Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators

1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran 2Department of Mathematics, Faculty of Sciences, University of Istanbul, Vezneciler, 34134 Istanbul, Turkey 3Department of Mathematics and Statistics, Faculty of Science, Tshwane University of Technology, Arcadia Campus, Building 2-117, Nelson Mandela Drive, Pretoria 0001, South Africa 4Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

The diffusion equations are important in many processes in science and engineering, for example, the diffusion of a dissolved substance in the solvent liquids and neutrons in a nuclear reactor and Brownian motion, while wave equations characterize the motion of a vibrating string (see [25,26] and the references therein).
The diffusion equation on Cantor sets (called local fractional diffusion equation) was recently described in [27] as where  2 denotes the fractal diffusion constant which is, in essence, a measure for the efficiency of the spreading of the underlying substance, while local fractional wave equation is written in the following form [28,29]: 2

Mathematical Problems in Engineering
The local fractional Laplace operator is given by [28,29] as follows: We notice that the local fractional diffusion equation yields and the local fractional wave equation has the following form: where 1/ 2 is a constant.This equation describes vibrations in a fractal medium.
The quantity (, ) is interpreted as the local fractional deviation at the time  from the position at rest of the point with rest position given by , , and .The above fractal derivatives were considered as the local fractional operators [30,31].
The paper is organized as follows.In Section 2, we introduce the notions of local fractional calculus theory used in this paper.In Section 3, we give the local fractional Laplace variational iteration method.Section 4 presents the solutions for diffusion and wave equations in Cantor set conditions.Section 5 is devoted to our conclusions.[28,29,32,33]) Definition 1. Suppose that there is the relation
The formulas of local fractional derivatives of special functions used in the paper are as follows: The formulas of local fractional integrals of special functions used in the paper are as follows: ,  ∈ . (10)

Local Fractional Laplace Variational Iteration Method
Let us consider the following local fractional partial differential equations: where   is the linear local fractional operator,   is the linear local fractional operator of order less than   , and (, ) is a source term of the nondifferential function.
For initial value problems of ( 14), we can start with We now take Yang-Laplace transform of (15); namely, or Take the local fractional variation of (18), which is given by By using computation of ( 19), we get Hence, from (20), we get where Therefore, we get Taking the inverse version of the Yang-Laplace transform, we have ,  ∈ . ( Mathematical Problems in Engineering In view of (24), we obtain Therefore, we have the following iteration algorithm: or where the initial value reads as follows: Thus, the local fractional series solution of ( 14) is

Applications to Diffusion and Wave Equations on Cantor Sets
In this section, four examples for diffusion and wave equations on Cantor sets will demonstrate the efficiency of local fractional Laplace variational iteration method.
Example 1.Let us consider the following diffusion equation on Cantor set: with the initial value condition Using relation (26), we structure the iterative relation as In view of ( 28), the initial value reads as follows: Hence, we get the first approximation; namely, Thus, The second approximation reads as follows: Therefore, we get Consequently, the local fractional series solution is The result is the same as the one which is obtained by the local fractional series expansion method [38].
Example 2. Let us consider the following diffusion equation on Cantor set: with the initial value conditions being as follows: Using relation (26), we structure the iterative relation as follows: In view of ( 28), the initial value reads as follows: Hence, we get the first approximation; namely, Thus, ) . (44) The second approximation reads as follows: Therefore, we get Consequently, the local fractional series solution is The result is the same as the one which is obtained by the local fractional series expansion method [38].
Example 3. Let us consider the following wave equation on Cantor set: with the initial value conditions being as follows: Using relation (26), we structure the iterative relation as In view of ( 28), the initial value reads as follows: Hence, we get the first approximation; namely, Thus, ) . ( The second approximation reads as follows: Therefore, we get Consequently, the local fractional series solution is The result is the same as the one which is obtained by the local fractional Adomian decomposition method and local fractional variational iteration method in [34].
Example 4. Let us consider the following wave equation on Cantor set: with the initial value conditions being as follows: Using relation (26), we structure the iterative relation as In view of ( 28), the initial value reads as follows: Hence, we get the first approximation; namely, ) . ( The second approximation reads as follows:  (65)

Conclusions
The local fractional Laplace variational iteration method was applied to the diffusion and wave equations defined on Cantor sets with the fractal conditions.The local fractional Laplace variational iteration method was proved to be effective and very reliable for analytic purposes.Further, the same problems are solved by local fractional expansion series method (LFESM), local fractional variational iteration method (LFVIM), and local fractional Adomian decomposition method (LFADM).The results obtained by the four methods are in agreement and, hence, this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations.