)

Figure 4: -norm and -norm of errors for Example 3; here, (a) and (b).

Acknowledgments

The authors are grateful to the referees for their valuable comments. This work is supported by the National Natural Science Foundation of China (11101109 and 11271102), the Natural Science Foundation of Hei-long-jiang Province of China (A201107), and SRF for ROCS, SEM.

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