Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013View this Special Issue
Research Article | Open Access
Application of the Local Fractional Series Expansion Method and the Variational Iteration Method to the Helmholtz Equation Involving Local Fractional Derivative Operators
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.
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 . 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 . The explicit solution for the Helmholtz equation was considered in  by using the homotopy perturbation method. The domain decomposition method for the Helmholtz equation was presented in . 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 . 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 . The homotopy perturbation method for multidimensional fractional Helmholtz equation was considered in .
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 . The mechanics of quasi-brittle materials with a fractal microstructure with the local fractional derivative was presented in . The anomalous diffusion modeling by fractal and fractional derivatives was considered in . The local fractional wave and heat equations were discussed in [36, 37]. Newtonian mechanics on fractals subset of real-line was investigated in . In , 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  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.
Using separation of variables in nondifferentiable functions, the three-dimensional Helmholtz equation involving local fractional derivative operators was suggested by the following expression : 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 ): The three-dimensional inhomogeneous Helmholtz equation is given by (see ) where is a local fractional continuous function.
The two-dimensional local fractional inhomogeneous Helmholtz equation is considered as follows (see ): 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 , a multiterm separated function of independent variables and reads as where and are local fractional continuous functions.
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
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.
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.
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).
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