# A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators

Authors: Yong-Ju Yang; Dumitru Baleanu; Xiao-Jun Yang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The local fractional variational iteration method for local fractional Laplace equation is investigated in this paper. The operators are described in the sense of local fractional operators. The obtained results reveal that the method is very effective.


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## Body

## 1. Introduction

As it is known, the partial differential equations [1, 2] and fractional differential equations [3–5] appear in many areas of science and engineering. As a result, various kinds of analytical methods and numerical methods were developed [6–8]. For example, the variational iteration method [9–15] was applied to solve differential equations [16–18], integral equations [19], and numerous applications to different nonlinear equations in physics and engineering. Also, the fractional variational iteration method [20–23] and the fractional complex transform [24–27] were discussed recently. The efficient techniques have successfully addressed a wide class of nonlinear problems for differential equations; see [28–36] and the references therein. We notice that the developed methods are very convenient, efficient, and accurate.Recently, the local fractional variational iteration method [37] is derived from local fractional operators [38–48]. The method, which accurately computes the solutions in a local fractional series form or in an exact form, presents interest to applied sciences for problems where the other methods cannot be applied properly.In this paper, we investigate the application of local fractional variational iteration method for solving the local fractional Laplace equations [49] with the different fractal conditions.This paper is organized as follows.In Section2, the basic mathematical tools are reviewed. Section 3 presents briefly the local fractional variational iteration method based on local fractional variational for fractal Lagrange multipliers. Section 4 presents solutions to the local fractional Laplace equations with differential fractal conditions.


## 2. Mathematical Fundamentals

In this section, we present few mathematical fundamentals of local fractional calculus and introduce the basic notions of local fractional continuity, local fractional derivative, and local fractional integral of nondifferential functions.


### 2.1. Local Fractional Continuity

Lemma 1 (see [42]).
LetF be a subset of the real line and a fractal. If f:(F,d)→(Ω′,d′) is a bi-Lipschitz mapping, then there is, for constants ρ,τ>0 and F⊂R,
(1)ρsHs(F)≤Hs(f(F))≤τsHs(F)
such that for all x1,x2∈F,
(2)ρα|x1-x2|α≤|f(x1)-f(x2)|≤τα|x1-x2|α.
As a direct result of Lemma 1, one has [42]
(3)|f(x1)-f(x2)|≤τα|x1-x2|α
such that
(4)|f(x1)-f(x2)|<εα,
where α is fractal dimension of F.
Suppose that there is [38–43]
(5)|f(x)-f(x0)|<εα
with |x-x0|<δ, for ε,δ>0 and ε,δ∈R, then f(x) is called local fractional continuous at x=x0 and it is denoted by
(6)limx→x0f(x)=f(x0).
Suppose that the functionf(x) is satisfied the condition (5) for x∈(a,b), and hence it is called a local fractional continuous on the interval (a,b), denoted by
(7)f(x)∈Cα(a,b).


### 2.2. Local Fractional Derivatives and Integrals

Suppose thatf(x)∈Cα(a,b), then the local fractional derivative of f(x) of order α at x=x0 is given by [37–43]
(8)Dx(α)f(x0)=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)).There is [38–40]
(9)f(x)∈Dx(α)(a,b)
if
(10)f(α)(x)=Dx(α)f(x)
for any x∈(a,b).Local fractional derivative of high order is written in the form [38–40]
(11)f(kα)(x)=Dx(α)⋯Dx(α)︷ktimesf(x),
and local fractional partial derivative of high order is [38–40]
(12)∂kα∂xkαf(x)=∂α∂xα⋯∂α∂xα︷ktimesf(x).Let a functionf(x) satisfy the condition (7). Local fractional integral of f(x) of order α in the interval [a,b] is given by [37–43]
(13)Iab(α)f(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α,
where Δtj=tj+1-tj, Δt=max{Δt1,Δt2,Δtj,…}, and [tj,tj+1], j=0,…,N-1, t0=a, tN=b, is a partition of the interval [a,b]. For other definition of local fractional derivative, see [44–48].There exists [38–40]
(14)f(x)∈Ix(α)(a,b)
if
(15)f(α)(x)=Iax(α)f(x)
for any x∈(a,b).Local fractional multiple integrals off(x) is written in the form [40]
(16)Ix0x(kα)f(x)=Ix0x(α)⋯Ix0x(α)︷ktimesf(x)
if (7) is valid for x∈(a,b).


## 2.1. Local Fractional Continuity

Lemma 1 (see [42]).
LetF be a subset of the real line and a fractal. If f:(F,d)→(Ω′,d′) is a bi-Lipschitz mapping, then there is, for constants ρ,τ>0 and F⊂R,
(1)ρsHs(F)≤Hs(f(F))≤τsHs(F)
such that for all x1,x2∈F,
(2)ρα|x1-x2|α≤|f(x1)-f(x2)|≤τα|x1-x2|α.
As a direct result of Lemma 1, one has [42]
(3)|f(x1)-f(x2)|≤τα|x1-x2|α
such that
(4)|f(x1)-f(x2)|<εα,
where α is fractal dimension of F.
Suppose that there is [38–43]
(5)|f(x)-f(x0)|<εα
with |x-x0|<δ, for ε,δ>0 and ε,δ∈R, then f(x) is called local fractional continuous at x=x0 and it is denoted by
(6)limx→x0f(x)=f(x0).
Suppose that the functionf(x) is satisfied the condition (5) for x∈(a,b), and hence it is called a local fractional continuous on the interval (a,b), denoted by
(7)f(x)∈Cα(a,b).


