About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 259125, 6 pages
http://dx.doi.org/10.1155/2013/259125
Research Article

Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators

1College of Science, Hebei United University, Tangshan 063009, China
2College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China
3School of Civil Engineering and Architecture, Chongqing Jiaotong University, Chongqing 400074, China
4Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4
5Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China

Received 31 July 2013; Accepted 17 October 2013

Academic Editor: Bashir Ahmad

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.

Abstract

We investigate solutions of the Helmholtz equation involving local fractional derivative operators. We make use of the series expansion method and the variational iteration method, which are based upon the local fractional derivative operators. The nondifferentiable solution of the problem is obtained by using these methods.

1. Introduction

The Helmholtz equation is known to arise in several physical problems such as electromagnetic radiation, seismology, and acoustics. It is a partial differential equation, which models the normal and nonfractal physical phenomena in both time and space [1]. It is an important differential equation, which is usually investigated by means of some analytical and numerical methods (see [211] and the references therein). For example, the FEM solution for the Helmholtz equation in one, two, and three dimensions was investigated in [2, 3]. The variational iteration method was used to solve the Helmholtz equation in [4]. The explicit solution for the Helmholtz equation was considered in [5] by using the homotopy perturbation method. The domain decomposition method for the Helmholtz equation was presented in [6]. The boundary element method for the Helmholtz equation was considered in [7, 8]. The modified Fourier-Galerkin method for the Helmholtz equations was applied in [9]. The Green’s function for the two-dimensional Helmholtz equation in periodic domains was suggested in [10, 11].

Fractional calculus theory [1226] has been applied to deal with the differentiable models from the practical engineering discipline, which are the anomalous and fractal physical phenomena. The fractional Helmholtz equations were considered in [2729]. In this work, there are two methods to deal with such problems. For example, an analytic solution for the fractional Helmholtz equation in terms of the Mittag-Leffler function was investigated in [28]. The homotopy perturbation method for multidimensional fractional Helmholtz equation was considered in [29].

Local fractional calculus theory [3044] has been used to process the nondifferentiable problems in natural phenomena. Taking an example, the local fractional Fokker-Planck equation was proposed in [30]. The mechanics of quasi-brittle materials with a fractal microstructure with the local fractional derivative was presented in [31]. The anomalous diffusion modeling by fractal and fractional derivatives was considered in [35]. The local fractional wave and heat equations were discussed in [36, 37]. Newtonian mechanics on fractals subset of real-line was investigated in [38]. In [39], the Helmholtz equation on the Cantor sets involving local fractional derivative operators was proposed. There are some other methods to handle the local fractional differential equations, such as local fractional series expansion method [40] and variational iteration method [4144].

The main objective of the present paper is to solve the Helmholtz equation involving the local fractional derivative operators by means of the local fractional series expansion method and the variational iteration method. The structure of the paper is as follows. In Section 2, we describe the Helmholtz equation involving the local fractional derivative operators. In Section 3, we give analysis of the methods used. In Section 4, we apply the local fractional series expansion method to deal with the Helmholtz equation. In Section 5, we apply the local fractional variational iteration method to deal with the Helmholtz equation. Finally, in Section 6, we present our conclusions.

2. Helmholtz Equations within Local Fractional Derivative Operators

The Helmholtz equation involving local fractional derivative operators was proposed.

Let us denote the local fractional derivative as follows [36, 37, 3944]: where .

Using separation of variables in nondifferentiable functions, the three-dimensional Helmholtz equation involving local fractional derivative operators was suggested by the following expression [39]: where the operator involved is a local fractional derivative operator.

In this case, the two-dimensional Helmholtz equation involving local fractional derivative operators is expressed as follows (see [39]): The three-dimensional inhomogeneous Helmholtz equation is given by (see [39]) where is a local fractional continuous function.

The two-dimensional local fractional inhomogeneous Helmholtz equation is considered as follows (see [39]): where is a local fractional continuous function.

The previous local fractional Helmholtz equations with local fractional derivative operators are applied to describe the governing equations in fractal electromagnetic radiation, seismology, and acoustics.

3. Analysis of the Methods Used

3.1. The Local Fractional Series Expansion Method

Let us consider a given local fractional differential equation where is a linear local fractional derivative operator of order with respect to .

By the local fractional series expansion method [40], a multiterm separated function of independent variables and reads as where and are local fractional continuous functions.

In view of (7), we have so that Making use of (9), we get In view of (10), we have Hence, from (11), the recursion reads as follows: By using (12), we arrive at the following result:

3.2. The Local Fractional Variational Iteration Method

Let us consider the following local fractional operator equation: where is linear local fractional derivative operator of order , is a lower-order local fractional derivative operator, and is the inhomogeneous source term.

By using the local fractional variational iteration method [4144], we can construct a correctional local fractional functional as follows: where the local fractional operator is defined as follows [36, 37, 4144]: and a partition of the interval is ,    and ,   , .

Following (15), we have The extremum condition of is given by [37, 41, 42] In view of (18), we have the following stationary conditions: So, from (19), we get The initial value is given by In view of (20), we have Finally, from (22), we obtain the solution of (14) as follows:

4. Local Fractional Series Expansion Method for the Helmholtz Equation

Let us consider the following Helmholtz equation involving local fractional derivative operators: We now present the initial value conditions as follows: Using relation (12), we have where Hence, we get the following iterative relations: From (28), we have From (29), we get the following terms: Hence, we obtain

5. Local Fractional Variational Iteration Method for the Helmholtz Equation

We now consider (24) with the initial and boundary conditions in (25) by using the local fractional variational iteration method.

