Abstract and Applied Analysis

Volume 2014 (2014), Article ID 176395, 7 pages

http://dx.doi.org/10.1155/2014/176395

## Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

^{1}School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China^{2}Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran

Received 17 November 2013; Accepted 9 December 2013; Published 2 January 2014

Academic Editor: Hossein Jafari

Copyright © 2014 Shun-Qin Wang 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 propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

#### 1. Introduction

Fractional differential equations with arbitrary orders [1] have attracted more and more attention to their extensive applications in various areas, such as physics, applied mathematics, and biology [2–8]. As a result, great deal of methods for solving the fractional differential equations are developed [9–21], such as the heat balance integral method [9, 10], the homotopy analysis method [11], the variational iteration method [12], the homotopy decomposition method [13, 14], and the Adomian decomposition method [15, 16].

The fractional differential equations were considered in sense of the Caputo derivative, the Riemann-Liouville derivative, and the Grünwald-Letnikov derivative [17]. However, they do not deal with the nondifferentiable functions defined on Cantor sets. Local fractional derivative [18, 19] is the best method for describing the nondifferential problems defined on Cantor sets. For example, the heat equations arising in fractal transient conduction were investigated in [19–22]. The Helmholtz and diffusion equations on the Cantor sets within local fractional derivative were discussed [23]. The Navier-Stokes equations on Cantor sets were suggested in [24]. There are some methods for solving the local fractional differential equations, such as the local fractional variational iteration method [20], the Yang-Fourier transform [21], the Yang-Laplace transform [22], the local fractional Fourier series method [25], and the local fractional Adomian decomposition method [26].

In this paper, our aims are to present the coupling method of local fractional series method and Yang-Laplace transform, which is called as the local fractional function decomposition method, and to use it to solve the differential equations with local fractional derivative. The organization of the manuscript is as follows. In Section 2, the basic mathematical tools are introduced. In Section 3, the local fractional function decomposition method for solving the differential equations with local fractional derivative is investigated. In Section 4, several examples are considered. Finally, in Section 5 the conclusions are given.

#### 2. Mathematical Fundamentals

In this section, we introduce the basic notions of local fractional continuity, local fractional derivative, local fractional Fourier series, and special function in fractal space [18, 19], which are used in the paper.

*Definition 1. *Suppose that there is [19]
with , for and ; then is called local fractional continuous at and it is denoted by .

*Definition 2. *Suppose that the function satisfies condition (1), for ; it is so-called local fractional continuous on the interval , denoted by

*Definition 3. *In fractal space, let , local fractional derivative of of order at is given by [19]
where .

Local fractional derivative of high order and local fractional partial derivative are defined, respectively, in the following forms [18, 19]:

*Definition 4. *In fractal space, the Mittage Leffler function, sine function, cosine function, hyperbolic sine function, and hyperbolic cosine function are, respectively, defined by [18, 19]

*Definition 5. *Let be -periodic. For and , the local fraction Fourier series of is defined as (see [18, 25])
where
are the local fraction Fourier coefficients.

*Definition 6. *Let . The Yang-Laplace transforms of is given by [18, 22]
where the latter integral converges and .

*Definition 7. *The inverse formula of the Yang-Laplace transforms of is given by [18, 22]:
where ; fractal imaginary unit and .

#### 3. Local Fractional Function Decomposition Method

In this section we will present the local fractional function decomposition method.

At first, we present the local fractional differential equation with constants , , , and with boundary and initial conditions Now we discuss the solution of (10).

According to the decomposition of the local fractional function, with respect to the system , the following functions coefficients can be given by where Substituting (12) into (10) implies that Suppose that the Yang-Laplace transforms of functions and are and , respectively. Then we obtain That is Hence, we have Let Hence, we get Then, making use of (8) and (9) and rearranging integration sequence, we have the following several formulas about and .

(I) Suppose that where

Then, we get When we get

(II) If then we have When we arrive at

(III) Let where .

Then, we have When we obtain The above results are the desired solutions.

#### 4. Illustrative Examples

In order to illustrate the above results in Section 3, we give the following several examples.

