Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

Abstract

We perform a comparison between the fractional iteration and decomposition methods applied to the wave equation on Cantor set. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

1. Introduction

Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger’s, the telegraph, the Advection, the Burgers, the KdV, the Boussinesq, and the Fisher equations and others [1].

Recently, the fractional calculus theory was recognized to be a good tool for modeling complex problems demonstrating its applicability in numerical scientific disciplines. Boundary value problems for the fractional differential equations have been the focus of several studies due to their frequent appearance in various areas, such as fractional diffusion and wave [2], fractional telegraph [3], fractional KdV [4], fractional Schrödinger [5], fractional evolution [6], fractional Navier-Stokes [7], fractional Heisenberg [8], fractional Klein-Gordon [9], and fractional Fisher equations [10].

Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations (see, e.g., [1–36] and the references therein). The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16, 17], the local fractional decomposition [18], and the generalized local fractional Fourier transform [19] methods.

Recently, the wave equation on Cantor sets (local fractional wave equation) was given by [35] 𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑡^2𝛼−𝑎^2𝛼𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑥^2𝛼=0, (1) where the operators are local fractional ones [16–19, 35, 36].

Following (1), a wave equation on Cantor sets was proposed as follows [36]: 𝜕^2𝛼𝑢(𝑥,𝑡)𝜕𝑡^2𝛼−𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑢(𝑥,𝑡)𝜕𝑥^2𝛼=0, (2) where 𝑢(𝑥,𝑡) is a fractal wave function.

In this paper, our purpose is to compare the local fractional variational iteration and decomposition methods for solving the local fractional differential equations. For illustrating the concepts we adopt one example for solving the wave equation on Cantor sets with local fractional operator.

Bearing these ideas in mind, the paper is organized as follows. In Section 2, we present basic definitions and provide some properties of local fractional derivative and integration. In Section 3, we introduce the local fractional variational iteration and the decomposition methods. In Section 4, we discuss one application. Finally, in Section 5 we outline the main conclusions.

2. Mathematical Tools

We recall in this section the notations and some properties of the local fractional operators [15–19, 35, 36].

Definition 1 (see [15–19, 35, 36]). The function 𝑓(𝑥) is local fractional continuous, if it is valid for 𝑓(𝑥)−𝑓𝑥₀<𝜀^𝛼, (3) where |𝑥−𝑥₀|<𝛿, for 𝜀>0 and 𝜀∈𝑅.

We notice that there are existence conditions of local fractional continuities that operating functions are right-hand and left-hand local fractional continuity. Meanwhile, the right-hand local fractional continuity is equal to its left-hand local fractional continuity. For more details, see [35].

Following (4), we have [15–19, 35, 36] 𝜌^𝛼𝑥−𝑥₀^𝛼≤𝑓(𝑥)−𝑓𝑥₀≤𝜅^𝛼𝑥−𝑥₀^𝛼 (4) with |𝑥−𝑥₀|<𝛿, for 𝜀, 𝛿>0 and 𝜀, 𝛿, 𝜅, 𝜌∈𝑅.

For a fractal set 𝐹, there is a fractal measure [35] 𝐻^𝛼𝐹∩𝑥,𝑥₀=𝑥−𝑥₀^𝛼, (5) where 𝑓(𝑥) presents a bi-Lipschitz mapping with fractal dimension 𝛼 and 𝐻^𝛼 denotes a Hausdorff dimension.

We verify that there is a measure 𝐻¹𝐹∩𝑥,𝑥₀=𝑥−𝑥₀ (6) in the case of 𝛼=1 and 𝑓(𝑥) is a Lipschitz mapping. If 𝐹 is a Cantor set, we have 𝐻^ln2/ln3(𝐹∩(𝑥,𝑥₀))=(𝑥−𝑥₀)^ln2/ln3 with 𝛼=ln2/ln3.

