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

and Applied Analysis 3 3.2. The Local Fractional Variational Iteration Method. Let us consider the following local fractional operator equation: Lαu + Rαu = g (t) , (14) where Lα is linear local fractional derivative operator of order 2α, Rα is a lower-order local fractional derivative operator, and g(t) 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: un+1 (x) = un (x) + 0I (α) x × {η (s) [Lαun (s) + Rα?̃?n (s) − g (s)]} , (15) where the local fractional operator is defined as follows [36, 37, 41–44]: a I b f (x) = 1 Γ (1 + α) ∫ b


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][3][4][5][6][7][8][9][10][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].
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.
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.

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 2 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:

The Local Fractional Variational Iteration Method.
Let us consider the following local fractional operator equation: where is linear local fractional derivative operator of order 2 , is a lower-order local fractional derivative operator, and ( ) is the inhomogeneous source term.

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: (0, ) = 0, Using relation (12), we have Hence, we get the following iterative relations: From (28), we have From (29), we get the following terms: Hence, we obtain

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) 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.

Conclusions
In this work, the nondifferentiable solution for the Helmholtz equation involving local fractional derivative operators is Abstract and Applied Analysis  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.