Abstract and Applied Analysis

Volume 2014 (2014), Article ID 372741, 6 pages

http://dx.doi.org/10.1155/2014/372741

Research Article

Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

Ai-Min Yang,1,2 Yu-Zhu Zhang,2,3 Carlo Cattani,⁴ Gong-Nan Xie,⁵ Mohammad Mehdi Rashidi,⁶ Yi-Jun Zhou,⁷ and Xiao-Jun Yang⁸

¹College of Science, Hebei United University, Tangshan 063009, China

²College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China

³College of Metallurgy and Energy, Hebei United University, Tangshan 063009, China

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

⁵School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710048, China

⁶Department of Mechanical Engineering, Bu-Ali Sina University, P.O. Box 65175-4161, Hamedan, Iran

⁷Qinggong College, Hebei United University, Tangshan 063009, China

⁸Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 5 January 2014; Revised 8 February 2014; Accepted 8 February 2014; Published 12 March 2014

Academic Editor: Ali H. Bhrawy

Copyright © 2014 Ai-Min Yang 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.

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

1. Introduction

The Klein-Gordon equation [1] has been applied to mathematical physics such as solid-state physics, nonlinear optics, and quantum field theory. Some of the analytical methods for solving the Klein-Gordon equation include the variational iteration method [2], the tanh and the sine-cosine methods [3], the decomposition method [4], the differential transform method [5], and the homotopy-perturbation method [6].

Recently, the solutions for the fractional Klein-Gordon equation with the Caputo fractional derivative were considered in [7–9]. Golmankhaneh et al. used the homotopy-perturbation method to obtain solution for the fractional Klein-Gordon equation [7]. Kurulay [8] pointed out the solution for the fractional Klein-Gordon equation by using the homotopy analysis method. Gepreel and Mohamed [9] presented the solution for nonlinear space-time fractional Klein-Gordon equation by the homotopy analysis method.

When some domains cannot be described by smooth functions, both the classical approach and the fractional approach based on Riemann-Liouville (or Caputo) derivatives are unacceptable. In such cases, the local fractional calculus is an efficient technique for modeling these physical problems [10–23]. Using the fractional complex transform method [20], one transforms the classical Klein-Gordon equation into the Klein-Gordon equation on Cantor sets in the following form:

subject to the initial value conditions:

where the operator is the local fractional derivative operator, which is defined by [16–23]

with and and are the local fractional continuous functions and are the mixed terms of nonlinear and liner functions.

In view of (1)-(2), the linear Klein-Gordon equation on Cantor sets:

subject to the initial value conditions:

is under consideration, where and are local fractional continuous functions.

On the other hand, the local fractional series expansion method was applied to solve the wave and diffusion equations on Cantor sets [21], the local fractional Schrödinger equation in the one-dimensional Cantorian system [22], and the local fractional Helmholtz equation [23]. In this paper, our aim is to investigate a new application of this technology to solve the linear Klein-Gordon equations on Cantor sets. The paper is organized as follows. In Section 2, the idea of local fractional series expansion method is given. In Section 3, the solutions for linear Klein-Gordon equations on Cantor sets are presented. Finally, Section 4 is the conclusions.

2. The Local Fractional Series Expansion Method

In order to illustrate the idea of the local fractional series expansion method [21–23], we consider the local fractional differential operator equation in the following form:

where is the linear local fractional operator and is a local fractional continuous function.

From (6), the multiterm separated functions with respect to , read as

where and are the local fractional continuous functions.

From (7), we have

so that

In view of (9), we obtain

Making use of (10), we have

so that

Hence, from (12) we get

We now rewrite (4) in the local fractional operator form as follows:

subject to the initial value conditions:

where the linear local fractional operator is defined as follows:

Hence, (16) is a special case of (6) and it is used with the linear Klein-Gordon equations on Cantor sets in next section.

3. Analytical Solutions for Linear Klein-Gordon Equations on Cantor Sets

In this section, we present the nondifferentiable solutions for linear Klein-Gordon equations on Cantor sets.

Example 1. Let us consider the Klein-Gordon equations on Cantor sets in the following form:

subject to the initial value conditions:

From (12) and (18), we can structure the following iterative formulas:

Hence, we can calculate

and so on.

Therefore, we have

and the corresponding graph is illustrated in Figure 1.

Figure 1: The plot of for the parameter .

Example 2. We consider the following Klein-Gordon equations on Cantor sets:

subject to the initial value conditions:

From (12) and (23), we get the following iterative formulas:

Hence, we get

and so on.

Hereby, we obtain the solution of (22):

and the corresponding graph is depicted in Figure 2.

Figure 2: The plot of for the parameter .

Example 3. We present the following Klein-Gordon equations on Cantor sets:

subject to the initial value conditions:

From (12) and (27)-(28), we get the following iterative formulas:

From (29) we obtain

and so on.

Therefore, we obtain the exact solution of (27)

and its graph is shown in Figure 3.

Figure 3: The plot of for the parameter .

Example 4. The Klein-Gordon equation on Cantor sets is presented as

and the initial value conditions are written as

From (12) and (27)-(28), the following iterative formulas are as follows:

From (29), we give

and so on.

Therefore, we give the exact solution of (32):

and its graph is shown in Figure 4.

Figure 4: The plot of for the parameter .

4. Conclusions

In this work the Klein-Gordon equations on Cantor sets within the local fractional differential operator had been analyzed using the local fractional series expansion method. The nondifferentiable solutions for local fractional Klein-Gordon equations were obtained. The present method is a powerful mathematical tool for solving the local fractional linear differential equations.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by 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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