Research Article | Open Access

Yuan Cao, Wei-Gang Ma, Lian-Chuan Ma, "Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets", *Abstract and Applied Analysis*, vol. 2014, Article ID 803693, 6 pages, 2014. https://doi.org/10.1155/2014/803693

# Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets

**Academic Editor:**Xiao-Jun Yang

#### Abstract

The analytical solutions for the diffusion equations on Cantor sets with the nondifferentiable terms are discussed by using the local fractional functional method, which is a coupling method for local fractional Fourier series and Laplace transform.

#### 1. Introduction

The local fractional calculus [1, 2], as a new branch of fractional calculus, was successfully applied to describe the fractal problems from science and engineering. For example, the local fractional Fokker-Planck equation [3], the local fractional diffusion equations defined on Cantor sets [4, 5], the local fractional wave equation defined on Cantor sets [6, 7], the local fractional Korteweg-de Vries equation [8], the local fractional Schrödinger equation [9], local fractional Navier-Stokes equations on cantor sets [10], the local fractional Laplace equation [11], the local fractional heat-conduction equation [12–16], the local fractional differential equations arising in the fractal forest gap [17], and others [18–21] were discussed.

In this paper, we consider the local fractional diffusion equations defined on Cantor sets [5] given by subject to the initial-boundary conditions where the local fractional partial derivatives denote and , and are the local fractional continuous functions. In the high-speed railway healthy monitor system, the problems of diffusion equations with the nondifferentiable terms always exist in fault diagnosing of high-speed trains and their control systems, so we solve this by the local fractional diffusion equations defined on Cantor sets. The local fractional function decomposition method structured in [11, 22], which is a coupling method of the local fractional Fourier series [21, 22] and the Yang-Laplace transform [14, 16, 18, 22], was used to solve the inhomogeneous local fractional wave equations defined on Cantor sets. The main aim of this paper is to discuss the local fractional diffusion equations defined on Cantor sets by the local fractional functional method.

The paper is organized as follows. In Section 2 the basic theory of the local fractional calculus and the Yang-Laplace transform were given. In Section 3, the local fractional functional method is analyzed. Section 4 presents the applications for the local fractional diffusion equations defined on Cantor sets. Finally, the conclusions are given in Section 5.

#### 2. Preliminaries

In this section, we present the basic theory of the local fractional calculus and the local fractional Laplace transform.

*Definition 1 (see [1, 5–7]). *The local fractional derivative of at is given as follows:
where .

The local fractional partial derivative of order is defined as follows [1]:
and the local fractional partial derivative of high order [1] is
where .

*Definition 2 (see [1, 8–12]). *Let us consider a partition of the interval , which is denoted as and with and . Local fractional integral of in the interval is defined as follows:

*Definition 3 (see [1, 5, 11, 16, 21]). *The Mittag Leffler, sine and cosine functions defined on Cantor sets are given as follows:
for .

*Definition 4 (see [11, 20–22]). *Let be -periodic. For , local fraction Fourier series of is given as
where the local fraction Fourier coefficients are as follows:

*Definition 5 (see [14, 16, 18, 22]). *Let . The local fractional Laplace transform of is given as
The inverse formula local fractional Laplace transform of is given as [14, 16, 18, 22]
where is local fractional continuous, and .

There is the following formula [14, 16, 18, 22]: The basic properties of the local fractional calculus and the local fractional Laplace transform were listed in [1, 14, 16, 18, 22].

#### 3. Analysis of the Local Fractional Functional Method

In this section, we introduce the local fractional functional method for the local fractional diffusion equations defined on Cantor sets [11, 22].

Let us consider the nondifferentiable decomposition of the function with the nondifferentiable systems . There are the following functional coefficients of (1) and (2), which are given as follows: where If we submit (14) into (1) and (2), then we have Taking the local fractional Laplace transform of (16) gives which leads to We can rewrite (18) as where The inverse formula local fractional Laplace transform of (19) gives where Hence, the solution of (1) reads as follows: where

#### 4. The Exact Solutions for Local Fractional Diffusion Equations Defined on Cantor Sets

In this section we give two examples for initial boundary problems for local fractional diffusion equations defined on Cantor sets.

