Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation

and Applied Analysis 3 The discrete form of above system is ( 1 12 − c 1 − c 2 + c 3 ) u k j+1 + ( 5 6 + 2c 2 ) u k j + ( 1 12 + c 1 − c 2 − c 3 ) u k j−1 = ( 1 12 − c 1 + c 2 π (1−α) 1 − c 3 π (1−α) 1 ) u k−1 j+1 + ( 5 6 − 2c 2 π (1−α) 1 ) u k−1 j + ( 1 12 + c 1 + c 2 π (1−α) 1 + c 3 π (1−α) 1 ) u k−1 j−1

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.

Lemma 1. Suppose that
]  (  , ) =  (  , ) ; then a fourth-order difference scheme for the above equation is given as follows: where A and L are two difference operators and are defined by in which  is an unit operator and     and  2  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.

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 A are where  = 1, 2, . . .,  − 1.
Note that  ≥ 1 and   > 0; if then we easily know that then At the moment, we obtain det(A) ̸ = 0; that is to say, the matrix A is invertible.Hence, difference equation ( 17) has a unique solution.

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  (1−)  ( = 0, 1, . ..) satisfy [8] as follows: 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  = 2/.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.
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 E  () and   () can be expanded to the following Fourier series, respectively: where The same as before, we also have 2 = ( Based on the above analysis, we can assume that E   and    have the following forms: respectively.Substituting the above two expressions into (48) yields Lemma 7. Let   ( = 0, 1, . . ., ) be the solution of (55); then there exists a positive constant  3 , so that           ≤  3  exp ( ( − 1) )      1     ,  = 0, 1, . . ., . (56) Proof.From E 0 = 0, we have In view of the convergence of the series of (53), there is a positive constant  1 , such that For  = 1, from (55), we have where  =  1  2  √  exp().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.

Numerical Example
In this section, a numerical example is presented to confirm our theoretical results.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  = 0.5 and  = 0.8 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.

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 ( + ℎ 4 ).The numerical results demonstrate the effectiveness of the proposed scheme.