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

and Applied Analysis 3 Let Γ(⋅) denote the gamma function. For any positive integer n and real number θ (n − 1 < θ < n), there are different definitions of fractional derivatives with order θ 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: C 0 D θ t V (t) = 1 Γ (n − θ) ∫ t 0 1 (t − τ) θ−n+1 ( d n dτ V (τ)) dτ, (7) (ii) the right Caputo derivative: C t D θ T V (t) = (−1) n


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][2][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][7][8][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][13][14][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: (, ) = −  (−Δ) /2  (, ) −   (−Δ) /2  (, ) , ( Abstract and Applied Analysis for example, where  is a solute concentration,   and   represent the dispersion coefficient and the average fluid velocity.Here, the fractional Laplacian operator −(−Δ) /2 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 −(−Δ) /2 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] 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 socalled Bagley-Torvik equation [28] is where , , and  are certain constants and  is a given function.The Basset equation is, in [6], where 0 <  < 1, , 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][31][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 wavediffusion 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 wellknown 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.

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 0 <  < 1, 1 <  < 2,  ∈ [0, ], and  ∈ [0, ] 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: (0, ) =  () ,  ∈ (0, ) .
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 Γ(⋅) denote the gamma function.For any positive integer  and real number  ( − 1 <  < ), 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.
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  ∞ (0, ) denote the space of infinitely differentiable functions on (0, ), and let  ∞ 0 (0, ) denote the space of infinitely differentiable functions with compact support in (0, ).Let  2 (Q) be the space of measurable functions whose square is the Lebesgue integrable in Q, which may denote a domain Q =  or Ω or  × Ω, where  = [0, ] denotes the time domain and Ω = [0, ] denotes the space domain.The inner product and norm of  2 (Q) are defined by For any real  > 0, we define the spaces    0 (Q) and    0 (Q) to be the closure of  ∞ 0 (Q) with respect to the norms ‖ V‖    0 (Q) and ‖ V‖    0 (Q) , respectively, where In the usual Sobolev space   0 (Q), we also have the definition From [18], for  > 0,  ̸ =  − 1/2, the spaces    0 (Q),    0 (Q), and   0 (Q) are equal, and their seminorms are all equivalent to | ⋅ |   0 (Q) .
We now give some results for fractional operators on these spaces.
Then, one obtains Here,  /2,/2 ( × Ω) 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  > 0, in [38], where  −1 [; 0]() denotes the Taylor polynomial for  of order  − 1, centered at 0, where Based on (25), we make Therefore, that led to the following weak formulation of ( 5).Let  /2,/2 ( × Ω)  be the dual space of  /2,/2 ( × Ω).For  ∈  /2,/2 ( × Ω)  , we find (, ) ∈  /2,/2 ( × Ω) such that where the bilinear form A(⋅, ⋅) is, based on Lemmas 2 and 3, and the functional F(⋅) is given by Lemma 4 (see [17]).For real  > 0, V ∈  ∞ 0 (R), 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  1 ,  2 such that  ≤  1  and  ≤  2  (i.e., (1/ 1 ) ≤  ≤  2 ).
We next prove the coercivity of the bilinear operator A(⋅, ⋅).Note that for all  ∈  ∞ 0 (Q), where φ is the extension of  by zero outside of (0, ).Thus, we find that That is the same for fractional operators with   ,  = 1, . . ., .
From the above analysis, we have By using the well-known Lax-Milgram theorem, there exists a unique solution  ∈  /2,/2 ( × Ω) such that (28) holds.

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.

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 Ω = [0, ] be an interval in one-dimensional space.All of the results in this section can be generalized into the cases of high dimension.
Theorem 11.For 0 <   < ⋅ ⋅ ⋅ <  1 <  < 1, let  ℎ and  be the solutions of (63) and (61), respectively.Then, it holds Proof.We write where  =  ℎ −  ℎ ,  =  ℎ  − .The second term is easily bounded by Lemma 10 and has the obvious estimate In order to estimate , for all  ∈  ℎ , we get Choosing  = () and integrating on both sides with respect to  on [0, ], we obtain ( ( 0   )  () ,  ())  2 (Ω) 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  ℎ (, ⋅) ∈  ℎ such that (63) holds.

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.

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 Δ and space step size Δ, which have been shown in Theorems 9 and 11.
where  is a positive constant.From (89), we can choose righthand side function (, ) to satisfy (88).
Choosing  = 0.9 and  1 = 0.2 in time fractional operators and  = 0.4 = 1.6 in the space Riesz fractional operators, we can obtain the numerical approximation to the exact solution of (88) on finite domain [0, 0.5] × [0, 1], with space step size Δ = 0.05 and time step size Δ = 0.01.In Figure 1, one can see that the numerical solution matches well with the exact solution.results have been shown in Figure 2, where the exact solution is noted by lines and numerical solution is noted by squares.
In the second test, we check the convergence rates of numerical solutions with respect to the fractional orders  1 , , , and .We fix  1 = 0.2,  = 0.8,  = 0.8, and  = 1.8 and choose Δ = 0.001 which is small enough such  that the space discretization errors are negligible as compared with the time errors.Choosing step size Δ = 1/2  ( = 1, . . ., 5), 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 Δ = 0.001 and choosing Δ = 1/2  ( = 1, . . ., 5).From Theorem 11, the convergence rate should be equal to or less than 1.1 (i.e., −/2 for  = 2 and  = 1.8).In Table 2, the numerical results match well with such conclusion.Here, we also report both the  2norm and  1 -norm of errors in Figure 4.

Table 1 :
Convergence rate in time for Example 3.

Table 2 :
Convergence rate in space for Example 3.