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

Authors: Ai-Min Yang; Zeng-Shun Chen; H. M. Srivastava; Xiao-Jun Yang
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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]:
(1)f(α)(x0)=dαf(x)dxα|x=x0=limx→x0Δα(f(x)-f(x0))(x-x0)α,
where Δα(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]:
(2)∂2αM(x,y,z)∂x2α+∂2αM(x,y,z)∂y2α+∂2αM(x,y,z)∂z2α+ω2αM(x,y,z)=0,
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]):
(3)∂2αM(x,y)∂x2α+∂2αM(x,y)∂y2α+ω2αM(x,y)=0.
The three-dimensional inhomogeneous Helmholtz equation is given by (see [39])
(4)∂2αM(x,y,z)∂x2α+∂2αM(x,y,z)∂y2α+∂2αM(x,y,z)∂z2α+ω2αM(x,y,z)=f(x,y,z),
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]):
(5)∂2αM(x,y)∂x2α+∂2αM(x,y)∂y2α+ω2αM(x,y)=f(x,y),
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(6)ut2α=Lαu,
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
(7)u(x,t)=∑i=0∞Ti(t)Xi(x),
where Ti(t) and Xi(x) are local fractional continuous functions.In view of (7), we have
(8)Ti(t)=tiαΓ(1+iα),
so that
(9)u(x,t)=∑i=0∞tiαΓ(1+iα)Xi(x).
Making use of (9), we get
(10)ut2α=∑i=0∞1Γ(1+iα)tiαXi+2(x),Lαu=Lα[∑i=0∞tiαΓ(1+iα)Xi(x)]=∑i=0∞tiαΓ(1+iα)(LαXi)(x).
In view of (10), we have
(11)∑i=0∞1Γ(1+iα)tiαXi+2(x)=∑i=0∞tiαΓ(1+iα)(LαXi)(x).
Hence, from (11), the recursion reads as follows:
(12)Xi+2(x)=(LαXi)(x).
By using (12), we arrive at the following result:
(13)u(x,t)=∑i=0∞tiαΓ(1+iα)Xi(x).


### 3.2. The Local Fractional Variational Iteration Method

Let us consider the following local fractional operator equation:(14)Lαu+Rαu=g(t),
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:
(15)un+1(x)=un(x)+I0x(α)×{η(s)[Lαun(s)+Rαu~n(s)-g~(s)]},
where the local fractional operator is defined as follows [36, 37, 41–44]:
(16)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α
and a partition of the interval [a,b] is Δtj=tj+1-tj,Δt=max{Δt1,Δt2,Δtj,…},    and j=0,…,N-1,   t0=a, tN=b.Following (15), we have
(17)δαun+1(x)=δαun(x)+I0x(α)δα×{η(s)[Lαun(s)+Rαu~n(s)-g~(s)]}.
The extremum condition of un+1 is given by [37, 41, 42]
(18)δαun+1=0.
In view of (18), we have the following stationary conditions:
(19)1-η(s)(α)|s=x=0,η(s)|s=x=0,η(s)(2α)|s=x=0.
So, from (19), we get
(20)η(s)=(s-x)αΓ(1+α).
The initial value u0(x) is given by
(21)u0(x)=u(0)+xαΓ(1+α)u(α)(0).
In view of (20), we have
(22)un+1(x)=un(x)+I0x(α)(s-x)αΓ(1+α)×{Lαun(s)+Rαu~n(s)-g~(s)}.
Finally, from (22), we obtain the solution of (14) as follows:
(23)u=limn→∞un.


## 3.1. The Local Fractional Series Expansion Method

Let us consider a given local fractional differential equation(6)ut2α=Lαu,
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
(7)u(x,t)=∑i=0∞Ti(t)Xi(x),
where Ti(t) and Xi(x) are local fractional continuous functions.In view of (7), we have
(8)Ti(t)=tiαΓ(1+iα),
so that
(9)u(x,t)=∑i=0∞tiαΓ(1+iα)Xi(x).
Making use of (9), we get
(10)ut2α=∑i=0∞1Γ(1+iα)tiαXi+2(x),Lαu=Lα[∑i=0∞tiαΓ(1+iα)Xi(x)]=∑i=0∞tiαΓ(1+iα)(LαXi)(x).
In view of (10), we have
(11)∑i=0∞1Γ(1+iα)tiαXi+2(x)=∑i=0∞tiαΓ(1+iα)(LαXi)(x).
Hence, from (11), the recursion reads as follows:
(12)Xi+2(x)=(LαXi)(x).
By using (12), we arrive at the following result:
(13)u(x,t)=∑i=0∞tiαΓ(1+iα)Xi(x).