*Example 1. *The local fractional Laplace differential equation is written in the following form [18, 19]:
subjected to the boundary and initial conditions described by
From (33), the final solution can be easily deduced as follows:

*Example 2. *We consider the following inhomogeneous wave equation with local fractional derivative:
subjected to the boundary and initial conditions
In order to find its solution, we suppose that
which leads to
Contrasting (37) with (35), we directly get
According to (30) and (32), we can derive
Conclusively, we get
Thus, we obtain

*Example 3. *The inhomogeneous wave equation with local fractional differential operator is written in the following form:
The boundary and initial conditions are described by
In order to find the solution of (46), we set
Hence, we get
Let
Making use of (21) and (23),we can write
When
we obtain
Conclusively, we arrive at
Hence, we obtain the solution of (46) in the following form:

*Example 4. *The inhomogeneous wave equation with local fractional differential operator is written in the form
The boundary and initial conditions are presented as follows:
Let
We can write
Obviously, we have
From (25) and (27) we obtain
Hence, the nondifferentiable solution of (56) reads as

#### 5. Conclusions

In this work we proposed the local fractional function decomposition method. The applications of the methods for solving the inhomogeneous wave equations with local fractional derivative are discussed in detail. The new technique is an efficient mathematical tool for the scientists to deal with local fractional differential equations.

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 609040410), the foundation and advanced technology research program of Henan province (112300410300), and the Nanyang Normal University (NYNU200749).

#### References

- R. D. Driver,
*Ordinary and Delay Differential Equations*, vol. 20 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1977. View at Publisher · View at Google Scholar · View at MathSciNet - I. Podlubny,
*Fractional differential equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science, Amsterdam, The Netherlands, 2006. View at MathSciNet - A. M. Wazwaz,
*Partial Differential Equations: Methods and Applications*, Balkema, Rotterdam, The Netherlands, 2002. - J. Fan and J. H. He, “Fractal derivative model for air permeability in hierarchic porous media,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 354701, 7 pages, 2012. View at Publisher · View at Google Scholar - N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,”
*Rheologica Acta*, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. Karami, A. Babakhani, and D. Baleanu, “Existence results for a class of fractional differential equations with periodic boundary value conditions and with delay,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 176180, 8 pages, 2013. View at Publisher · View at Google Scholar - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - 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 - J. Hristov, “A short-distance integral-balance solution to a strong subdiffusion equation: a weak power-law profile,”
*International Review of Chemical Engineering*, vol. 2, no. 5, pp. 555–563, 2010. View at Google Scholar - H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2006–2012, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Jafari and H. Tajadodi, “He's variational iteration method for solving fractional Riccati differential equation,”
*International Journal of Differential Equations*, vol. 2010, Article ID 764738, 8 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Atangana and A. Secer, “The time-fractional coupled-Korteweg-de-Vries equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 947986, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Atangana and A. Kılıçman, “Analytical solutions of boundary values problem of 2D and 3D poisson and biharmonic equations by homotopy decomposition method,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 380484, 9 pages, 2013. View at Publisher · View at Google Scholar - C. Li and Y. Wang, “Numerical algorithm based on Adomian decomposition for fractional differential equations,”
*Computers and Mathematics with Applications*, vol. 57, no. 10, pp. 1672–1681, 2009. View at Publisher · View at Google Scholar · View at Scopus - V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Scopus - C. Li, D. Qian, and Y. Chen, “On Riemann-Liouville and Caputo derivatives,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 562494, 15 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic Publisher, Hong Kong, 2011. - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - 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 Publisher · View at Google Scholar - A. M. Yang, Y. Z. Zhang, and Y. Long, “The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar,”
*Thermal Science*, vol. 17, no. 3, pp. 707–713, 2013. View at Publisher · View at Google Scholar - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at Publisher · View at Google Scholar - Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 754248, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M.-S. Hu, R. P. Agarwal, and X.-J. Yang, “Local fractional Fourier series with application to wave equation in fractal vibrating string,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 567401, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · 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 Series A*, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar