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

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]: 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (1) where Δ^𝛼(𝑓(𝑥)−𝑓(𝑥₀))≅Γ(1+𝛼)Δ(𝑓(𝑥)−𝑓(𝑥₀)).

Using separation of variables in nondifferentiable functions, the three-dimensional Helmholtz equation involving local fractional derivative operators was suggested by the following expression [39]: 𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑦^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑧^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦,𝑧=0, (2) 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]): 𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑦^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦=0. (3) The three-dimensional inhomogeneous Helmholtz equation is given by (see [39]) 𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑦^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦,𝑧𝜕𝑧^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦,𝑧=𝑓𝑥,𝑦,𝑧, (4) where 𝑓(𝑥,𝑦,𝑧) is a local fractional continuous function.

The two-dimensional local fractional inhomogeneous Helmholtz equation is considered as follows (see [39]): 𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑀𝑥,𝑦𝜕𝑦^2𝛼+𝜔^2𝛼𝑀𝑥,𝑦=𝑓𝑥,𝑦, (5) where 𝑓(𝑥,𝑦) 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 𝑢_𝑡^2𝛼=𝐿_𝛼𝑢, (6) where 𝐿 is a linear local fractional derivative operator of order 2𝛼 with respect to 𝑥.

By the local fractional series expansion method [40], a multiterm separated function of independent variables 𝑡 and 𝑥 reads as 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑇_𝑖(𝑡)𝑋_𝑖(𝑥), (7) where 𝑇_𝑖(𝑡) and 𝑋_𝑖(𝑥) are local fractional continuous functions.

In view of (7), we have 𝑇_𝑖(𝑡)=𝑡^𝑖𝛼Γ(),1+𝑖𝛼 (8) so that 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥). (9) Making use of (9), we get 𝑢_𝑡^2𝛼=^∞_𝑖=01Γ()𝑡1+𝑖𝛼^𝑖𝛼𝑋_𝑖+2(𝑥),𝐿_𝛼𝑢=𝐿_𝛼^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝑋_𝑖(𝑥). (10) In view of (10), we have ^∞_𝑖=01Γ()𝑡1+𝑖𝛼^𝑖𝛼𝑋_𝑖+2(𝑥)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝐿1+𝑖𝛼^𝛼𝑋_𝑖(𝑥). (11) Hence, from (11), the recursion reads as follows: 𝑋_𝑖+2(𝑥)=𝐿_𝛼𝑋_𝑖(𝑥). (12) By using (12), we arrive at the following result: 𝑢(𝑥,𝑡)=^∞_𝑖=0𝑡^𝑖𝛼Γ()𝑋1+𝑖𝛼^𝑖(𝑥). (13)

3.2. The Local Fractional Variational Iteration Method

Let us consider the following local fractional operator equation: 𝐿_𝛼𝑢+𝑅_𝛼𝑢=𝑔(𝑡), (14) where 𝐿_𝛼 is linear local fractional derivative operator of order 2𝛼, 𝑅_𝛼 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: 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+₀𝐼_𝑥^(𝛼)×𝜂(𝑠)𝐿_𝛼𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠), (15) where the local fractional operator is defined as follows [36, 37, 41–44]: _𝑎𝐼_𝑏^(𝛼)𝑓(𝑥)=1Γ(1+𝛼)^𝑏_𝑎𝑓(𝑡)(𝑑𝑡)^𝛼=1Γ(1+𝛼)lim_Δ𝑡→0^{𝑗=𝑁−1}_𝑗=0𝑓𝑡_𝑗Δ𝑡_𝑗^𝛼 (16) and a partition of the interval [𝑎,𝑏] is Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗,Δ𝑡=max{Δ𝑡₁,Δ𝑡₂,Δ𝑡_𝑗,…},    and 𝑗=0,…,𝑁−1,   𝑡₀=𝑎, 𝑡_𝑁=𝑏.

Following (15), we have 𝛿^𝛼𝑢_𝑛+1(𝑥)=𝛿^𝛼𝑢_𝑛(𝑥)+₀𝐼_𝑥^(𝛼)𝛿^𝛼×𝜂(𝑠)𝐿_𝛼𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠). (17) The extremum condition of 𝑢_𝑛+1 is given by [37, 41, 42] 𝛿^𝛼𝑢_𝑛+1=0. (18) In view of (18), we have the following stationary conditions: 1−𝜂(𝑠)^(𝛼)_𝑠=𝑥=0,𝜂(𝑠)_𝑠=𝑥𝜂(𝑠)=0,_{(2𝛼)𝑠=𝑥}=0. (19) So, from (19), we get 𝜂(𝑠)=(𝑠−𝑥)^𝛼Γ().1+𝛼 (20) The initial value 𝑢₀(𝑥) is given by 𝑢₀(𝑥)(0)+𝑥=𝑢^𝛼Γ()𝑢1+𝛼^(𝛼)(0). (21) In view of (20), we have 𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+₀𝐼_𝑥^(𝛼)(𝑠−𝑥)^𝛼Γ()×𝐿1+𝛼_𝛼𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)−𝑔(𝑠). (22) Finally, from (22), we obtain the solution of (14) as follows: 𝑢=lim_𝑛→∞𝑢_𝑛. (23)

