## Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013

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

**Academic Editor:**Bashir Ahmad

#### 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 [2–11] 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 [12–26] 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 [27–29]. 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 [30–44] 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 [41–44].

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, 39–44]: 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 [41–44], we can construct a correctional local fractional functional as follows: where the local fractional operator is defined as follows [36, 37, 41–44]: 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.

#### 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

- 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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: Google Scholar - 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: Google Scholar - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet - 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 Site | 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 - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begerll House, West Redding, Conn, USA, 2006. - 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 Site | MathSciNet - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, Oxford, UK, 2008. View at: MathSciNet - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models*, World Scientific Publishing, Singapore, 2010. View at: Publisher Site | Zentralblatt MATH | MathSciNet - R. Herrmann,
*Fractional Calculus: An Introduction for Physicists*, World Scientific Publishing, Singapore, 2011. - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics: Recent Advances*, World Scientific Publishing, Singapore, 2012. View at: MathSciNet - S. Das,
*Functional Fractional Calculus*, Springer, Berlin, Germany, 2nd edition, 2011. View at: Publisher Site | MathSciNet - V. E. Tarasov,
*Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media*, Springer, Berlin, Germany, 2011. - A. B. Malinowska and D. F. M. Torres,
*Introduction to the Fractional Calculus of Variations*, Imperial College Press, London, UK, 2012. View at: MathSciNet - 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 Site | MathSciNet - D. Baleanu, J. A. T. Machado, and A. C. Luo,
*Fractional Dynamics and Control*, Springer, New York, NY, USA, 2012. View at: Publisher Site | MathSciNet - 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 Site | Google Scholar - M. S. Samuel and A. Thomas, “On fractional Helmholtz equations,”
*Fractional Calculus & Applied Analysis*, vol. 13, no. 3, pp. 295–308, 2010. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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 Site | Google Scholar | MathSciNet - 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, 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 Site | Google Scholar - 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 - 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 - 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, 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 - 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 Publisher, New York, NY, USA, 2012. - 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 - 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 Site | Google Scholar | MathSciNet - 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 - 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, article 89, pp. 1–11, 2013. View at: Publisher Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. 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.