Abstract and Applied Analysis

Volume 2014 (2014), Article ID 396469, 6 pages

http://dx.doi.org/10.1155/2014/396469

Research Article

Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method

Shao-Hong Yan,1,2 Xiao-Hong Chen,¹ Gong-Nan Xie,³ Carlo Cattani,⁴ and Xiao-Jun Yang⁵

¹College of Geophysics and Information Engineering, China University of Petroleum, Beijing 102249, China

²College of Science, Hebei United University, Tangshan 063000, China

³School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710048, China

⁴Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy

⁵Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 4 February 2014; Accepted 6 March 2014; Published 31 March 2014

Academic Editor: Ali H. Bhrawy

Copyright © 2014 Shao-Hong Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The local fractional decomposition method is applied to approximate the solutions for Fokker-Planck equations on Cantor sets with local fractional derivative. The obtained results give the present method that is very effective and simple for solving the differential equations on Cantor set.

1. Introduction

The Fokker-Planck equation [1–16] plays an important role in describing the system dynamics. For example, the Langevin approach for microscopic dynamics [1], the dynamics of energy cascade in turbulence [2], the population dynamics [3], the chaotic universe dynamics [4], the fatigue crack growth dynamics [5], the fission dynamics [6], the dynamics of distributions of heavy quarks [7] and financial returns [8], the spin relaxation dynamics [9], the electron dynamics of plasmas and semiconductors [10], the critical dynamics [11], and the fiber dynamics [12] had been investigated by using the Fokker-Planck equation (see [13, 14] and the cited references therein) and its 3 solution was presented by the different methods. There are some methods for solving the differential equations, such as Adomian decomposition, Homotopy perturbation, Variational iteration, Metropolis Monte Carlo, multiscale finite element and finite difference methods (see, e.g., [15–20]), and others [21–25].

In recent years, fractional calculus was applied to model many fractal dynamical systems [26–30]. The fractional Fokker-Planck equation [26–38], as one of dynamical equations, has interested many researchers. Its solution was also investigated in [39–41]. However, the above fractional Fokker-Planck equation did not describe the nondifferentiable behaviors of dynamical systems because of the limit of the fractional operators. In order to overcome the above problems, the local fractional calculus was developed and applied to the fractal phenomenon in science and engineering [42–53]. Local fractional Fokker-Planck equation [45], which was an analog of a diffusion equation with local fractional derivative, was suggested as follows:

where the local fractional operator was the Kolwankar-Gangal local fractional differential operator. In [46], the Fokker-Planck equation on a Cantor set with local fractional derivative was presented as follows:

where the local fractional partial differential operator of order () was defined as [42–44]

with

Analytical and approximate solutions for local fractional differential equations were presented by different researchers (see [43, 46–53] and the cited references therein). Applications of local fractional decomposition method were presented (see [51–53] and the cited references therein). Our main purpose of the paper is to apply the local fractional decomposition method to solve the Fokker-Planck equations on a Cantor set.

In this paper, Section 2 gives the recent results for local fractional integral operator. The local fractional decomposition method is analyzed in Section 3. The approximate solutions are presented in Section 4. Finally, conclusions are given in Section 5.

2. The Local Fractional Integral Operator

In this section, we introduce the local fractional integral operator and its recent results.

Definition 1 (see [42, 46–53]). Let the function , if it is valid for

where , for and .

Definition 2 (see [42, 43, 46–53]). Let . The local fractional integral of of order in the interval is defined as

where the partitions of the interval are denoted by , , and with and .

Some properties of local fractional integrals operator used in the paper are listed as follows [42, 51]:

The formulas of local fractional integrals operator used in the paper are listed as follows [42]:

Definition 3 (see [42, 43, 46–53]). Let . The local fractional derivative of of order in the interval is defined as

where

The formulas of local fractional differential operator used in the paper are listed as follows [42]:

3. Analysis of the Method

In this section, the local fractional decomposition method for a class of differential equations defined on Cantor is given.

We now write (2) in the following form:

where is a th local fractional differential operator with respect to , is a th local fractional differential operator with respect to , and   is th  local fractional differential operator with respect to .

The initial condition reads as follows:

We now define the th-fold local fractional integral operator in the form

In view of (13), we structure

Hence, from (15), we have

where the nondifferentiable term is obtained from the initial condition.

Making use of (16), for , we give the recurrence relationship in the following form:

subject to the initial value

Finally, the approximation form of the solution reads as

4. The Approximate Solutions

In this section, we investigate the approximate solutions for Fokker-Planck equations on Cantor sets with local fractional derivative by using the local fractional decomposition method.

Example 1. Let us consider the following Fokker-Planck equation on a Cantor set with local fractional derivative in the form

subject to the initial value

In view of (17), we have the recurrence formulas in the form

From (23), we obtain the following approximate formulas:

Hence, the nondifferentiable solution for (20) with the initial value (21) is

and its graph is shown in Figure 1.

Figure 1: Graph of with the parameter .

Example 2. We consider the Fokker-Planck equation on a Cantor set with local fractional derivative

together with initial condition

From (17), we get the recurrence formulas in the form

Making use of (29), we reach the following formulas:

Therefore, the nondifferentiable solution of (26) with the initial value (27) reads as follows:

with plot shown in Figure 2.

Figure 2: Graph of with the parameter .

Example 3. We present the Fokker-Planck equation on a Cantor set with local fractional derivative

and suggest the initial condition

In view of (17), the recurrence formulas can be written as

From (35), we get the following approximate equalities:

Therefore, the nondifferentiable solution of (32) with the initial value (33) reads as follows:

together with plot shown in Figure 3.

Figure 3: Graph of with the parameter .

Example 4. We suggest the Fokker-Planck equation on a Cantor set with local fractional derivative

and the initial condition is

From (17), the recurrence formulas read as follows:

Hereby, from (41), we have the following formulas:

So the nondifferentiable solution of (32) with the initial value (33) is

together with plot illustrated in Figure 4.

Figure 4: Graph of with the parameter .

5. Conclusions

In this work, we had used the local fractional decomposition method to solve the Fokker-Planck equations on Cantor sets which were described by the local fractional differential operator. The nondifferentiable solutions were obtained. The obtained results show that the present method is a very effective and powerful mathematical tool for finding the nondifferentiable solutions for the local fractional differential equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 51274270), and the National Natural Science Foundation of Hebei Province (no. E2013209215).

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