Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
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New Trends on Fractional and Functional Differential Equations

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Volume 2014 |Article ID 372741 | 6 pages | https://doi.org/10.1155/2014/372741

Application of Local Fractional Series Expansion Method to Solve Klein-Gordon Equations on Cantor Sets

Academic Editor: Ali H. Bhrawy
Received05 Jan 2014
Revised08 Feb 2014
Accepted08 Feb 2014
Published12 Mar 2014

Abstract

We use the local fractional series expansion method to solve the Klein-Gordon equations on Cantor sets within the local fractional derivatives. The analytical solutions within the nondifferential terms are discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems of the liner differential equations on Cantor sets.

1. Introduction

The Klein-Gordon equation [1] has been applied to mathematical physics such as solid-state physics, nonlinear optics, and quantum field theory. Some of the analytical methods for solving the Klein-Gordon equation include the variational iteration method [2], the tanh and the sine-cosine methods [3], the decomposition method [4], the differential transform method [5], and the homotopy-perturbation method [6].

Recently, the solutions for the fractional Klein-Gordon equation with the Caputo fractional derivative were considered in [79]. Golmankhaneh et al. used the homotopy-perturbation method to obtain solution for the fractional Klein-Gordon equation [7]. Kurulay [8] pointed out the solution for the fractional Klein-Gordon equation by using the homotopy analysis method. Gepreel and Mohamed [9] presented the solution for nonlinear space-time fractional Klein-Gordon equation by the homotopy analysis method.

When some domains cannot be described by smooth functions, both the classical approach and the fractional approach based on Riemann-Liouville (or Caputo) derivatives are unacceptable. In such cases, the local fractional calculus is an efficient technique for modeling these physical problems [1023]. Using the fractional complex transform method [20], one transforms the classical Klein-Gordon equation into the Klein-Gordon equation on Cantor sets in the following form: subject to the initial value conditions: where the operator is the local fractional derivative operator, which is defined by [1623] with and and are the local fractional continuous functions and are the mixed terms of nonlinear and liner functions.

In view of (1)-(2), the linear Klein-Gordon equation on Cantor sets: subject to the initial value conditions: is under consideration, where and are local fractional continuous functions.

On the other hand, the local fractional series expansion method was applied to solve the wave and diffusion equations on Cantor sets [21], the local fractional Schrödinger equation in the one-dimensional Cantorian system [22], and the local fractional Helmholtz equation [23]. In this paper, our aim is to investigate a new application of this technology to solve the linear Klein-Gordon equations on Cantor sets. The paper is organized as follows. In Section 2, the idea of local fractional series expansion method is given. In Section 3, the solutions for linear Klein-Gordon equations on Cantor sets are presented. Finally, Section 4 is the conclusions.

2. The Local Fractional Series Expansion Method

In order to illustrate the idea of the local fractional series expansion method [2123], we consider the local fractional differential operator equation in the following form: where is the linear local fractional operator and is a local fractional continuous function.

From (6), the multiterm separated functions with respect to , read as where and are the local fractional continuous functions.

From (7), we have so that In view of (9), we obtain Making use of (10), we have so that Hence, from (12) we get We now rewrite (4) in the local fractional operator form as follows: subject to the initial value conditions: where the linear local fractional operator is defined as follows: Hence, (16) is a special case of (6) and it is used with the linear Klein-Gordon equations on Cantor sets in next section.

3. Analytical Solutions for Linear Klein-Gordon Equations on Cantor Sets

In this section, we present the nondifferentiable solutions for linear Klein-Gordon equations on Cantor sets.

Example 1. Let us consider the Klein-Gordon equations on Cantor sets in the following form: subject to the initial value conditions: From (12) and (18), we can structure the following iterative formulas: Hence, we can calculate and so on.
Therefore, we have and the corresponding graph is illustrated in Figure 1.

Example 2. We consider the following Klein-Gordon equations on Cantor sets: subject to the initial value conditions: From (12) and (23), we get the following iterative formulas: Hence, we get and so on.
Hereby, we obtain the solution of (22): and the corresponding graph is depicted in Figure 2.

