Research Article | Open Access

Ai-Min Yang, Xiao-Jun Yang, Zheng-Biao 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. https://doi.org/10.1155/2013/351057

# Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

**Academic Editor:**Dumitru Baleanu

#### Abstract

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

#### 1. Introduction

Fractional calculus theory [1–3] has been applied to a wide class of complex problems encompassing physics, biology, mechanics, and interdisciplinary areas [4–9]. Various methods, for example, the Adomian decomposition method [10], the Rach-Adomian-Meyers modified decomposition method [11], the variational iteration method [12, 13], the homotopy perturbation method [13, 14], the fractal Laplace and Fourier transforms [15], the homotopy analysis method [16], the heat-balance integral method [17–19], the fractional variational iteration method [20–22], the fractional subequation method [23, 24], and the generalized Exp-function method [25], have been utilized to solve fractional differential equations [3, 15].

The characteristics of fractal materials have local and fractal behaviors well described by nondifferential functions. However, the classic fractional calculus is not valid for differential equation on Cantor sets due to its no-local nature. In contrast, the local fractional calculus is one of the best candidates for dealing with such problems [26–44]. The local fractional calculus theory has played crucial applications in several fields, such as theoretical physics, transport problems in fractal media described by nondifferential functions. There are some versions of the local fractional calculus where different approaches in definition of the local fractional derivative exist, among them the local fractional derivative of Kolwankar et al. [32–38], the fractal derivative of Chen et al. [39, 40], the fractal derivative of Parvate et al. [41, 42], the modified Riemann-Liouville of Jumarie [43, 44], and versions described in [45–52].

In order to deal with local fractional ordinary and partial differential equations, there are some developed technologies, for example, the local fractional variational iteration method [45, 46], the local fractional Fourier series method [47, 48], the Cantor-type cylindrical-coordinate method [49], the Yang-Fourier transform [50, 51], and the Yang-Laplace transform [52].

The local fractional derivative is defined as follows [26–31, 45–52]: where , and is satisfied with the condition [26, 47] so that [26–31] with , for and .

The main idea of this paper is to present the local fractional series expansion method for effective solutions of wave and diffusion equations on Cantor sets involving local fractional derivatives. The paper has been organized as follows. Section 2 gives a local fractional series expansion method. Some illustrative examples are shown in Section 3. The conclusions are presented in Section 4.

#### 2. Analysis of the Method

Let us consider the local fractional differential equation where is a linear local operator with respect to , .

In accordance with the results in [28, 47], there are multiterm separated functions of independent variables and , namely, where and are local fractional continuous functions.

Moreover, there is a nondifferential series term where is a coefficient.

In view of (6), we may present the solution in the form
Then, following (7), we have
Hence,
In view of (9), we have
Hence, from (10) we can obtain a recursion; namely,
with ; we arrive at the following relation:
with ; we may rewrite (11) as
By the recursion formulas, we can obtain the solution of (4) as
The convergent condition is
This approach is termed *the local fractional series expansion method* (LFSEM)

#### 3. Applications to Wave and Diffusion Equations on Cantor Sets

In this section, four examples for wave and diffusion equations on Cantor sets will demonstrate the efficiency of LFSEM.

*Example 1. *Let us consider the diffusion equation on Cantor set
with the initial condition
Following (12), we have recursive formula
Hence, we get
and so on.

Therefore, through (19) we get the solution

*Example 2. *Let us consider the diffusion equation on Cantor set
with the initial condition
Following (12), we get
By using the recursive formula (23), we get consequently
As a direct result of these recursive calculations, we arrive at

*Example 3. *Let us consider the following wave equation on Cantor sets:
with the initial condition
In view of (14), we obtain
Hence, using the relations (29), the recursive calculations yield
and so on.

Finally, we obtain

*Example 4. *Let us consider the wave equation on Cantor sets [26, 30]
where is a constant.

The initial condition is
By using (14) we have
Then, through the iterative relations (35), we have
Therefore, we obtain
where
For more details concerning (38), we refer to [26–28].

#### 4. Conclusions

In this work, the local fractional series expansion method is demonstrated as an effective method for solutions of a wide class of problems. Analytical solutions of the wave and diffusion equations on Cantor sets involving local fractional derivatives are successfully developed by recurrence relations resulting in convergent series solutions. In this context, the suggested method is a potential tool for development of approximate solutions of local fractional differential equations with fractal initial value conditions, which, of course, draws new problems beyond the scope of the present work.

#### Acknowledgments

The first author was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 11126213 and no. 61170317), and the National Natural Science Foundation of Hebei Province (no. E2013209123). The third author is supported in part by NSF11061028 of China and Yunnan Province NSF Grant no. 2011FB090.

