Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.

The fractional differential equations were considered in sense of the Caputo derivative, the Riemann-Liouville derivative, and the Grünwald-Letnikov derivative [17].However, they do not deal with the nondifferentiable functions defined on Cantor sets.Local fractional derivative [18,19] is the best method for describing the nondifferential problems defined on Cantor sets.For example, the heat equations arising in fractal transient conduction were investigated in [19][20][21][22].The Helmholtz and diffusion equations on the Cantor sets within local fractional derivative were discussed [23].The Navier-Stokes equations on Cantor sets were suggested in [24].There are some methods for solving the local fractional differential equations, such as the local fractional variational iteration method [20], the Yang-Fourier transform [21], the Yang-Laplace transform [22], the local fractional Fourier series method [25], and the local fractional Adomian decomposition method [26].
In this paper, our aims are to present the coupling method of local fractional series method and Yang-Laplace transform, which is called as the local fractional function decomposition method, and to use it to solve the differential equations with local fractional derivative.The organization of the manuscript is as follows.In Section 2, the basic mathematical tools are introduced.In Section 3, the local fractional function decomposition method for solving the differential equations with local fractional derivative is investigated.In Section 4, several examples are considered.Finally, in Section 5 the conclusions are given.

Mathematical Fundamentals
In this section, we introduce the basic notions of local fractional continuity, local fractional derivative, local fractional Fourier series, and special function in fractal space [18,19], which are used in the paper.Definition 1. Suppose that there is [ where Local fractional derivative of high order and local fractional partial derivative are defined, respectively, in the following forms [18,19]: ( Definition 5. Let () be 2-periodic.For  ∈  and () ∈   (−∞, +∞), the local fraction Fourier series of () is defined as (see [18,25]) where are the local fraction Fourier coefficients.
The Yang-Laplace transforms of () is given by [18,22] where the latter integral converges and   ∈   .

Local Fractional Function Decomposition Method
In this section we will present the local fractional function decomposition method.At first, we present the local fractional differential equation with constants  1 ,  2 ,  3 , 0 <  ≤ 1/2 and with boundary and initial conditions Now we discuss the solution of (10).
Abstract and Applied Analysis 3 According to the decomposition of the local fractional function, with respect to the system {sin    (/)  }, the following functions coefficients can be given by where Substituting ( 12) into (10) implies that Suppose that the Yang-Laplace transforms of functions V  () and   () are   () and   (), respectively.Then we obtain That is Hence, we have Hence, we get Then, making use of ( 8) and ( 9) and rearranging integration sequence, we have the following several formulas about V 1, () and V 2, ().
(I) Suppose that where When we get (II) If then we have When we arrive at where we obtain The above results are the desired solutions.

Illustrative Examples
In order to illustrate the above results in Section 3, we give the following several examples.
Example 1.The local fractional Laplace differential equation is written in the following form [18,19]: subjected to the boundary and initial conditions described by  (, 0) = sin  (  ) ,    (, )   = sin  (  ) ,  (0, ) =  (, ) = 0. (34) From (33), the final solution can be easily deduced as follows: Example 2. We consider the following inhomogeneous wave equation with local fractional derivative: subjected to the boundary and initial conditions In order to find its solution, we suppose that which leads to Contrasting (37) with (35), we directly get According to (30) and (32), we can derive (49)