## 2.2. Local Fractional Derivatives and Integrals

Suppose thatf(x)∈Cα(a,b), then the local fractional derivative of f(x) of order α at x=x0 is given by [37–43]
(8)Dx(α)f(x0)=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)).There is [38–40]
(9)f(x)∈Dx(α)(a,b)
if
(10)f(α)(x)=Dx(α)f(x)
for any x∈(a,b).Local fractional derivative of high order is written in the form [38–40]
(11)f(kα)(x)=Dx(α)⋯Dx(α)︷ktimesf(x),
and local fractional partial derivative of high order is [38–40]
(12)∂kα∂xkαf(x)=∂α∂xα⋯∂α∂xα︷ktimesf(x).Let a functionf(x) satisfy the condition (7). Local fractional integral of f(x) of order α in the interval [a,b] is given by [37–43]
(13)Iab(α)f(x)=1Γ(1+α)∫abf(t)(dt)α=1Γ(1+α)limΔt→0∑j=0j=N-1f(tj)(Δtj)α,
where Δtj=tj+1-tj, Δt=max{Δt1,Δt2,Δtj,…}, and [tj,tj+1], j=0,…,N-1, t0=a, tN=b, is a partition of the interval [a,b]. For other definition of local fractional derivative, see [44–48].There exists [38–40]
(14)f(x)∈Ix(α)(a,b)
if
(15)f(α)(x)=Iax(α)f(x)
for any x∈(a,b).Local fractional multiple integrals off(x) is written in the form [40]
(16)Ix0x(kα)f(x)=Ix0x(α)⋯Ix0x(α)︷ktimesf(x)
if (7) is valid for x∈(a,b).


## 3. Local Fractional Variational Iteration Method

In this section, we introduce the local fractional variational iteration method derived from the local fractional variational approach for fractal Lagrange multipliers [40].Let us consider the local fractional variational approach in the one-dimensional case through the following local fractional functional, which reads [40]
(17)I(y)=Iab(α)f(x,y(x),y(α)(x)),
where y(α)(x) is taken in local fractional differential operator and a≤x≤b.The local fractional variational derivative is given by [40]
(18)δαI=Iab(α){(∂f∂y-dαdxα(∂f∂y(α)))η(x)},
where δα is local fractional variation signal and η(a)=η(b)=0.The nonlinear local fractional equation reads as(19)Lαu+Nαu=0,
where Lα and Nα are linear and nonlinear local fractional operators, respectively.Local fractional variational iteration algorithm can be written as [37]
(20)un+1(t)=un(t)+It0t(α){ξα[Lαun(s)+Nαun(s)]}.
Here, we can construct a correction functional as follows [37]:
(21)un+1(t)=un(t)+It0t(α){ξα[Lαun(s)+Nαu~n(s)]},
where u~n is considered as a restricted local fractional variation and ξα is a fractal Lagrange multiplier; that is, δαu~n=0 [37, 40].Having determined the fractal Lagrangian multipliers, the successive approximationsun+1, n≥0, of the solution u will be readily obtained upon using any selective fractal function u0. Consequently, we have the solution
(22)u=limn→∞un.
Here, this technology is called the local fractional variational method [37]. We notice that the classical variation is recovered in case of local fractional variation when the fractal dimension is equal to 1. Besides, the convergence of local fractional variational process and its algorithms were taken into account [37].


