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

and Applied Analysis 3 where ?̃? n is considered as a restricted local fractional variation; that is, δ?̃? n = 0 (for more details, see [35]). For n = 2, we have


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].
Following (1), a wave equation on Cantor sets was proposed as follows [36]: where (, ) 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.

Mathematical Tools
We recall in this section the notations and some properties of the local fractional operators [15-19, 35, 36].
We notice that there are existence conditions of local fractional continuities that operating functions are righthand and left-hand local fractional continuity.Meanwhile, the right-hand local fractional continuity is equal to its lefthand local fractional continuity.For more details, see [35].
For a fractal set , there is a fractal measure [35] where () presents a bi-Lipschitz mapping with fractal dimension  and   denotes a Hausdorff dimension.We verify that there is a measure in the case of  = 1 and () is a Lipschitz mapping.If  is a Cantor set, we have  ln 2/ ln 3 ( ∩ (,  0 )) = ( −  0 ) ln 2/ ln 3 with  = ln 2/ ln 3.

Analytical Methods
In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation as follows: where  ()  is linear local fractional operators, respectively, with  = 1, 2 and   is linear local fractional operators of order less than  ()  .

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] Here, we can construct a correction functional as follows [16,17]: Abstract and Applied Analysis where ũ is considered as a restricted local fractional variation; that is,   ũ = 0 (for more details, see [35]).
For  = 2, we have so that iteration is expressed as Finally, the solution is

Local Fractional Decomposition Method.
When  ()  in ( 10) is a local fractional differential operator of order 2, we denote it as By defining the -fold local fractional integral operator we get (2)    () =  (−2)     () . Thus, where the term () is to be determined from the fractal initial conditions.Therefore, we get the iterative formula as follows: with  0 () = ().

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: Using ( 14) we have the iterative formula where the initial value is given by Thus, after computing ( 23) we obtain Hence, from (27) we obtain the solution of (3) as Here, from (21) we get Therefore, from (29) we give the components as follows: Consequently, the exact solution is given by where The solution of (2) for  = ln 2/ ln 3 is depicted in Figure 1.

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.