Example 3. We present the following Klein-Gordon equations on Cantor sets: subject to the initial value conditions: From (12) and (27)-(28), we get the following iterative formulas: From (29) we obtain and so on.
Therefore, we obtain the exact solution of (27) and its graph is shown in Figure 3.

Example 4. The Klein-Gordon equation on Cantor sets is presented as and the initial value conditions are written as From (12) and (27)-(28), the following iterative formulas are as follows: From (29), we give and so on.
Therefore, we give the exact solution of (32): and its graph is shown in Figure 4.

4. Conclusions

In this work the Klein-Gordon equations on Cantor sets within the local fractional differential operator had been analyzed using the local fractional series expansion method. The nondifferentiable solutions for local fractional Klein-Gordon equations were obtained. The present method is a powerful mathematical tool for solving the local fractional linear differential equations.

Conflict of Interests

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

Acknowledgments

This work was supported by National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 51274270), and the National Natural Science Foundation of Hebei Province (no. E2013209215).

References

  1. A.-M. Wazwaz, “Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 1005–1013, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. E. Yusufoğlu, “The variational iteration method for studying the Klein-Gordon equation,” Applied Mathematics Letters, vol. 21, no. 7, pp. 669–674, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. A.-M. Wazwaz, “The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1179–1195, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. S. M. El-Sayed, “The decomposition method for studying the Klein-Gordon equation,” Chaos, Solitons & Fractals, vol. 18, no. 5, pp. 1025–1030, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. A. S. V. Ravi Kanth and K. Aruna, “Differential transform method for solving the linear and nonlinear Klein-Gordon equation,” Computer Physics Communications, vol. 180, no. 5, pp. 708–711, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. M. S. H. Chowdhury and I. Hashim, “Application of homotopy-perturbation method to Klein-Gordon and sine-Gordon equations,” Chaos, Solitons & Fractals, vol. 39, no. 4, pp. 1928–1935, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. 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 Site | Google Scholar
  8. M. Kurulay, “Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method,” Advances in Difference Equations, vol. 2012, no. 1, article 187, pp. 1–8, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  9. K. A. Gepreel and M. S. Mohamed, “Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation,” Chinese Physics B, vol. 22, no. 1, Article ID 010201, 2013. View at: Google Scholar
  10. 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
  11. 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 Site | Google Scholar
  12. A. K. Golmankhaneh and D. Baleanu, “On a new measure on fractals,” Journal of Inequalities and Applications, vol. 2013, no. 1, article 522, 2013. View at: Google Scholar
  13. A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “Lagrangian and Hamiltonian mechanics on fractals subset of real-line,” International Journal of Theoretical Physics, vol. 52, no. 11, pp. 4210–4217, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. K. G. Alireza, 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
  15. A. S. Balankin, “Stresses and strains in a deformable fractal medium and in its fractal continuum model,” Physics Letters A, vol. 377, no. 38, pp. 2535–2541, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  16. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  17. 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: Google Scholar
  18. S. Q. Wang, Y. J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 176395, 7 pages, 2014. View at: Publisher Site | Google Scholar
  19. J.-H. He, “Exp-function method for fractional differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 14, no. 6, pp. 363–366, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  20. X.-J. Yang, D. Baleanu, and J.-H. He, “Transport equations in fractal porous media within fractional complex transform method,” Proceedings of the Romanian Academy A, vol. 14, no. 4, pp. 287–292, 2013. View at: Google Scholar
  21. 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: Publisher Site | Google Scholar | MathSciNet
  22. Y. Zhao, D.-F. Cheng, and X.-J. Yang, “Approximation solutions for local fractional Schrödinger equation in the one-dimensional Cantorian system,” Advances in Mathematical Physics, vol. 2013, Article ID 291386, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  23. A. M. Yang, Z. S. Chen, H. M. Srivastava, and X. -J. Yang, “Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators,” Abstract and Applied Analysis, vol. 2013, Article ID 259125, 6 pages, 2013. View at: Publisher Site | Google Scholar

Copyright © 2014 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.


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