In the article titled “Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay” [1], there are a number of minor typographical errors introduced to the equations during the typesetting of the article. The corrected article is as follows.
Abstract
A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method, and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.
1. Introduction
In this paper, we consider the linearized fractional Crank-Nicolson-Galerkin finite element method for solving the nonlinear time fractional parabolic problems with time delay where is a bounded convex and convex polygon in (or polyhedron in ) and is the delay term. denotes the Riemann-Liouville fractional derivative, defined by
The nonlinear fractional parabolic problems with time delay have attracted significant attention because of their widely range of applications in various fields, such as biology, physics, and engineering [1–9]. Recently, plenty of numerical methods were presented for solving the linear time fractional diffusion equations. For instance, Chen et al. [10] used finite difference methods and the Kansa method to approximate time and space derivatives, respectively. Dehghan et al. [11] presented a full discrete scheme based on the finite difference methods in time direction and the meshless Galerkin method in space direction and proved that the scheme was unconditionally stable and convergent. Murio [12] and Zhuang [13] proposed a fully implicit finite difference numerical scheme and obtained unconditionally stability. Jin et al. [14] derived the time fractional Crank-Nicolson scheme to approximate Riemann-Liouville fractional derivative. Li et al. [15] used a transformation to develop some new schemes for solving the time-fractional problems. The new schemes admit some advantages for both capturing the initial layer and solving the models with small parameter . More studies can be found in [16–32].
Recently, it has been one of the hot spots in the investigations of different numerical methods for the nonlinear time fractional problems. For the analysis of the L1-type methods, we refer readers to the paper [33–40]. For the analysis of the convolution quadrature methods or the fractional Crank-Nicolson scheme, we refer to the recent papers [41–46]. The key role in the convergence analysis of the schemes is the fractional Grönwall type inequations. However, as pointed out in [47–49], the similar fractional Grönwall type inequations can not be directly applied to study the convergence of numerical schemes for the nonlinear time fractional problems with delay.
In this paper, we present a linearized numerical scheme for solving the nonlinear fractional parabolic problems with time delay. The time Riemann-Liouville fractional derivative is approximated by fractional Crank-Nicolson type time-stepping scheme, the spatial derivative is approximated by using the standard Galerkin finite element method, and the nonlinear term is approximated by the extrapolation method. To study the numerical behavior of the fully discrete scheme, we construct a novel discrete fractional type Grönwall inequality. With the inequality, we consider the convergence of the numerical methods for the nonlinear fractional parabolic problems with time delay.
The rest of this article is organized as follows. In Section 2, we present a linearized numerical scheme for the nonlinear time fractional parabolic problems with delay and main convergence results. In Section 3, we present a detailed proof of the main results. In Section 4, numerical examples are given to confirm the theoretical results. Finally, the conclusions are presented in Section 5.
2. Fractional Crank-Nicolson-Galerkin FEMs
Denote is a shape regular, quasi-uniform triangulation of the into -simplexes. Let {diam }. Let be the finite-dimensional subspace of consisting of continuous piecewise function on . Let be the time step size, where is a positive integer. Denote , , .
The approximation to the Riemann-Liouville fractional derivative at point is given by [14]: where
For simplicity, denote , , , .
With the notation, the fully discrete scheme is to find such that and the initial condition
where is Ritz projection operator which satisfies following equality [50]
We present the main convergence results here and leave its proof in the next section.
Theorem 1. Suppose the system (1) has a unique solution satisfying and the source term satisfies the Lipschitz condition where is a constant independent of , and , and are given positive constants. Then, there exists a positive constant such that for , the following estimate holds that where is a positive constant independent of and .
Remark 2. The main contribution of the present study is that we obtain a discrete fractional Grönwall’s Grönwall's inequality. Thanks to the inequality, the convergence of the fully discrete scheme for the nonlinear time fractional parabolic problems with delay can be obtained.
Remark 3. At present, the convergence of the proposed scheme is proved without considering the weak singularity of the solutions. In fact, if the initial layer of the problem is taken into account, some corrected terms are added at the beginning. Then, the scheme can be of order two in the temporal direction for nonsmooth initial data and some incompatible source term. However, we still have the difficulties to get the similar discrete fractional Grönwall’s inequality. We hope to leave the challenging problems in future.