Definition 2 (see [15–19, 35, 36]). The local fractional derivative of 𝑓(𝑥) at 𝑥=𝑥₀ is defined as [16–20] 𝑓^(𝛼)𝑥₀=𝑑^𝛼𝑓(𝑥)𝑑𝑥^𝛼_𝑥=𝑥0=lim_𝑥→𝑥0Δ^𝛼𝑓(𝑥)−𝑓𝑥₀𝑥−𝑥₀^𝛼, (7) where Δ^𝛼𝑓(𝑥)−𝑓𝑥₀≅Γ(1+𝛼)Δ𝑓(𝑥)−𝑓𝑥₀. (8)

We find that the existence condition for local fractional derivative of 𝑓(𝑥) is that the right-hand local fractional derivative is equal to the left-hand local fractional derivative (see, e.g., [16, 35] and the references therein).

Definition 3 (see [15–19, 35, 36]). A partition of the interval [𝑎,𝑏] is denoted as (𝑡_𝑗,𝑡_𝑗+1), 𝑗=0,…,𝑁−1, 𝑡₀=𝑎, and 𝑡_𝑁=𝑏 with Δ𝑡_𝑗=𝑡_𝑗+1−𝑡_𝑗 and Δ𝑡=max{Δ𝑡₀,Δ𝑡₁,Δ𝑡_𝑗,…}. Local fractional integral of 𝑓(𝑥) in the interval [𝑎,𝑏] is given by _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. (9)

If the functions are local fractional continuous then the local fractional derivatives and integrals exist. That is to say, operating functions have nondifferentiable and fractal properties (see [35] and the references therein).

Some properties of local fractional derivative and integrals are given in [35].

3. Analytical Methods

In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation as follows: 𝐿_𝛼^(𝑛)𝑢+𝑅_𝛼𝑢=0, (10) where 𝐿_𝛼^(𝑛) is linear local fractional operators, respectively, with 𝑛=1,2 and 𝑅_𝛼 is linear local fractional operators of order less than 𝐿_𝛼^(𝑛).

3.1. Local Fractional Variational Iteration Method

The local fractional variational iteration algorithm is given by [16, 17] on the line of the formalism suggested in [35] 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+1Γ(1+𝛼)×^𝑡₀𝜆^𝛼Γ()𝐿1+𝛼^𝛼(𝑛)𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)(𝑑𝑠)^𝛼. (11) Here, we can construct a correction functional as follows [16, 17]: 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+1Γ(1+𝛼)×^𝑡₀𝜆^𝛼Γ()𝐿1+𝛼^𝛼(𝑛)𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)(𝑑𝑠)^𝛼, (12) where 𝑢_𝑛 is considered as a restricted local fractional variation; that is, 𝛿^𝛼𝑢_𝑛=0 (for more details, see [35]).

For 𝑛=2, we have 𝜆^𝛼=(𝑠−𝑡)^𝛼Γ(),1+𝛼 (13) so that iteration is expressed as 𝑢_𝑛+1(𝑡)=𝑢_𝑛(𝑡)+1Γ(1+𝛼)×^𝑡₀(𝑠−𝑡)^𝛼Γ()𝐿1+𝛼^𝛼(𝑛)𝑢_𝑛(𝑠)+𝑅_𝛼𝑢_𝑛(𝑠)(𝑑𝑠)^𝛼. (14) Finally, the solution is 𝑢(𝑥)=lim_𝑛→∞𝑢_𝑛(𝑥). (15)

3.2. Local Fractional Decomposition Method

When 𝐿_𝛼^(𝑛) in (10) is a local fractional differential operator of order 2𝛼, we denote it as 𝐿_𝛼^(𝑛)=𝐿^(2𝛼)_𝑥𝑥=𝜕^2𝛼𝜕𝑥^2𝛼,𝑅_𝛼𝑢(𝑡)=𝜕^𝛼𝜕𝑥^𝛼𝑢(𝑡)+𝑓(𝑡). (16) By defining the 𝑛-fold local fractional integral operator 𝐿_𝛼⁽⁻²⁾𝑚(𝑠)=₀𝐼_𝑥0^(𝛼)𝐼_𝑥^(𝛼)𝑚(𝑠) (17) we get 𝐿_𝛼⁽⁻²⁾𝐿_𝛼⁽²⁾𝑢(𝑠)=𝐿_𝛼⁽⁻²⁾𝑅_𝛼𝑢(𝑠). (18) Thus, 𝑢(𝑠)=𝑟(𝑥)+𝐿_𝛼⁽⁻²⁾𝑅_𝛼𝑢(𝑠), (19) where the term 𝑟(𝑥) is to be determined from the fractal initial conditions.

