Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
Special Issue

Fractional Integral Transform and Application

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Research Article | Open Access

Volume 2014 |Article ID 282190 | 10 pages | https://doi.org/10.1155/2014/282190

Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation

Academic Editor: Adem Kilicman
Received26 Dec 2013
Revised08 Feb 2014
Accepted16 Feb 2014
Published27 Mar 2014

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 [37]. Meanwhile, some effective numerical techniques are developed; see [817].

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.


The maximum errorTemporal convergence orderSpatial convergence order

0.1

0.2

0.3

0.4

0.5

0.6