Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations
A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.
In recent years, the numerical treatment and supporting analysis of fractional order differential equations has become an important research topic that offers great potential. The FEMs for fractional partial differential equations have been studied by many authors (see [1–3]). All of these papers only considered single-term fractional equations, where they only had one fractional differential operator. In this paper, we consider the MT-FPDEs, which include more than one fractional derivative. Some authors also considered solving linear problems with multiterm fractional derivatives (see [4, 5]). This motivates us to consider their effective numerical solutions for such MT-FPDEs, which have been proposed in [6, 7].
Let , where is the space dimension. We consider the MT-FPDEs with the Caputo time fractional derivatives as follows: where the operator is defined as with and . Here denotes the left Caputo fractional derivative with respect to the time variable and denotes the Laplace operator with respect to the space variable .
Some numerical methods have been considered for solving the multiterm fractional differential equations. In , Liu et al. investigate some effective numerical methods for time fractional wave-diffusion and diffusion equations: where and are arbitrary positive constants and is a sufficiently smooth function. The authors consider the implicit finite difference methods (FDMs) and prove that it is unconditionally stable. The error estimate of the FDM is , where and are the time and space step size, respectively. They also investigate the fractional predictor-corrector methods (FPCMs) of the Adams-Moulton methods for multiterm time fractional differential equations (1) with order by solving the equivalent Volterra integral equations. The error estimate of the FPCM is . In recent years, there are some articles for the predictor-correction method for initial-value problems (see [9–14]). For the application of the FDMs, there have been many research articles as follows. In [15–20], Simos et al. investigate the numerical methods for solving the Schrödinger equation. In [21–24], the Runge-Kutta methods are considered and applied to get the numerical solution of orbital problems. For long-time integration, the Newton-Cotes formulae are considered in [25–27].
In , Badr investigate the FEM for linear multiterm fractional differential equations with one variable as follows: where are known functions. The author gives the details of the modified Galerkin method for the above equations and makes the numerical example for checking the numerical method. In , Ford et al. consider the FEM for (5) with singular fractional order and obtain the error estimate . In this paper, we follow the work in  and consider the FEM for solving MT-FPDEs (1)–(3). Then, we prove the stability and convergence of the FEM for MT-FPDEs and make the error estimate.
The paper is organized as follows. In Section 2, the weak formulation of the MT-FPDEs is given and the existence and uniqueness results for such problems are proved. In Section 3, we consider the convergence rate of time discretization of MT-FPDEs, based on the Diethelm fractional backward difference method (DFBDM). In Section 4, we propose an FEM based on the weak formulation and carry out the error analysis. In Section 5, the stability of this method is proven. Finally, the numerical examples are considered for matching well with the main conclusions.
2. Existence and Uniqueness
Let denote the gamma function. For any positive integer and , the Caputo derivative are the Riemann-Liouville derivative are, respectively, defined as follows .(i)The left Caputo derivatives: (ii)The left Riemann-Liouville derivatives: (iii)The right Riemann-Liouville derivatives:
Let denote the space of infinitely differentiable functions on and denote the space of infinitely differentiable functions with compact support in . We use the expression to mean that when is a positive real number and use the expression to mean that . Let be the space of measurable functions whose square is the Lebesgue integrable in which may denote a domain , or . Here time domain and 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 , for , , the spaces , , and are equal, and their seminorms are all equivalent to . We first recall the following results.
Lemma 1 (see ). Let , . Then for any , then
From , we define the following space:
Here is a Banach space with respect to the following norm: where , endowed with the norm
Based on the relation equation between the left Caputo and the Riemann-Liouville derivative in , we can translate the Caputo problem to the Riemann-Liouville problem. Then, we consider the weak formulation of (1) as follows. For , find such that where the bilinear form is, by Lemma 1, and the functional is , .
Based on the main results in Subsection 3.2 in , we can prove the following existence and uniqueness theorem.
Theorem 2. Assume that and . Then the system (17) has a unique solution in . Furthermore,
Proof. The existence and uniqueness of the solution of (17) is guaranteed by the well-known Lax-Milgram theorem. The continuity of the bilinear form and the functional is obvious. Now we need to prove the coercivity of in the space . From the equivalence of , and , for all , using the similar proof process in , we obtain Then we take in (17) to get by the Schwarz inequality and the Poincaré inequality.
