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Solving Fokker-Planck Equations on Cantor Sets Using Local Fractional Decomposition Method
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.
The Fokker-Planck equation [1–16] plays an important role in describing the system dynamics. For example, the Langevin approach for microscopic dynamics , the dynamics of energy cascade in turbulence , the population dynamics , the chaotic universe dynamics , the fatigue crack growth dynamics , the fission dynamics , the dynamics of distributions of heavy quarks  and financial returns , the spin relaxation dynamics , the electron dynamics of plasmas and semiconductors , the critical dynamics , and the fiber dynamics  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 , 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 , 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.
The formulas of local fractional integrals operator used in the paper are listed as follows :
Definition 3 (see [42, 43, 46–53]). Let . The local fractional derivative of of order in the interval is defined as
The formulas of local fractional differential operator used in the paper are listed as follows :
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.
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.
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).
A. Naert, R. Friedrich, and J. Peinke, “Fokker-Planck equation for the energy cascade in turbulence,” Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 56, no. 6, pp. 6719–6722, 1997.View at: Google Scholar
M. Saad and T. Goudon, “On a Fokker-Planck equation arising in population dynamics,” Revista Matemática Complutense, vol. 11, no. 2, pp. 353–372, 1998.View at: Google Scholar
A. Tsurui and H. Ishikawa, “Application of the Fokker-Planck equation to a stochastic fatigue crack growth model,” Structural Safety, vol. 4, no. 1, pp. 15–29, 1986.View at: Google Scholar
J. R. Nix, A. J. Sierk, H. Hofmann, F. Scheuter, and D. Vautherin, “Stationary Fokker-Planck equation applied to fission dynamics,” Nuclear Physics, Section A, vol. 424, no. 2, pp. 239–261, 1984.View at: Google Scholar
D. B. Walton and J. Rafelski, “Equilibrium distribution of heavy quarks in Fokker-Planck dynamics,” Physical Review Letters, vol. 84, no. 1, pp. 31–34, 2000.View at: Google Scholar
C. P. Enz, “Fokker-Planck description of classical systems with application to critical dynamics,” Physica A: Statistical Mechanics and its Applications, vol. 89, no. 1, pp. 1–36, 1977.View at: Google Scholar
L. P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization, World Scientific, 2000.
J. Biazar, K. Hosseini, and P. Gholamin, “Homotopy perturbation method Fokker-Planck equation,” International Mathematical Forum, vol. 3, no. 19, pp. 945–954, 2008.View at: Google Scholar
K. Kikuchi, M. Yoshida, T. Maekawa, and H. Watanabe, “Metropolis Monte Carlo method as a numerical technique to solve the Fokker-Planck equation,” Chemical Physics Letters, vol. 185, no. 3-4, pp. 335–338, 1991.View at: Google Scholar
A. H. Bhrawy and M. Alshomrani, “A shifted Legendre spectral method for fractional-order multi-point boundary value problems,” Advances in Difference Equations, vol. 2012, pp. 1–19, 2012.View at: Google Scholar
A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,” Boundary Value Problems, vol. 2012, no. 1, pp. 1–13, 2012.View at: Google Scholar
J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific, 2012.
D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer, 2012.
V. E. Tarasov, Fractional Dynamics, Springer, 2010.
G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2005.
K. Kim and Y. S. Kong, “Anomalous behaviors in fractional Fokker-Planck equation,” Journal of the Korean Physical Society, vol. 40, no. 6, pp. 979–982, 2002.View at: Google Scholar
I. M. Sokolov, “Thermodynamics and fractional Fokker-Planck equations,” Physical Review E, Statistical, Nonlinear, and Soft Matter Physics, vol. 63, no. 5, pp. 561111–561118, 2001.View at: Google Scholar
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
X. -J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, China, 2011.
X.-J. Yang, H. M. Srivastava, J. H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,” Physics Letters A, vol. 377, no. 28–30, pp. 1696–1700, 2013.View at: Google Scholar
K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998.View at: Google Scholar
X.-J. Yang and D. Baleanu, “Local fractional variational iteration method for Fokker-Planck equation on a Cantor set,” Acta Universitaria, vol. 23, no. 2, pp. 3–8, 2013.View at: Google Scholar
X.-J. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on Cantor space-time,” Proceedings of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013.View at: Google Scholar
D. Baleanu, J. A. T. Machado, C. Cattani, M. C. Baleanu, and X.-J. Yang, “Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators,” Abstract and Applied Analysis, vol. 2014, Article ID 535048, 6 pages, 2014.View at: Publisher Site | Google Scholar