4. Local Fractional Series Expansion Method for the Helmholtz Equation

Let us consider the following Helmholtz equation involving local fractional derivative operators: 𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢𝑥,𝑦𝜕𝑦^2𝛼=𝑢𝑥,𝑦. (24) We now present the initial value conditions as follows: 𝑢𝜕0,𝑦=0,𝜕𝑥^𝛼𝑢0,𝑦=𝐸_𝛼𝑦^𝛼. (25) Using relation (12), we have 𝑢_𝑖+2𝑦=𝐿_𝛼𝑢_𝑖𝑦,𝑢₀𝑦𝑢=𝑢0,𝑦=0,₁𝑦=𝜕𝜕𝑥^𝛼𝑢0,𝑦=𝐸_𝛼𝑦^𝛼, (26) where 𝐿_𝛼𝑢_𝑖=𝑢_𝑖−𝜕^2𝛼𝑢_𝑖𝜕𝑦^2𝛼. (27) Hence, we get the following iterative relations: 𝑢_𝑖+2𝑦=𝑢_𝑖−𝜕^2𝛼𝑢_𝑖𝜕𝑦^2𝛼𝑦,𝑢₀𝑦𝑢=𝑢0,𝑦=0,_𝑖+2𝑦=𝑢_𝑖−𝜕^2𝛼𝑢_𝑖𝜕𝑦^2𝛼𝑦,𝑢₁𝑦=𝐸_𝛼𝑦^𝛼. (28)(29) From (28), we have 𝑢₀𝑦=𝑢₂𝑦=𝑢₄𝑦=⋯=0. (30) From (29), we get the following terms: 𝑢₁𝑦=𝐸_𝛼𝑦^𝛼,𝑢₃𝑦=𝑢₁−𝜕^2𝛼𝑢₁𝜕𝑦^2𝛼𝑦=𝐸_𝛼𝑦^𝛼−𝐸_𝛼𝑦^𝛼𝑢=0,₅𝑦𝑢=0,₇𝑦=⋯=0. (31) Hence, we obtain 𝑢=𝑥𝑥,𝑦^𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (32)

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 𝑢_𝑛+1𝑥,𝑦=𝑢_𝑛𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢_𝑛𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢_𝑛𝑥,𝑦𝜕𝑦^2𝛼−𝑢_𝑛𝑥,𝑦, (33) where the initial value is given by 𝑢₀=𝑥𝑥,𝑦^𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (34) Therefore, from (34) we have 𝑢₁𝑥,𝑦=𝑢₀𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢₀𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢₀𝑥,𝑦𝜕𝑦^2𝛼−𝑢₀=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (35) The second approximate term reads as follows: 𝑢₂𝑥,𝑦=𝑢₁𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢₁𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢₁𝑥,𝑦𝜕𝑦^2𝛼−𝑢₁=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (36) The third approximate term reads as follows: 𝑢₃𝑥,𝑦=𝑢₂𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢₂𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢₂𝑥,𝑦𝜕𝑦^2𝛼−𝑢₂=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (37) Other approximate terms are presented as follows: 𝑢₄𝑥,𝑦=𝑢₃𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢₃𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢₃𝑥,𝑦𝜕𝑦^2𝛼−𝑢₃=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼,𝑢₅𝑥,𝑦=𝑢₄𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢₄𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢₄𝑥,𝑦𝜕𝑦^2𝛼−𝑢₄=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼⋮𝑢_𝑛𝑥,𝑦=𝑢_𝑛−1𝑥,𝑦+₀𝐼_𝑦^(𝛼)𝑠−𝑦^𝛼Γ()×𝜕1+𝛼_2𝛼𝑢_𝑛−1𝑥,𝑦𝜕𝑥^2𝛼+𝜕^2𝛼𝑢_𝑛−1𝑥,𝑦𝜕𝑦^2𝛼−𝑢_𝑛−1=𝑥𝑥,𝑦_𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼 (38) and so on.

So, we get 𝑢𝑥,𝑦=lim_𝑛→∞𝑢_𝑛=𝑥𝑥,𝑦^𝛼Γ()𝐸1+𝛼^𝛼𝑦^𝛼. (39)

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.

Figure 1 

Graph of 𝑢(𝑥,𝑦) for 𝛼=ln2/ln3.

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).