3. Time Discretization and Convergence
In this section, we consider DFBDM for the time discretization of (1)–(3), which is introduced in  for fractional ordinary differential equations. We can obtain the convergence order for the time discretization for the MT-FPDEs. Let , . Let , , and denote the one-variable functions as , , and , respectively. Then (1) can be written in the abstract form, for , , with initial value . Now we have
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, for , we have
Let , we can write (21) as Denote as the approximation of and . We obtain the following equation:
Lemma 3 (see ). For , let the sequence be given by and . Then, , for .
Let denote the error in . Then we have the following error estimate.
Proof. Subtracting (27) from (26), we obtain the error equation
Note that . Denote Let denote the -norm, then we have
Note that is a positive definite elliptic operator with all of eigenvalues . Since and , we have Hence,
Denote and where . By induction and Lemma 3, then we have
4. Space Discretization and Convergence
In this section, we will consider the space discretization for MT-FPDEs (1) and show the complete process and details of numerical scheme. The variational form of (1) is to find , such that, for all ,
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 spaces with the accuracy of order , that is, 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 (1) is to find the approximate solution and for each such that
Let . After the time discretization, we have
In terms of the basis , choosing , writing and inserting it into (38), one obtains
Let . From (40), we obtain a vector equation where initial condition is , is the mass matrix, is stiffness matrix as , and is a vector valued function. Then, we can obtain the solution at .
Let be the elliptic projection, defined by , for all .
5. Stability of the Numerical Method
In this section, we analyze the stability of the FEM for MT-FPEDs (1)–(3). Now we do some preparations before proving the stability of the method. Based on the definition of coefficients in Section 3, we can obtain the following lemma easily.
Lemma 7. For , the coefficients , satisfy the following properties: (i) and for , (ii).
Now we report the stability theorem of this FEM for MT-FPDEs in this section as follows.
Theorem 8. The FEM defined as in (38) is unconditionally stable.
Proof. In (38), let at and the right hand . We have
Using, Cauchy-Schwarz inequality, for and Lemma 7, we get
We prove the stability of (37) by induction. Since when , we have The induction basis is presupposed. For the induction step, we have . Then using this result, by Lemma 7, we obtain
Here . After squaring at both sides of the above inequality, we obtain .
6. Numerical Experiments
In this section, we present the numerical examples of MT-FPDEs to demonstrate the effectiveness of our theoretical analysis. The main purpose is to check the convergence behavior of numerical solutions with respect to and , which have been shown in Theorem 4 and Theorem 6. It is noted that the method in  is a special case of the method in our paper for fractional partial differential equation with single fractional order. So, we just need to compare FEM in our paper with other existing methods in [8, 28].
Example 9. For , , consider the MT-FPDEs with two variables as follows:
where the right-side function . The exact solution is .
We use this example to check the convergence rate (c. rate) and CPU time (CPUT) of numerical solutions with respect to the fractional orders and .
In the first test, we fix , and and choose which is small enough such that the space discretization errors are negligible as compared with the time errors. Choosing (), we report that the convergence rate of FDM in time is nearly 1.15 in Table 1, which matches well with the result of Theorem 4. On the other hand, Table 2 shows that an approximate convergence rate is 2, by fixing and choosing (), which matches well with the result of Theorem 6. In the second test, we give the convergence rate when , for in Table 3, and in Table 4, respectively. We also report the -norm and -norm of errors in Figures 1 and 2, respectively.
Fixing , , and in (46), we compare the error and CPUT calculated by the FEM in this paper with the FDM in  and the FPCM in . From Table 5, it can be seen that the FEM in this paper is computationally effective.
Example 10. Consider the following multierm fractional differential problem:
where . For and , the exact solution is .
For the problem (47), our method in this paper is just the DFBDM in Section 3. Therefore, we only need to compare M1 with the FEM in  (FEM2). In Table 6, although the convergence rate of FEM2 is higher than that of DFBDM, the error and CPUT of DFBDM are smaller than those of FEM2.
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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