Applying the iterative relation equation (22), we get where the initial value is given by Therefore, from (34) we have The second approximate term reads as follows: The third approximate term reads as follows: Other approximate terms are presented as follows: and so on.

So, we get

The result is the same as the one which is obtained by the local fractional series expansion method. The nondifferentiable solution is shown in Figure 1.

259125.fig.001
Figure 1: Graph of for .

6. Conclusions

In this work, the nondifferentiable solution for the Helmholtz equation involving local fractional derivative operators is investigated by using the local fractional series expansion method and the variational iteration method. By using these two markedly different methods, the same solution is obtained. These two approaches are remarkably efficient to process other linear local fractional differential equations as well.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (nos. 11126213 and 61170317), and the National Natural Science Foundation of the Hebei Province (nos. A2012209043 and E2013209215).

References

  1. R. Kreß and G. F. Roach, “Transmission problems for the Helmholtz equation,” Journal of Mathematical Physics, vol. 19, no. 6, pp. 1433–1437, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. A. Deraemaeker, I. Babuška, and P. Bouillard, “Dispersion and pollution of the FEM solution for the helmholtz equation in one, two and three dimensions,” International Journal for Numerical Methods in Engineering, vol. 46, no. 4, pp. 471–499, 1999. View at Scopus
  3. A. Hannukainen, M. Huber, and J. Schöberl, “A mixed hybrid finite element method for the Helmholtz equation,” Journal of Modern Optics, vol. 58, no. 5-6, pp. 424–437, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Rafei and D. D. Ganji, “Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 3, pp. 321–328, 2006. View at Scopus
  6. J.-D. Benamou and B. Desprès, “A domain decomposition method for the Helmholtz equation and related optimal control problems,” Journal of Computational Physics, vol. 136, no. 1, pp. 68–82, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. M. Grigoriev and G. F. Dargush, “A fast multi-level boundary element method for the Helmholtz equation,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 3–5, pp. 165–203, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Tomioka, S. Nisiyama, M. Itagaki, and T. Enoto, “Internal field error reduction in boundary element analysis for Helmholtz equation,” Engineering Analysis with Boundary Elements, vol. 23, no. 3, pp. 211–222, 1999. View at Scopus
  9. O. F. Næss and K. S. Eckhoff, “A modified Fourier-Galerkin method for the Poisson and Helmholtz equations,” Journal of Scientific Computing, vol. 17, no. 1–4, pp. 529–539, 2002. View at Scopus
  10. C. M. Linton, “The Green's function for the two-dimensional Helmholtz equation in periodic domains,” Journal of Engineering Mathematics, vol. 33, no. 4, pp. 377–401, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Dienstfrey, F. Hang, and J. Huang, “Lattice sums and the two-dimensional, periodic Green's function for the Helmholtz equation,” Proceedings of the Royal Society A, vol. 457, no. 2005, pp. 67–85, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  14. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  15. 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
  16. R. L. Magin, Fractional Calculus in Bioengineering, Begerll House, West Redding, Conn, USA, 2006.
  17. J. Sabatier, O. P. Agrawal, and J. A. T. 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
  18. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008. View at MathSciNet
  19. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific Publishing, Singapore, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scientific Publishing, Singapore, 2011.
  21. J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, World Scientific Publishing, Singapore, 2012. View at MathSciNet
  22. S. Das, Functional Fractional Calculus, Springer, Berlin, Germany, 2nd edition, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  23. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, Germany, 2011.
  24. A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, UK, 2012. View at MathSciNet
  25. H. Sheng, Y. Chen, and T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Signals and Communication Technology, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. D. Baleanu, J. A. T. Machado, and A. C. Luo, Fractional Dynamics and Control, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. E. Goldfain, “Fractional dynamics, Cantorian space-time and the gauge hierarchy problem,” Chaos, Solitons & Fractals, vol. 22, no. 3, pp. 513–520, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. M. S. Samuel and A. Thomas, “On fractional Helmholtz equations,” Fractional Calculus & Applied Analysis, vol. 13, no. 3, pp. 295–308, 2010. View at Zentralblatt MATH · View at MathSciNet
  29. P. K. Gupta, A. Yildirim, and K. N. Rai, “Application of He's homotopy perturbation method for multi-dimensional fractional Helmholtz equation,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22, no. 3-4, pp. 424–435, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. A. Carpinteri, B. Chiaia, and P. Cornetti, “On the mechanics of quasi-brittle materials with a fractal microstructure,” Engineering Fracture Mechanics, vol. 70, no. 16, pp. 2321–2349, 2003. View at Publisher · View at Google Scholar · View at Scopus
  32. 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
  33. 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
  34. 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
  35. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. X.-J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, 2011.
  37. X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
  38. 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.
  39. 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
  40. 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 · View at Google Scholar · View at MathSciNet
  41. 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.
  42. 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, article 89, pp. 1–11, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  43. Y.-J. Yang, D. Baleanu, and X.-J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,” Abstract and Applied Analysis, vol. 2013, Article ID 202650, 6 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. J.-H. He, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,” Nonlinear Science Letters A, vol. 4, no. 1, pp. 15–20, 2013.