Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 913202, 7 pages

http://dx.doi.org/10.1155/2014/913202

Research Article

Local Fractional Variational Iteration Method for Inhomogeneous Helmholtz Equation within Local Fractional Derivative Operator

Xian-Jin Wang,¹ Yang Zhao,² Carlo Cattani,³ and Xiao-Jun Yang⁴

¹Foundation College, Ningbo Dahongying University, Ningbo 315175, China

²Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

³Department of Mathematics, University of Salerno, Via Giovanni Paolo II, Fisciano, 84084 Salerno, Italy

⁴Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 27 May 2014; Accepted 6 June 2014; Published 26 June 2014

Academic Editor: Abdon Atangana

Copyright © 2014 Xian-Jin Wang 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.

The inhomogeneous Helmholtz equation within the local fractional derivative operator conditions is investigated in this paper. The local fractional variational iteration method is applied to obtain the nondifferentiable solutions and the graphs of the illustrative examples are also shown.

1. Introduction

Helmholtz equation has played an important role in the partial differential equations arising in mathematical physics [1, 2]. In computing the solution of Helmholtz equation, some analytical and numerical methods were presented. For example, Ihlenburg and Babuška used the finite element method to deal with the Helmholtz equation [3]. Momani and Abuasad suggested the variational iteration method to solve the Helmholtz equation [4]. Rafei and Ganji reported the homotopy perturbation method to report the solution to the Helmholtz equation [5]. Bayliss et al. considered the iterative method to discuss the Helmholtz equation [6]. Benamou and Desprès reported the domain decomposition method for solving the Helmholtz equation [7]. Linton presented Green’s function method for the Helmholtz equation [8]. Singer and Turkel proposed the finite difference method to solve the Helmholtz equation [9]. Otto and Larsson applied the second-order method to discuss the Helmholtz equation [10].

Recently, the fractional calculus [11, 12] was developed and applied to present some models in the fields, such as the fractional-order digital control systems [13], the fractional-order viscoelasticity [14], the fractional-order quantum mechanics [15], and fractional-order dynamics [16]. More recently, Samuel and Thomas report the fractional Helmholtz equations [17] and some methods for solving the fractional differential equations were reported in [18–23]. However, we are faced with the problem that there must be some calculus to deal with the nondifferentiable solution for Helmholtz equation, which was structured within the local fractional derivative [24–34]. In this paper, we consider the local fractional inhomogeneous Helmholtz equation in two-dimensional case [31, 32]:

where is a local fractional continuous function and the local fractional partial derivative is defined as follows [24]:

with

The local fractional inhomogeneous Helmholtz equation in three-dimensional case was suggested as follows [29, 30]:

where is a local fractional continuous function. Here, we use the local fractional variational iteration method [30–34] to solve the local fractional inhomogeneous Helmholtz equation in two-dimensional case. The structure of this paper is as follows. In Section 2, we analyze the local fractional variational iteration method. In Section 3, we present some illustrative examples. Finally, the conclusion is given in Section 4.

2. Analysis of the Local Fractional Variational Iteration Method

Here, we give the analysis of the local fractional variational iteration method as follows. We first consider the local fractional linear partial differential equation:

where denotes linear local fractional derivative operator of order , denotes a lower-order local fractional derivative operator, and is the nondifferentiable source term.

Let the local fractional operator be defined as [24, 30–34]

with the partitions of the interval , , , and , .

We now structure the correctional local fractional functional in the form

Making the local fractional variation, we have

such that the following stationary conditions are given as

In view of (9), we obtain the fractal Lagrange multiplier, which is given by

From (7) and (10), we reach at the local fractional variational iteration algorithm

where the nondifferentiable initial value is suggested as

Therefore, from (11), we write the solution of (7) as follows:

3. Some Illustrative Examples

In this section, we give some illustrative examples for solving the local fractional inhomogeneous Helmholtz equation in two-dimensional case.

We present the following local fractional inhomogeneous Helmholtz equation:

subject to the initial-boundary conditions:

Making use of (11), we structure the local fractional variational iteration algorithm as follows:

where the initial value is presented as

whose plot is shown in Figure 1.

Figure 1: The graph of the initial value (17) where .

In view of (16) and (17), we arrive at the first approximation:

The second approximation is

Making best of (16) and (19), the third approximation reads as

From (16) and (20), we obtain the fourth approximation of (14) given as

As similar manner, from (21), we arrive at the fifth approximate formula:

Hence, we have the local fractional series solution of (14):

From (13), we get the exact solution of (14) given as

and its plot is illustrated in Figure 2.

Figure 2: The graph of exact solution of (14) where .

Example 1. We suggest the following local fractional inhomogeneous Helmholtz equation:

and the initial-boundary conditions read as

From (11), we set up the local fractional variational iteration algorithm as follows:

where the initial value is suggested as

Appling (27) and (28) gives the first approximate solution:

Using (27) and (29), we obtain the second approximate term, which is expressed as follows:

In view of (27) and (30), we obtain the third approximation, which reads as follows:

Therefore, we arrive at the approximate term

which leads to the exact solution of (25) given as

and its plot is illustrated in Figure 3.

Figure 3: The plot of exact solution of (25) where .

4. Conclusions

In this work, the boundary value problems for the inhomogeneous Helmholtz equation within local fractional derivative operator were discussed by using the local fractional variational iteration method. Their nondifferentiable solutions are obtained and the graphs of the solutions with fractal dimension are also given.

Conflict of Interests

The authors declare that they have no competing interests in this paper.

Acknowledgment

This work was supported by the Zhejiang Provincial Natural Science Foundation (LY13A010007).

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