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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 912464, 5 pages

http://dx.doi.org/10.1155/2014/912464
Research Article

Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method

College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450000, China

Received 22 June 2014; Accepted 1 July 2014; Published 13 July 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Yu Zhang. 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

The initial-boundary value problems for the local fractional differential equation are investigated in this paper. The local fractional Fourier series solutions with the nondifferential terms are obtained. Two illustrative examples are given to show efficiency and accuracy of the presented method to process the local fractional differential equations.

1. Introduction

In various fields of physics, mathematics, and engineering, because of the different operators, there are classical differential equations [1], fractional differential equation [24], and local fractional differential equations [5, 6]. There are more techniques to achieve analytical approximations to the solutions to differential equations in mathematical physics, such as the decomposition method [7], the variational iteration method [8], the homotopy perturbation method [9], the heat-balance integral method [10], the Fourier transform [11], the Laplace transform [11], and the references therein.

Recently, a new Fourier series (local fractional Fourier series) via local fractional operator was proposed [6] and had various applications in the applied fields such as fractal wave problems in fractal string [12, 13] and the heat-conduction problems arising in fractal heat transfer [14, 15]. For a detailed description of the local fractional Fourier series method, we refer the readers to the recent works [1416]. This is the main advantage of local fractional differential equations in comparison with classical integer-order and fractional-order models.

In the present paper we consider the local fractional differential equation: subject to the initial-boundary value conditions: where the operators are described by the local fractional differential operators [5, 6, 1215]. The paper is organized as follows. In Section 2, the basic theory of the local fractional calculus and local fractional Fourier series is presented. In Section 3, we discuss the initial-boundary problems for local fractional differential equation. Finally, Section 4 is devoted to the conclusions.

2. Analysis of the Method

In this section, we present the basic theory of the local fractional calculus and analyze the local fractional Fourier series method.

Definition 1. Let be a subset of the real line and be a fractal. The mass function can be written as [5] The following properties are present as follows [5].(i)If , then .(ii)If and , then .If is a bi-Lipschitz mapping, then we have [5, 12] such that In view of (5), we have such that where is the fractal dimension of . This result is directly deduced from fractal geometry and relates to the fractal coarse-grained mass function , which reads [5, 13] with where is dimensional Hausdorff measure.

Definition 2. If there is [5, 6, 1215] with , for and , then is called local fractional continuous at .

If is local fractional continuous on the interval , then we can write it in the form [5, 6, 12]

Definition 3. Local fractional derivative of of order at is defined as follows [5, 6, 1215]: where .

From (12) we can rewrite the local fractional derivative as where

Definition 4. The partition of the interval is , , and with and . Local fractional integral of of order in the interval is given by [5, 6, 1215]

Following (14), we have where If are Cantor sets, we can get the derivative and integral on Cantor sets.

Some properties of local fractional integrals are listed as follows:

Definition 5. Local fractional trigonometric Fourier series of is given by [6, 1216] for and .

The local fractional Fourier coefficients of (19) can be computed by If , then we get where the local fractional Fourier coefficients can be computed by

3. The Initial-Boundary Problems for the Local Fractional Differential Equation

In this section, we consider (1) with the various initial-boundary conditions.

Example 6. The initial-boundary condition (2) becomes Let in (1). Separation of the variables yields Setting we obtain Hence, we have their solutions, which read Therefore, a solution is written in the form where .

For the given condition there is , so that For the given condition we obtain Thus, from (33) we deduce that To satisfy the condition (23), (34) is written in the form Then, we derive

Example 7. Let us consider (1) with the initial-boundary value condition, which becomes Following (35), we have where Hence, we get

4. Conclusions

In this work, the initial-boundary value problems for the local fractional differential equation are discussed by using the local fractional Fourier series method. Analytical solutions for the local fractional differential equation with the nondifferentiable conditions are obtained.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

  1. A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam, The Netherlands, 2002.
  2. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  3. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  4. J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  5. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  6. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.
  7. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, The Netherlands, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 141–156, 2008.
  9. H. Jafari and S. Momani, “Solving fractional diffusion and wave equations by modified homotopy perturbation method,” Physics Letters. A, vol. 370, no. 5-6, pp. 388–396, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. J. Hristov, “Approximate solutions to fractional subdiffusion equations,” European Physical Journal Special Topics, vol. 193, no. 1, pp. 229–243, 2011. View at Publisher · View at Google Scholar · View at Scopus
  11. 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
  12. M. 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 MathSciNet
  13. Y. Yang, D. Baleanu, and X. Yang, “Analysis of fractal wave equations by local fractional Fourier series method,” Advances in Mathematical Physics, vol. 2013, Article ID 632309, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  14. Y. Z. Zhang, A. M. Yang, and X. J. Yang, “1-D heat conduction in a fractal medium: a solution by the local fractional Fourier series method,” Thermal Science, vol. 17, no. 3, pp. 953–956, 2013.
  15. A. M. Yang, C. Cattani, C. Zhang, et al., “Local fractional Fourier series solutions for non-homogeneous heat equations arising in fractal heat flow with local fractional derivative,” 5, vol. 2014, Article ID 514639, 5 pages, 2014. View at Publisher · View at Google Scholar
  16. Z. Y. Chen, C. Cattani, and W. P. Zhong, “Signal processing for nondifferentiable data defined on cantor sets: a local fractional fourier series approach,” Advances in Mathematical Physics, vol. 2014, Article ID 561434, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet