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.
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.
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.
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.
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).