Abstract and Applied Analysis

Volume 2014 (2014), Article ID 535048, 6 pages

http://dx.doi.org/10.1155/2014/535048

Research Article

Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

Dumitru Baleanu,1,2,3 J. A. Tenreiro Machado,⁴ Carlo Cattani,⁵ Mihaela Cristina Baleanu,⁶ and Xiao-Jun Yang⁷

¹Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

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

³Institute of Space Sciences, Magurele, Bucharest, Romania

⁴Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal

⁵Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy

⁶Mihail Sadoveanu Theoretical High School, District 2, Street Popa Lazar No. 8, 021586 Bucharest, Romania

⁷Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China

Received 9 November 2013; Accepted 8 December 2013; Published 2 January 2014

Academic Editor: Ming Li

Copyright © 2014 Dumitru Baleanu 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 perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

1. Introduction

Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger’s, the telegraph, the Advection, the Burgers, the KdV, the Boussinesq, and the Fisher equations and others [1].

Recently, the fractional calculus theory was recognized to be a good tool for modeling complex problems demonstrating its applicability in numerical scientific disciplines. Boundary value problems for the fractional differential equations have been the focus of several studies due to their frequent appearance in various areas, such as fractional diffusion and wave [2], fractional telegraph [3], fractional KdV [4], fractional Schrödinger [5], fractional evolution [6], fractional Navier-Stokes [7], fractional Heisenberg [8], fractional Klein-Gordon [9], and fractional Fisher equations [10].

Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations (see, e.g., [1–36] and the references therein). The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16, 17], the local fractional decomposition [18], and the generalized local fractional Fourier transform [19] methods.

Recently, the wave equation on Cantor sets (local fractional wave equation) was given by [35]

where the operators are local fractional ones [16–19, 35, 36].

Following (1), a wave equation on Cantor sets was proposed as follows [36]:

where is a fractal wave function.

In this paper, our purpose is to compare the local fractional variational iteration and decomposition methods for solving the local fractional differential equations. For illustrating the concepts we adopt one example for solving the wave equation on Cantor sets with local fractional operator.

Bearing these ideas in mind, the paper is organized as follows. In Section 2, we present basic definitions and provide some properties of local fractional derivative and integration. In Section 3, we introduce the local fractional variational iteration and the decomposition methods. In Section 4, we discuss one application. Finally, in Section 5 we outline the main conclusions.

2. Mathematical Tools

We recall in this section the notations and some properties of the local fractional operators [15–19, 35, 36].

Definition 1 (see [15–19, 35, 36]). The function is local fractional continuous, if it is valid for

where , for and .

We notice that there are existence conditions of local fractional continuities that operating functions are right-hand and left-hand local fractional continuity. Meanwhile, the right-hand local fractional continuity is equal to its left-hand local fractional continuity. For more details, see [35].

Following (4), we have [15–19, 35, 36]

with , for , and , , , .

For a fractal set , there is a fractal measure [35]

where presents a bi-Lipschitz mapping with fractal dimension and denotes a Hausdorff dimension.

We verify that there is a measure

in the case of and is a Lipschitz mapping. If is a Cantor set, we have with .

Definition 2 (see [15–19, 35, 36]). The local fractional derivative of at is defined as [16–20]

where

We find that the existence condition for local fractional derivative of is that the right-hand local fractional derivative is equal to the left-hand local fractional derivative (see, e.g., [16, 35] and the references therein).

Definition 3 (see [15–19, 35, 36]). A partition of the interval is denoted as , , , and with and . Local fractional integral of in the interval is given by

If the functions are local fractional continuous then the local fractional derivatives and integrals exist. That is to say, operating functions have nondifferentiable and fractal properties (see [35] and the references therein).

Some properties of local fractional derivative and integrals are given in [35].

3. Analytical Methods

In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation as follows:

where is linear local fractional operators, respectively, with and is linear local fractional operators of order less than .

3.1. Local Fractional Variational Iteration Method

The local fractional variational iteration algorithm is given by [16, 17] on the line of the formalism suggested in [35]

Here, we can construct a correction functional as follows [16, 17]:

where is considered as a restricted local fractional variation; that is, (for more details, see [35]).

For , we have

so that iteration is expressed as

Finally, the solution is

3.2. Local Fractional Decomposition Method

When in (10) is a local fractional differential operator of order , we denote it as

By defining the -fold local fractional integral operator

we get

Thus,

where the term is to be determined from the fractal initial conditions.

Therefore, we get the iterative formula as follows:

with .

Hence, for , we have the following recurrence relationship:

Finally, the solution can be constructed as

For more details, see [18].

4. An Illustrative Example

In this section one example for wave equation is presented in order to demonstrate the simplicity and the efficiency of the above methods.

In (2), we consider the following initial and boundary conditions:

Using (14) we have the iterative formula

where the initial value is given by

Thus, after computing (23) we obtain

Hence, from (27) we obtain the solution of (3) as

Here, from (21) we get

Therefore, from (29) we give the components as follows:

Consequently, the exact solution is given by

where

The solution of (2) for is depicted in Figure 1.

Figure 1: Graph of for .

5. Conclusions

In this work, we developed a comparison between the variational iteration method and the decomposition method within local fractional operators. The two approaches constitute efficient tools to handle the approximation solutions for differential equations on Cantor sets with local fractional derivative. We notice that the fractional variational iteration method gives the several successive approximate formulas using the iteration of the correction local fractional functional. However, the local fractional decomposition method provides the components of the exact solution, which is local fractional continuous function, where these components are also local fractional continuous functions. Both the variational iteration method and the decomposition method within local fractional operators provide the solution in successive components. The methods are structured to get the local fractional series solution, which is a nondifferentiable function.

Conflict of Interests

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

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