Therefore, we get the iterative formula as follows: 𝑢(𝑥)=𝑢₀(𝑥)+𝐿_𝛼⁽⁻²⁾𝑅_𝛼𝑢(𝑠), (20) with 𝑢₀(𝑥)=𝑟(𝑥).

Hence, for 𝑛≥0, we have the following recurrence relationship: 𝑢_𝑛+1(𝑥)=𝐿_𝛼⁽⁻²⁾𝑅_𝛼𝑢_𝑛(𝑠),𝑢₀(𝑥)=𝑟(𝑥). (21) Finally, the solution can be constructed as 𝑢(𝑥)=lim_𝑛→∞𝜙_𝑛(𝑥)=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛(𝑥). (22) For more details, see [18].

4. An Illustrative Example

In this section one example for wave equation is presented in order to demonstrate the simplicity and the efficiency of the above methods.

In (2), we consider the following initial and boundary conditions: 𝜕^𝛼𝑢(𝑥,0)𝜕𝑡^𝛼()=𝑥=0,𝑢𝑥,0^2𝛼Γ().1+2𝛼 (23)

Using (14) we have the iterative formula 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢_𝑛(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝑥1+𝛼Γ(^2𝛼)𝜕1+2𝛼^2𝛼𝑢_𝑛(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼, (24) where the initial value is given by 𝑢₀()=𝑥𝑥,𝑡^2𝛼Γ().1+2𝛼 (25) Thus, after computing (23) we obtain 𝑢₁(𝑥,𝑡)=𝑢₀()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₀(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝑥1+𝛼Γ(^2𝛼)𝜕1+2𝛼^2𝛼𝑢₀(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼=𝑢₀()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()−𝑥1+𝛼Γ(^2𝛼))1+2𝛼(𝑑𝑠^𝛼=𝑥^2𝛼Γ()𝑡1+2𝛼1+^2𝛼Γ(),𝑢1+2𝛼₂(𝑥,𝑡)=𝑢₁()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₁(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝑥1+𝛼Γ(^2𝛼)𝜕1+2𝛼^2𝛼𝑢₁(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼=𝑢₁()+1𝑥,𝑡Γ()×1+𝛼_𝑡0(𝑠−𝑡)^𝛼Γ()−𝑥1+𝛼Γ(^2𝛼)𝑠1+2𝛼Γ(^2𝛼))1+2𝛼(𝑑𝑠^𝛼=𝑥^2𝛼Γ()𝑡1+2𝛼1+^2𝛼Γ()+𝑡1+2𝛼Γ(^4𝛼),𝑢1+4𝛼₃(𝑥,𝑡)=𝑢₂()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₂(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝑥1+𝛼Γ(^2𝛼)𝜕1+2𝛼^2𝛼𝑢₂(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼=𝑢₂()+1𝑥,𝑡Γ(1+𝛼)×^𝑡₀(𝑠−𝑡)^𝛼Γ()−𝑥1+𝛼Γ(^2𝛼)𝑡1+2𝛼Γ(^4𝛼))1+4𝛼(𝑑𝑠^𝛼=𝑥^2𝛼Γ(1+2𝛼)³_𝑖=0𝑡^2𝑖𝛼Γ(),𝑢1+2𝑖𝛼₄(𝑥,𝑡)=𝑢₃()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢₃(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)^𝑡₀𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑢₃(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼=𝑢₃()+1𝑥,𝑡Γ()×1+𝛼_𝑡0(𝑠−𝑡)^𝛼Γ()−𝑥1+𝛼Γ(^2𝛼)𝑡1+2𝛼Γ(^6𝛼))1+6𝛼(𝑑𝑠^𝛼=𝑥^2𝛼Γ(1+2𝛼)⁴_𝑖=0𝑡^2𝑖𝛼Γ(),⋮𝑢1+2𝑖𝛼_𝑛(𝑥,𝑡)=𝑢_𝑛−1()+1𝑥,𝑡Γ(1+𝛼)^𝑡₀(𝑠−𝑡)^𝛼Γ()𝜕1+𝛼^2𝛼𝑢_𝑛−1(𝑥,𝑠)𝜕𝑠^2𝛼(𝑑𝑠)^𝛼−1Γ(1+𝛼)×^𝑡₀(𝑠−𝑡)^𝛼Γ()𝑥1+𝛼Γ(^2𝛼)𝜕1+2𝛼^2𝛼𝑢_𝑛−1(𝑥,𝑠)𝜕𝑥^2𝛼(𝑑𝑠)^𝛼=𝑥^2𝛼Γ(1+2𝛼)^𝑛_𝑖=0𝑡^2𝑖𝛼Γ().1+2𝑖𝛼 (26)(27)

