Abstract and Applied Analysis

Volume 2014 (2014), Article ID 803693, 6 pages

http://dx.doi.org/10.1155/2014/803693

Research Article

Local Fractional Functional Method for Solving Diffusion Equations on Cantor Sets

Yuan Cao,1,2 Wei-Gang Ma,³ and Lian-Chuan Ma1,2

¹National Engineering Research Center of Rail Transportation Operation and Control System, Beijing Jiaotong University, Beijing 100044, China

²School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China

³Faculty of Computer Science and Engineering, Xi’an University of Technology, Xi’An 710048, China

Received 11 July 2014; Accepted 18 July 2014; Published 5 August 2014

Academic Editor: Xiao-Jun Yang

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.

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.

Figure 1: The solution of (1) with initial-boundary value (25) when and .

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.

Figure 2: The solution of (1) with initial-boundary value (30) when and .

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.

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