Abstract
A numerical method for the modified time fractional Fokker-Planck equation is proposed. Stability and convergence of the method are rigorously discussed by means of the Fourier method. We prove that the difference scheme is unconditionally stable, and convergence order is , where and are the temporal and spatial step sizes, respectively. Finally, numerical results are given to confirm the theoretical analysis.
1. Introduction
Fractional differential equations have attracted considerable attention due to their frequent appearance in various applications in fluid mechanics, biology, physics, and engineering [1, 2]. Usually, fractional differential equations do not have analytic solutions and can only be solved by some semianalytical and numeric techniques. Recently, several semianalytical methods, such as variational iteration method, homotopy perturbation method, Adomian decomposition method, homotopy analysis method, and collocation method, have been used to solve fractional differential equations [3–7]. Meanwhile, some effective numerical techniques are developed; see [8–17].
In the present paper, we are motivated to study the following modified time fractional Fokker-Planck equation [18]: where is a fractional diffusion coefficient, and is a fractional friction coefficient. denotes the temporal Riemann-Liouville derivative operator defined as [2] where is the gamma function.
The outline of the paper is as follows. In Section 2, an effective numerical method for solving the modified time fractional Fokker-Planck equation is proposed. The solvability, stability, and convergence of the numerical method are discussed in Sections 3 and 4, respectively. In Section 5, we give some numerical results demonstrating the convergence orders of the numerical method. Also a conclusion is given in Section 6.
2. The Construction of Numerical Method
Let and , where and are the uniform spatial and temporal mesh sizes, respectively, and , are two positive integers.
Lemma 1. Suppose that then a fourth-order difference scheme for the above equation is given as follows: where and are two difference operators and are defined by in which is an unit operator and and are average central and second central difference operators with respect to and are defined by
Proof. In view of Taylor expansion, we can obtain
Noting (4), we easily obtain
This completes the proof.
Combing (1), (4), and (5), we obtain
Using the relation of the Riemann-Liouville fractional derivative and Grünwald-Letnikov fractional derivative , we can approximate the Riemann-Liouville fractional derivative by [2] where .
For first-order derivative , we apply the following backward difference scheme:
Let be the numerical approximation of ; substituting (11) and (12) into (10) and omitting the error term , we can obtain the following difference scheme for solving (1):
The discrete form of above system is where , , , and .
The initial and boundary conditions can be discretized as
Obviously, the local truncation error of difference scheme (14) is .
3. The Solvability of the Difference Scheme
Firstly, let us denote
Then we can give the compact form of difference scheme (14) as follows: where
Theorem 2. Difference equation (17) is uniquely solvable.
Proof. It is well known that the eigenvalues of the matrix are
where .
Note that and ; if
then we easily know that .
If
then
At the moment, we obtain ; that is to say, the matrix is invertible. Hence, difference equation (17) has a unique solution.
4. Stability and Convergence Analysis
In this section, we analyze the stability and convergence of difference scheme (17) by the Fourier method [8]. Firstly, we give the stability analysis.
Lemma 3. The coefficients satisfy [8] as follows:(1), , , ;(2) .
Let be the approximate solution of (14) and define respectively.
So, we can easily obtain the following roundoff error equation: Now, we define the grid functions then can be expanded in a Fourier series: where
We introduce the following norm: and according to the Parseval equality we obtain Through the above analysis, we can suppose that the solution of (25) has the following form: where .
Substituting the above expression into (25) one gets
Lemma 4. The following relation holds: where
Proof. Because of
we obtain
Furthermore, we can rewrite the above inequality as
that is,
This completes the proof of Lemma 4.
Lemma 5. Supposing that is the solution of (33), then we have
Proof. For , from (33), we get
In the light of Lemma 4, it is clear that
Now, we suppose that
For , from (33) with Lemmas 3 and 4, we have
This finishes the proof of Lemma 5.
Lemma 6. Difference scheme (14) is unconditionally stable.
Proof. According to Lemma 5, we obtain
which means that difference scheme (14) is unconditionally stable.
Next, we give the convergence analysis. Suppose
and denote
From (14), we obtain
Similar to the stability analysis method, we define the grid functions
then and can be expanded to the following Fourier series, respectively:
where
The same as before, we also have
Based on the above analysis, we can assume that and have the following forms:
respectively. Substituting the above two expressions into (48) yields
Lemma 7. Let be the solution of (55); then there exists a positive constant , so that
Proof. From , we have
In view of the convergence of the series of (53), there is a positive constant , such that
For , from (55), we have
Noticing (58), then
Now, we suppose that
Then when , we obtain
This completes the proof.
Theorem 8. Difference scheme (14) is convergent, and the convergence order is .
Proof. Firstly, we know that there are exist positive constants , such that
Using (52) and (64) with Lemma 7, we get
Due to ,
so,
where . This ends the proof.
Remark 9. From above discussion, we know that difference scheme is an implicit scheme and it is unconditionally stable and convergent. If we take in (1), then we can obtain an explicit scheme and it is conditionally stable and convergent.
5. Numerical Example
In this section, a numerical example is presented to confirm our theoretical results.
Example 10. Consider the following equation:
with the initial and boundary conditions
where . The analytical solution of this equation is .
The maximum error, temporal, and spatial convergence orders by difference scheme (14) for various are listed in Table 1. From the obtained results, we can draw the following conclusions: the experimental convergence orders are approximately 1 and 4 in temporal and spatial directions, respectively. Figures 1 and 2 show the comparison of the numerical solution with the analytical solution at and for different temporal and spatial mesh sizes.
By Table 1 and Figures 1 and 2, it can be seen that the numerical solution is in excellent agreement with the analytical solution. These results confirm our theoretical analysis.
6. Conclusion
In this paper, a computationally effective numerical method is proposed for simulating the modified time fractional Fokker-Planck equation. It has proven the unconditional stability and solvability of proposed scheme. Also we showed that the method is convergent with order . The numerical results demonstrate the effectiveness of the proposed scheme.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work was partially supported by Tianshui Normal University “QingLan” Talent Engineering Funds.