Application of Local Fractional Series Expansion Method to Solve

and Applied Analysis 3 subject to the initial value conditions: u (x, 0) = 0, ∂ α ∂u u (x, 0) = x 2α Γ (1 + 2α) . (18) From (12) and (18), we can structure the following iterative formulas: ψ i+2 (x) = ( ∂ 2α ψ i ∂x + ψ i ) (x) ,


Introduction
The Klein-Gordon equation [1] has been applied to mathematical physics such as solid-state physics, nonlinear optics, and quantum field theory.Some of the analytical methods for solving the Klein-Gordon equation include the variational iteration method [2], the tanh and the sine-cosine methods [3], the decomposition method [4], the differential transform method [5], and the homotopy-perturbation method [6].
Recently, the solutions for the fractional Klein-Gordon equation with the Caputo fractional derivative were considered in [7][8][9].Golmankhaneh et al. used the homotopyperturbation method to obtain solution for the fractional Klein-Gordon equation [7].Kurulay [8] pointed out the solution for the fractional Klein-Gordon equation by using the homotopy analysis method.Gepreel and Mohamed [9] presented the solution for nonlinear space-time fractional Klein-Gordon equation by the homotopy analysis method.
In view of (1)-( 2), the linear Klein-Gordon equation on Cantor sets: subject to the initial value conditions: is under consideration, where () and () are local fractional continuous functions.
On the other hand, the local fractional series expansion method was applied to solve the wave and diffusion equations on Cantor sets [21], the local fractional Schrödinger equation in the one-dimensional Cantorian system [22], and the local fractional Helmholtz equation [23].In this paper, our aim is to investigate a new application of this technology to solve the linear Klein-Gordon equations on Cantor sets.The paper is organized as follows.In Section 2, the idea of local fractional series expansion method is given.In Section 3, the solutions for linear Klein-Gordon equations on Cantor sets are presented.Finally, Section 4 is the conclusions.

The Local Fractional Series Expansion Method
In order to illustrate the idea of the local fractional series expansion method [21][22][23], we consider the local fractional differential operator equation in the following form: where   is the linear local fractional operator and  is a local fractional continuous function.From (6), the multiterm separated functions with respect to ,  read as where   () and   () are the local fractional continuous functions.From (7), we have so that In view of (9), we obtain Making use of ( 10), we have so that Hence, from (12) we get We now rewrite (4) in the local fractional operator form as follows: subject to the initial value conditions: where the linear local fractional operator is defined as follows: subject to the initial value conditions: From ( 12) and ( 18), we can structure the following iterative formulas: Hence, we can calculate and so on.Therefore, we have and the corresponding graph is illustrated in Figure 1.
Example 2. We consider the following Klein-Gordon equations on Cantor sets: subject to the initial value conditions: (, 0) = 0. From ( 12) and ( 23), we get the following iterative formulas: Hence, we get and so on.Hereby, we obtain the solution of (22): and the corresponding graph is depicted in Figure 2.
Example 3. We present the following Klein-Gordon equations on Cantor sets: subject to the initial value conditions: From ( 12) and ( 27)-( 28), we get the following iterative formulas: From (29) we obtain and so on.
Therefore, we obtain the exact solution of ( 27) and its graph is shown in Figure 3.
Example 4. The Klein-Gordon equation on Cantor sets is presented as and the initial value conditions are written as From ( 12) and ( 27)-( 28), the following iterative formulas are as follows: From (29), we give and so on.Therefore, we give the exact solution of (32): and its graph is shown in Figure 4.

Conclusions
In this work the Klein-Gordon equations on Cantor sets within the local fractional differential operator had been analyzed using the local fractional series expansion method.The nondifferentiable solutions for local fractional Klein-Gordon equations were obtained.The present method is a powerful mathematical tool for solving the local fractional linear differential equations.