Advances in Mathematical Physics

Volume 2013 (2013), Article ID 632309, 6 pages

http://dx.doi.org/10.1155/2013/632309

Research Article

Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

Yong-Ju Yang,¹ Dumitru Baleanu,2,3,4 and Xiao-Jun Yang⁵

¹School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China

²Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey

³Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

⁴Institute of Space Sciences, Magurele, 077125 Bucharest, Romania

⁵Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 12 May 2013; Accepted 13 June 2013

Academic Editor: H. Srivastava

Copyright © 2013 Yong-Ju 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.

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations.

1. Introduction

Fractional calculus deals with derivative and integrals of arbitrary orders [1]. During the last four decades, fractional calculus has been applied to almost every field of science and engineering [2–6]. In recent years, there has been a great deal of interest in fractional differential equations [7]. As a result, various kinds of analytical methods were developed [8–18]. For example, there are the exp-function method [8], the variational iteration method [9, 10], the homotopy perturbation method [11], the homotopy analysis method [12], the heat-balance integral method [13], the fractional variational iteration method [14, 15], the fractional difference method [16], the finite element method [17], the fractional Fourier and Laplace transforms [18], and so on.

Recently, local fractional calculus was applied to deal with problems for nondifferentiable functions; see [19–26] and the references therein. There are also analytical methods for solving the local fractional differential equations, which are referred to in [27–34]. The local fractional series method [32–34] was applied to process the local fractional wave equation in fractal vibrating [32] and local fractional heat-conduction equation [33].

More recently, the wave equation on the Cantor sets was considered as [21, 28]

Local damped wave equation was written in the form [30]

and local fractional dissipative wave equation in fractal strings was [31]

In this paper, we investigate the application of local fractional series method for solving the following local fractional wave equation:

where initial and boundary conditions are presented as

The organization of the paper is as follows. In Section 2, the basic concepts of local fractional calculus and local fractional Fourier series are introduced. In Section 3, we present a local fractional Fourier series solution of wave equation with local fractional derivative. Two examples are shown in Section 4. Finally, Section 5 is devoted to our conclusions.

2. Mathematical Tools

In this section, we present some concepts of local fractional continuity, local fractional derivative, and local fractional Fourier series.

Definition 1 (see [21, 28, 30–32]). Suppose that there is

with , for , and , . Then is called local fractional continuous at , where with .

Suppose that the function satisfies the above properties of the local fractional continuity. Then the condition (6) for is denoted as

where .

Definition 2 (see [19–21]). Let . Local fractional derivative of of order at is given by

where .

Definition 3 (see [19, 20, 32–34]). Let , and let be -periodic. For , local fraction Fourier series of is defined as

where the local fraction Fourier coefficients are

with local fractional integral given by [21, 29–34]

where , and , , , is a partition of the interval .

In view of (10), the weights of the fractional trigonometric functions are expressed as follows:

Lemma 4 (see [21]). If and are constant coefficients, then local fractional differential equation with constant coefficients

has a family of solution

with two constants and .

Proof. See [21].

3. Solution to Wave Equation with Local Fractional Derivative

If we have the particular solution of (4) in the form

then we get the equations

with the boundary conditions

Equation (4) has the solution

where and are all constant numbers.

According to (19), for and we get

Obviously , since otherwise .

Hence, we arrive at

where is an integer.

We notice

For and , following (17) implies that

where

Therefore,

We now suppose a local fractional Fourier series solution of (4):

Therefore,

where

with .

Submitting (26) to (5), we have

So,

Let

In view of (30) and (31), we rewrite

We now find the local fractional Fourier coefficients of and , respectively,

Following (34), we have

such that

Thus, we get the solution of (4):

where

with

with

4. Illustrative Examples

In order to illustrate the above result in this section, we give two examples.

Let us consider (4) subject to initial and boundary conditions

In view of (40), we have

such that

Hence,

where

In view of (4), our second example is initial and boundary conditions as follows:

Following (40), we get

Hence, we obtain

So,

with

We notice that fraction boundary condition is expressed as a Lebesgue-Cantor staircase function [21, 32]; namely,

where is any fractal set and the fractal dimension of is . For the graph of the Lebesgue-Cantor staircase function (52) is shown in Figure 1 when fractal dimension is .

Figure 1: For , graph of a Lebesgue-Cantor staircase function shown at .

5. Conclusions

The present work expresses the local fractional Fourier series solution to wave equations with local fractional derivative. Two examples are given to illustrat approximate solutions for wave equations with local fractional derivative resulting from local fractional Fourier series method. The results obtained from the local fractional analysis seem to be general since the obtained solutions go back to the classical one when fractal dimension ; namely, it is a process from fractal geometry to Euclidean geometry. Local fractional Fourier series method is one of very efficient and powerful techniques for finding the solutions of the local fractional differential equations. It is also worth noting that the advantage of the local fractional differential equations displays the nondifferential solutions, which show the fractal and local behaviors of moments. However, the classical Fourier series is used to handle the continuous functions.

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