#### References

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. View at: Publisher Site | Zentralblatt MATH | MathSciNet - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity*, Imperial College Press, London, UK, 2010. View at: Publisher Site | Zentralblatt MATH | MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. View at: Zentralblatt MATH | MathSciNet - R. L. Magin,
*Fractional Calculus in Bioengineering,*, Begerll House, West Redding, Conn, USA, 2006. - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics in Physics: Recent Advances*, World Scientific, Singapore, 2011. - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, Oxford, UK, 2008. View at: Zentralblatt MATH | MathSciNet - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. View at: Publisher Site | Zentralblatt MATH | MathSciNet - V. E. Tarasov,
*Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media*, Springer, Berlin, Germany, 2011. View at: Publisher Site | Zentralblatt MATH | MathSciNet - J. A. Tenreiro Machado, A. C. J. Luo, and D. Baleanu,
*Nonlinear Dynamics of Complex Systems: Applications in Physical, Biological and Financial Systems*, Springer, New York, NY, USA, 2011. - J. S. Duan, T. Chaolu, R. Rach, and L. Lu, “The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations,”
*Computers & Mathematics With Applications*, 2013. View at: Publisher Site | Google Scholar - J. S. Duan, T. Chaolu, and R. Rach, “Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach-Adomian-Meyers modified decomposition method,”
*Applied Mathematics and Computation*, vol. 218, no. 17, pp. 8370–8392, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Das, “Analytical solution of a fractional diffusion equation by variational iteration method,”
*Computers & Mathematics with Applications*, vol. 57, no. 3, pp. 483–487, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 910–919, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,”
*Physics Letters A*, vol. 365, no. 5-6, pp. 345–350, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. View at: Publisher Site | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | 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 Site | Google Scholar - J. Hristov, “Integral-balance solution to the stokes' first problem of a viscoelastic generalized second grade fluid,”
*Thermal Science*, vol. 16, no. 2, pp. 395–410, 2012. View at: Google Scholar - J. Hristov, “Transient flow of a generalized second grade fluid due to a constant surface shear stress: an approximate integral-balance solution,”
*International Review of Chemical Engineering*, vol. 3, no. 6, pp. 802–809, 2011. View at: Google Scholar - G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,”
*Physics Letters A*, vol. 374, no. 25, pp. 2506–2509, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science,”
*Computers & Mathematics with Applications*, vol. 62, no. 5, pp. 2273–2278, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. C. Wu, “A fractional variational iteration method for solving fractional nonlinear differential equations,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 2186–2190, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,”
*Physics Letters A*, vol. 375, no. 7, pp. 1069–1073, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 587179, 5 pages, 2013. View at: Publisher Site | Google Scholar - S. Zhang, Q. A. Zong, D. Liu, and Q. Gao, “A generalized expfunction method for fractional Riccati differential equations,”
*Communications in Fractional Calculus*, vol. 1, pp. 48–52, 2010. View at: Google Scholar - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X. J. Yang, “Local fractional integral transforms,”
*Progress in Nonlinear Science*, vol. 4, pp. 1–225, 2011. View at: Google Scholar - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, 2011. - W. P. Zhong, X. J. Yang, and F. Gao, “A Cauchy problem for some local fractional abstract differential equation with fractal conditions,”
*Journal of Applied Functional Analysis*, vol. 8, no. 1, pp. 92–99, 2013. View at: Google Scholar - W. H. Su, X. J. Yang, H. Jafari, and D. Baleanu, “Fractional complex transform method for wave equations on Cantor sets within local fractional differential operator,”
*Advances in Difference Equations*, vol. 2013, no. 1, pp. 97–107, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. S. Hu, D. Baleanu, and X. J. Yang, “One-phase problems for discontinuous heat transfer in fractal media,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 358473, 3 pages, 2013. View at: Publisher Site | Google Scholar - 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: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Carpinteri and A. Sapora, “Diffusion problems in fractal media defined on Cantor sets,”
*ZAMM Journal of Applied Mathematics and Mechanics*, vol. 90, no. 3, pp. 203–210, 2010. View at: Publisher Site | Google Scholar - A. Carpinteri and P. Cornetti, “A fractional calculus approach to the description of stress and strain localization in fractal media,”
*Chaos, Solitons and Fractals*, vol. 13, no. 1, pp. 85–94, 2002. View at: Publisher Site | Google Scholar - 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 Site | Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Chen, “Time-space fabric underlying anomalous diffusion,”
*Chaos, Solitons and Fractals*, vol. 28, no. 4, pp. 923–925, 2006. View at: Publisher Site | Google Scholar - W. Chen, H. Sun, X. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,”
*Romania Reports in Physics*, vol. 65, pp. 84–93, 2013. View at: Google Scholar - A. Parvate and A. D. Gangal, “Fractal differential equations and fractal-time dynamical systems,”
*Pramana*, vol. 64, no. 3, pp. 389–409, 2005. View at: Google Scholar - G. Jumarie, “Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions,”
*Chaos, Solitons and Fractals*, vol. 40, no. 3, pp. 1428–1448, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,”
*Applied Mathematics Letters*, vol. 22, no. 11, pp. 1659–1664, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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, no. 1, pp. 89–102, 2013. View at: Publisher Site | Google Scholar | 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. A. Anastassiou and O. Duman,
*Advances in Applied Mathematics and Approximation Theory*, Springer, New York, NY, USA, 2013. - X. J. Yang, H. M. Srivastava, J. H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,”
*Physics Letters A*, vol. 377, no. 28–30, pp. 1696–1700, 2013. View at: Google Scholar - X. Yang, M. Liao, and J. Chen, “A novel approach to processing fractal signals using the Yang-Fourier transforms,” in
*Proceedings of the International Workshop on Information and Electronics Engineering (IWIEE '12)*, vol. 29, pp. 2950–2954, March 2012. View at: Publisher Site | Google Scholar - W. Zhong, F. Gao, and X. Shen, “Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral,”
*Advanced Materials Research*, vol. 461, pp. 306–310, 2012. View at: Publisher Site | Google Scholar - W. P. Zhong and F. Gao, “Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with fractional derivative,” in
*Proceedings of the 3rd International Conference on Computer Technology and Development*, pp. 209–213, ASME, 2011. View at: Google Scholar

#### Copyright

Copyright © 2013 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.