3. Proof of the Main Results
In this section, we will present a detailed proof of the main result.
3.1. Preliminaries and Discrete Fractional Grönwall Inequality
Firstly, we review the definition of weights, denote . Then, we can get
Actually, it has been shown [51] that and process following properties. (1)The weights can be evaluated recursively, (2)The sequence are monotone increasing (3)The sequence are monotone decreasing, for and
Noticing the definition of , can be rewritten as
In fact, rearranging this identity yields
where .
Lemma 4 (see [51]). Consider the sequence given by
Then, satisfies the following properties (i)(ii)(iii)
Lemma 5 (see [51]). Consider the matrix
Then, satisfies the following properties (i)(ii)(iii)where , is a constant.
Theorem 6. Assuming and are nonnegative sequence, for , , if
then there exists a positive constant , for , the following holds where , is the Mittag-Leffler function, and .
Proof. By using the definition of in (13), we have
Multiplying equation (18) by and summing the index from to , we get
We change the order of summation and make use of the definition of to obtain and using Lemma 4, we have
Noticing is monotone decreasing, and using Lemma 4, we have
Substituting (20), (21), and (22) into (19), we can obtain
Applying Lemma 4, we have
Therefore
Denote
(23) can be rewritten as
Let , when , we have
Let , then (28) can be rewritten in the following matrix form where
Since the definition of , we have
Then,
Hence, (29) can be shown as follows
where .
Therefore,
According to Lemma 5, the result can be proved.
Lemma 7 (see [51]). For any sequence , the following inequality holds
Lemma 8 (see [52]). There exists a positive constant , independent of , for any , such that
3.2. Proof of Theorem 1
Now, we are ready to prove our main results.
Proof. Taking in the first equation (1), we can find that satisfies the following equation
for and , where
Now, we estimate the error of . Actually, from the definition of and and the regularity of the exact solution (8), we can obtain that where , , are constants.
Applying (39) and (40) and the Lipschitz condition which further implies that
here .
Denote
Substituting fully scheme (5) from equation (37) and using the property in (7), we can get that where
Setting and applying Cauchy-Schwarz inequality, it holds that
Noticing the fact and ,
Together with (9) and (36), we can arrive that Similarly, we have
where , .
Therefore
Substituting (43), (48), and (50) into (47) and the fact , we can get
where .
Applying Lemma 7, we have
In terms of the definition of and , we obtain
Using Theorem 6, we can find a positive constant such that , then
where is a nonnegative constant which only dependents on . In terms of the definition of , we have
Then, we complete the proof.
4. Numerical Examples
In this section, we give two examples to verify our theoretical results. The errors are all calculated in L2-norm.
Example 1. Consider the nonlinear time fractional Mackey-Glass-type equation where
The exact solution is given as
In order to test the convergence order in temporal direction, we fixed for and different . Similarly, to obtain the convergence order in spatial direction, we fixed for and different . Table 1 gives the errors and convergence orders in temporal direction by using the Q-FEM. Table 1 shows that the convergence order in temporal direction is 2. Similarly, Tables 2 and 3 give the errors and convergence orders in spatial direction by using the L-FEM and Q-FEM, respectively. These numerical results correspond to our theoretical convergence order.
Example 2. Consider the following nonlinear time fractional Nicholsons blowflies equation where the exact solution is given as
In this example, in order to test the convergence order in temporal and spatial direction, we solve this problem by using the L-FEM with and the Q-FEM with , respectively. Tables 4 and 5 show that the convergence orders in temporal and spatial direction are 2 and 3, respectively. The numerical results confirm our theoretical convergence order.
5. Conclusions
We proposed a linearized fractional Crank-Nicolson-Galerkin FEM for the nonlinear fractional parabolic equations with time delay. A novel fractional Grönwall type inequality is developed. With the help of the inequality, we prove convergence of the numerical scheme. Numerical examples confirm our theoretical results.
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