Hence, from (27) we obtain the solution of (3) as 𝑢(𝑥)=lim_𝑛→∞𝑢_𝑛(𝑥)=𝑥^2𝛼Γ(1+2𝛼)cosh_𝛼𝑡^𝛼. (28) Here, from (21) we get 𝑢_𝑛+1(𝑥,𝑡)=₀𝐼_𝑡0^(𝛼)𝐼_𝑡^(𝛼)𝑥^2𝛼Γ()𝜕1+2𝛼^2𝛼𝑢_𝑛(𝑥,𝑠)𝜕𝑥^2𝛼,𝑢₀()=𝑥𝑥,𝑡^2𝛼Γ().1+2𝛼 (29) Therefore, from (29) we give the components as follows: 𝑢₀()=𝑥𝑥,𝑡^2𝛼Γ(),𝑢1+2𝛼₁()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^2𝛼),𝑢1+2𝛼₂()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^4𝛼),𝑢1+4𝛼₃()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^3𝛼),𝑢1+6𝛼₄()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^8𝛼),⋮𝑢1+8𝛼_𝑛()=𝑥𝑥,𝑡^2𝛼Γ()𝑡1+2𝛼Γ(^2𝑛𝛼).1+2𝑛𝛼 (30) Consequently, the exact solution is given by 𝑢(𝑥,𝑡)=lim^∞_{𝑛→∞𝑛=0}𝑢_𝑛(𝑥,𝑡)=lim^∞_{𝑛→∞𝑛=0}𝑥^2𝛼Γ()𝑡1+2𝛼Γ(^2𝑛𝛼)=𝑥1+2𝑛𝛼_2𝛼Γ(1+2𝛼)cosh_𝛼𝑥^𝛼, (31) where cosh_𝛼𝑥^𝛼=^∞_𝑛=0𝑡^2𝑛𝛼Γ().1+2𝑛𝛼 (32) The solution of (2) for 𝛼=ln2/ln3 is depicted in Figure 1.

Figure 1 

Graph of 𝑢(𝑥,𝑡) for 𝛼=ln2/ln3.

5. Conclusions

In this work, we developed a comparison between the variational iteration method and the decomposition method within local fractional operators. The two approaches constitute efficient tools to handle the approximation solutions for differential equations on Cantor sets with local fractional derivative. We notice that the fractional variational iteration method gives the several successive approximate formulas using the iteration of the correction local fractional functional. However, the local fractional decomposition method provides the components of the exact solution, which is local fractional continuous function, where these components are also local fractional continuous functions. Both the variational iteration method and the decomposition method within local fractional operators provide the solution in successive components. The methods are structured to get the local fractional series solution, which is a nondifferentiable function.

Conflict of Interests

The authors declare that there is no conflict of interests regarding publication of this paper.