## 3.2. The Local Fractional Variational Iteration Method

Let us consider the following local fractional operator equation:(14)Lαu+Rαu=g(t),
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:
(15)un+1(x)=un(x)+I0x(α)×{η(s)[Lαun(s)+Rαu~n(s)-g~(s)]},
where the local fractional operator is defined as follows [36, 37, 41–44]:
(16)Ib(α)af(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α
and a partition of the interval [a,b] is Δtj=tj+1-tj,Δt=max{Δt1,Δt2,Δtj,…},    and j=0,…,N-1,   t0=a, tN=b.Following (15), we have
(17)δαun+1(x)=δαun(x)+I0x(α)δα×{η(s)[Lαun(s)+Rαu~n(s)-g~(s)]}.
The extremum condition of un+1 is given by [37, 41, 42]
(18)δαun+1=0.
In view of (18), we have the following stationary conditions:
(19)1-η(s)(α)|s=x=0,η(s)|s=x=0,η(s)(2α)|s=x=0.
So, from (19), we get
(20)η(s)=(s-x)αΓ(1+α).
The initial value u0(x) is given by
(21)u0(x)=u(0)+xαΓ(1+α)u(α)(0).
In view of (20), we have
(22)un+1(x)=un(x)+I0x(α)(s-x)αΓ(1+α)×{Lαun(s)+Rαu~n(s)-g~(s)}.
Finally, from (22), we obtain the solution of (14) as follows:
(23)u=limn→∞un.


## 4. Local Fractional Series Expansion Method for the Helmholtz Equation

Let us consider the following Helmholtz equation involving local fractional derivative operators:(24)∂2αu(x,y)∂x2α+∂2αu(x,y)∂y2α=u(x,y).
We now present the initial value conditions as follows:
(25)u(0,y)=0,∂∂xαu(0,y)=Eα(yα).
Using relation (12), we have
(26)ui+2(y)=(Lαui)(y),u0(y)=u(0,y)=0,u1(y)=∂∂xαu(0,y)=Eα(yα),
where
(27)Lαui=ui-∂2αui∂y2α.
Hence, we get the following iterative relations:
(28)ui+2(y)=(ui-∂2αui∂y2α)(y),u0(y)=u(0,y)=0,(29)ui+2(y)=(ui-∂2αui∂y2α)(y),u1(y)=Eα(yα).
From (28), we have
(30)u0(y)=u2(y)=u4(y)=⋯=0.
From (29), we get the following terms:
(31)u1(y)=Eα(yα),u3(y)=(u1-∂2αu1∂y2α)(y)=[Eα(yα)-Eα(yα)]=0,u5(y)=0,u7(y)=⋯=0.
Hence, we obtain
(32)u(x,y)=xαΓ(1+α)Eα(yα).


## 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
(33)un+1(x,y)=un(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αun(x,y)∂x2α+∂2αun(x,y)∂y2α-un(x,y)},
where the initial value is given by
(34)u0(x,y)=xαΓ(1+α)Eα(yα).
Therefore, from (34) we have
(35)u1(x,y)=u0(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αu0(x,y)∂x2α+∂2αu0(x,y)∂y2α-u0(x,y)}=xαΓ(1+α)Eα(yα).
The second approximate term reads as follows:
(36)u2(x,y)=u1(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αu1(x,y)∂x2α+∂2αu1(x,y)∂y2α-u1(x,y)}=xαΓ(1+α)Eα(yα).
The third approximate term reads as follows:
(37)u3(x,y)=u2(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αu2(x,y)∂x2α+∂2αu2(x,y)∂y2α-u2(x,y)}=xαΓ(1+α)Eα(yα).
Other approximate terms are presented as follows:
(38)u4(x,y)=u3(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αu3(x,y)∂x2α+∂2αu3(x,y)∂y2α-u3(x,y)}=xαΓ(1+α)Eα(yα),u5(x,y)=u4(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αu4(x,y)∂x2α+∂2αu4(x,y)∂y2α-u4(x,y)}=xαΓ(1+α)Eα(yα)⋮un(x,y)=un-1(x,y)+Iy(α)0(s-y)αΓ(1+α)×{∂2αun-1(x,y)∂x2α+∂2αun-1(x,y)∂y2α000-un-1(x,y)∂2αun-1(x,y)∂x2α+∂2αun-1(x,y)∂y2α}=xαΓ(1+α)Eα(yα)
and so on.So, we get(39)u(x,y)=limn→∞un(x,y)=xαΓ(1+α)Eα(yα).The result is the same as the one which is obtained by the local fractional series expansion method. The nondifferentiable solution is shown in Figure1.Figure 1
Graph ofu(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.



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