Abstract and Applied Analysis

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

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

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

^{1}Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey^{3}Institute of Space Sciences, Magurele, Bucharest, Romania^{4}Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal^{5}Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy^{6}Mihail Sadoveanu Theoretical High School, District 2, Street Popa Lazar No. 8, 021586 Bucharest, Romania^{7}Department 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.

#### 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.

#### References

- A. M. Wazwaz,
*Partial Differential Equations: Methods and Applications*, Elsevier, Balkema, The Netherlands, 2002. - W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,”
*Journal of Mathematical Physics*, vol. 30, no. 1, pp. 134–144, 1989. View at Google Scholar · View at Scopus - Z. G. Zhao and C. P. Li, “Fractional difference/finite element approximations for the time-space fractional telegraph equation,”
*Applied Mathematics and Computation*, vol. 219, no. 6, pp. 2975–2988, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - S. Momani, Z. Odibat, and A. Alawneh, “Variational iteration method for solving the space- and time-fractional KdV equation,”
*Numerical Methods for Partial Differential Equations*, vol. 24, no. 1, pp. 262–271, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - N. Laskin, “Fractional Schrödinger equation,”
*Physical Review E*, vol. 66, Article ID 056108, 2002. View at Google Scholar - Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - V. E. Tarasov, “Fractional Heisenberg equation,”
*Physics Letters A*, vol. 372, no. 17, pp. 2984–2988, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional KleinGordon equation,”
*Signal Processing*, vol. 91, no. 3, pp. 446–451, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z. B. Li, W. H. Zhu, and L. L. Huang, “Application of fractional variational iteration method to time-fractional Fisher equation,”
*Advanced Science Letters*, vol. 10, no. 1, pp. 610–614, 2012. View at Google Scholar - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at Publisher · View at Google Scholar · View at Scopus - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - C. Cattani, “Harmonic wavelet solution of Poisson's problem,”
*Balkan Journal of Geometry and Its Applications*, vol. 13, no. 1, pp. 27–37, 2008. View at Google Scholar · View at Scopus - C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,”
*Computers & Mathematics with Applications*, vol. 50, no. 8-9, pp. 1191–1210, 2005. View at Publisher · View at Google Scholar · View at Scopus - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, China, 2011. - X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at Google Scholar - W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,”
*Fixed Point Theory and Applications*, vol. 2013, article 89, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on Cantor time-space,”
*Proceeding of the Romanian Academy A*, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, no. 1, pp. 131–146, 2013. View at Google Scholar - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics: Recent Advances*, World Scientific, Singapore, 2012. - J. Sabatier, O. P. Agrawal, and J. T. Machado,
*Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering*, Springer, 2007. - R. Hilfer,
*Applications of Fractional Calculus in Physics*, Word Scientific, Singapore, 2000. - M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 657839, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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 Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,”
*Chaos*, vol. 6, no. 4, pp. 505–513, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 66–79, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,”
*Journal of Mathematical Analysis and Applications*, vol. 362, no. 1, pp. 17–33, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Carpinteri, B. Chiaia, and P. Cornetti, “Static-kinematic duality and the principle of virtual work in the mechanics of fractal media,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 191, no. 1-2, pp. 3–19, 2001. View at Publisher · View at Google Scholar · View at Scopus - A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,”
*Computers and Structures*, vol. 82, no. 6, pp. 499–508, 2004. View at Publisher · View at Google Scholar · View at Scopus - A. Carpinteri and P. Cornetti, “A fractional calculus approach to the description of stress and strain localization in fractal media,”
*Chaos, Solitons & Fractals*, vol. 13, no. 1, pp. 85–94, 2002. View at Publisher · View at Google Scholar · View at Scopus - F. Ben Adda and J. Cresson, “About non-differentiable functions,”
*Journal of Mathematical Analysis and Applications*, vol. 263, no. 2, pp. 721–737, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - W. Chen, “Time-space fabric underlying anomalous diffusion,”
*Chaos, Solitons & Fractals*, vol. 28, no. 4, pp. 923–929, 2006. View at Publisher · View at Google Scholar · View at Scopus - W. Chen, H. G. Sun, X. D. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,”
*Computers & Mathematics with Applications*, vol. 59, no. 5, pp. 1754–1758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,”
*Computers & Mathematics with Applications*, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 351057, 5 pages, 2013. View at Google Scholar · View at MathSciNet