## 4. Solutions to Local Fractional Laplace Equation in Fractal Timespace

The local fractional Laplace equation (see [38–40] and the references therein) is one of the important differential equations with local fractional derivatives. In the following, we consider solutions to local fractional Laplace equations in fractal timespace.Case 1.
Let us start with local fractional Laplace equation given by(23)∂2αT(x,t)∂t2α+∂2αT(x,t)∂x2α=0
and subject to the fractal value conditions
(24)∂α∂tαT(x,0)=0,T(x,0)=-Eα(xα).
A corrected local fractional functional for (24) reads as
(25)un+1(x,t)=un(x,t)+I0t(α){λαΓ(1+α)(∂2αTn(x,τ)∂τ2α+∂2αTn(x,τ)∂x2α)}.
Taking into account the properties of the local fractional derivative, we obtain
(26)δαun+1(x,t)=δαun(x,t)+δαI0t(α){λαΓ(1+α)(∂2αTn(x,τ)∂τ2α+∂2αTn(x,τ)∂x2α)}.
Hence, from (25)-(26) we get
(27)δαun+1(x,t)=δαun(x,t)+λαΓ(1+α)δαun(α)(x,t)|τ=t-[λαΓ(1+α)](α)δαun(x,t)|τ=t-(δαun(x,τ))I0t(α)(λαΓ(1+α))(2α)=δαun(x,t)+λαΓ(1+α)δαun(α)|τ=t-(λαΓ(1+α))(α)δαun(x,t)|τ=t+(δαun(x,τ))I0t(α)(λαΓ(1+α))(2α)=0.
As a result, from (27) we can derive
(28)(λαΓ(1+α))(2α)=0,λαΓ(1+α)|τ=t=0,(λαΓ(1+α))(α)=1.
We have λ=τ-t such that the fractal Lagrange multiplier reads as
(29)λαΓ(1+α)=(τ-t)αΓ(1+α).
From (24) we take the initial value, which reads as
(30)u0(x,t)=-Eα(xα).
By using (25) we structure a local fractional iteration procedure as
(31)un+1(x,t)=un(x,t)+I0t(α){(τ-t)αΓ(1+α)(∂2αTn(x,τ)∂τ2α+∂2αTn(x,τ)∂x2α)}.
Hence, we can derive the first approximation term as
(32)u1(x,t)=u0(x,t)+I0t(α){(τ-t)αΓ(1+α)(∂2αT0(x,τ)∂τ2α+∂2αT0(x,τ)∂x2α)}=-Eα(xα)+I0t(α){(τ-t)αΓ(1+α)(-Eα(xα))}=Eα(xα)(-1+t2αΓ(1+2α)).
The second approximation can be calculated in the similar way, which is
(33)u2(x,t)=u1(x,t)+I0t(α){(τ-t)αΓ(1+α)(∂2αT1(x,τ)∂τ2α+∂2αT1(x,τ)∂x2α)}=Eα(xα)(-1+t2αΓ(1+2α))+I0t(α){(τ-t)αΓ(1+α)(t2αEα(xα)Γ(1+2α))}=Eα(xα)(-1+t2αΓ(1+2α)-t4αΓ(1+4α)).
Proceeding in this manner, we get
(34)un(x,t)=Eα(xα)(∑k=0n(-1)kt2kαΓ(1+2kα)).
Thus, the final solution reads as
(35)u(x,t)=limn→∞un(x,t)=Eα(xα)(∑k=0∞(-1)kt2kαΓ(1+2kα))=-Eα(xα)cosα(tα).Case 2.
Consider the local fractional Laplace equation as(36)∂2αT(x,t)∂t2α+∂2αT(x,t)∂x2α=0
subject to fractal value conditions given by
(37)∂α∂tαT(x,0)=-Eα(xα),T(x,0)=0.
Now we can structure the same local fractional iteration procedure (31).
By using (36)-(37) we take an initial value as
(38)u0(x,t)=-tαEα(xα)Γ(1+α).
The first approximation term reads as
(39)u1(x,t)=u0(x,t)+I0t(α){(τ-t)αΓ(1+α)(∂2αT0(x,τ)∂τ2α+∂2αT0(x,τ)∂x2α)}=-tαEα(xα)Γ(1+α)+I0t(α){(τ-t)αΓ(1+α)(-tαEα(xα)Γ(1+α))}=-tαEα(xα)Γ(1+α)+t3αEα(xα)Γ(1+3α).
In the same manner, the second approximation is given by
(40)u2(x,t)=u1(x,t)+I0t(α){(τ-t)αΓ(1+α)(∂2αT1(x,τ)∂τ2α+∂2αT1(x,τ)∂x2α)}=-tαEα(xα)Γ(1+α)+t3αEα(xα)Γ(1+3α)+I0t(α){(τ-t)αΓ(1+α)(t3αEα(xα)Γ(1+3α))}=-tαEα(xα)Γ(1+α)+t3αEα(xα)Γ(1+3α)-t5αEα(xα)Γ(1+5α).
Finally, we can obtain the local fractional series solution as follows:
(41)un(x,t)=Eα(xα)(∑k=0n(-1)kt(2k+1)αΓ(1+(2k+1)α)).
Thus, the expression of the final solution is given by
(42)u(x,t)=limn→∞un(x,t)=Eα(xα)(∑i=0∞(-1)kt(2k+1)αΓ(1+(2k+1)α))=-Eα(xα)sinα(tα).
As is known, the Mittag-Leffler function in fractal space can be written in the form(43)|Eα(xα)-Eα(x0α)|≤Eα(x0α)|x-x0|α<εα,|sinα(tα)-sinα(t0α)|<|cosα(x0α)||t-t0|α<εα.
Hence, the fractal dimensions of bothEα(xα) and cosα(tα) are equal to α.


## 5. Conclusions

Local fractional calculus is set up on fractals and the local fractional variational iteration method is derived from local fractional calculus. This new technique is efficient for the applied scientists to process these differential and integral equations with the local fractional operators. The variational iteration method [9–19, 27] is derived from fractional calculus and classical calculus; the fractional variational iteration method [20–22, 27] is derived from the modified fractional derivative, while the local fractional variational iteration method [37] is derived from the local fractional calculus [37–43]. Other methods for local fractional ordinary and partial differential equations were considered in [27].In this paper, two outstanding examples of applications of the local fractional variational iteration method to the local fractional Laplace equations are investigated in detail. The reliable obtained results are complementary with the ones presented in the literature.



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