*Example 6. *The initial-boundary values of (1) read as follows:
Making use of (14), we obtain the following formulas:
which lead to the following parameters:
Therefore, (23) gives the nondifferentiable solution of (1) with initial-boundary values (25)
When , we get the nondifferentiable solution
and its graph is shown in Figure 1.

*Example 7. *We present the initial-boundary values of (1) as
Using the relation (14), we get
which reduce to
Using (23), we hence have the nondifferentiable solution of (1) with initial-boundary values (30), which is given as
For , the nondifferentiable solution rewrites as follows:
and its graph is illustrated in Figure 2.

#### 5. Conclusions

Local fractional calculus was applied to describe the physical problems because of nondifferentiable characteristics. In this work, the initial-boundary value problems for the diffusion equation on Cantor sets within the local fractional derivatives were investigated by using the local fractional functional method, which is a coupling method for local fractional Fourier series and Laplace transform based upon the nondifferentiable decomposition of the function with the nondifferentiable systems. The two examples are given to express the efficiency of the presented method and their graphs are also obtained. The results of this paper could provide the theory support to the problems diffusion equations with the nondifferentiable terms in health monitor of high-speed trains and their control systems.

#### Conflict of Interests

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

#### References

- X. J. Yang,
*Advanced Local Fractional C alculus and I ts A pplications*, World Science, New York, NY, USA, 2012. - A. Babakhani and V. D. Gejji, “On calculus of local fractional derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 66–79, 2002. View at: Publisher Site | Google Scholar | MathSciNet - 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: Publisher Site | Google Scholar | MathSciNet - A. Carpinteri and A. Sapora, “Diffusion problems in fractal media defined on Cantor sets,”
*ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik*, vol. 90, no. 3, pp. 203–210, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on Cantor space-time,”
*Proceedings of the Romanian Academy, Series A*, vol. 14, no. 2, pp. 127–133, 2013. View at: Google Scholar | MathSciNet - J. Ahmad, S. T. Mohyud-Din, and X. J. Yang, “Local fractional decomposition method on wave equation in fractal strings,”
*Mitteilungen Klosterneuburg*, vol. 64, no. 2, pp. 98–105, 2014. View at: Google Scholar - 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: Publisher Site | Google Scholar | MathSciNet - X. J. Yang, J. Hristov, and H. M. Srivastava, “Modelling fractal waves on shallow water surfaces via local fractional Korteweg-de Vries equation,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 278672, 10 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, no. 1, pp. 1–16, 2013. View at: Google Scholar - X. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of navier-stokes equations on cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. Yan, H. Jafari, and H. K. Jassim, “Local fractional Adomain d ecomposition and function decomposition methods for Laplace equation within local fractional operators,”
*Advances in Mathematical Physics*, vol. 2014, Article ID 161580, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - A. M. Yang, Y. Z. Zhang, and X. L. Zhang, “The nondifferentiable solution for local fractional Tricomi equation arising in fractal transonic flow by local fractional variational iteration method,”
*Advances in Mathematical Physics*, vol. 2014, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - A. Yang, C. Zhang, H. Jafari, C. Cattani, and Y. Jiao, “Picard successive approximation method for solving differential equations arising in fractal heat transfer with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 395710, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - Y. Z. Zhang, A. M. Yang, and Y. Long, “Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform,”
*Thermal Science*, vol. 18, no. 2, pp. 677–681, 2014. View at: Google Scholar - J. H. He and F. J. Liu, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,”
*Nonlinear Science Letters A*, vol. 4, no. 1, pp. 15–20, 2013. View at: Google Scholar - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at: Google Scholar - C. Long, Y. Zhao, and H. Jafari, “Mathematical models arising in the fractal forest gap via local fractional calculus,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 782393, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - C. G. Zhao, A. M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 386459, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - J. H. He, “Exp-function method for fractional differential equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 14, no. 6, pp. 363–366, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Z. Chen, C. Cattani, and W. P. Zhong, “Signal processing for non-differentiable data defined on Cantor sets: a local fractional fourier series approach,”
*Advances in Mathematical Physics*, vol. 2014, Article ID 561434, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - G. A. Anastassiou and O. Duman,
*Advances in Applied Mathematics and Approximation Theory*, Springer, New York, NY, USA, 2013. - S. Wang, Y. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 176395, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Yuan Cao 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.