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. 1. Introduction Fractional differential equations with arbitrary orders [1] have attracted more and more attention to their extensive applications in various areas, such as physics, applied mathematics, and biology [2–8]. As a result, great deal of methods for solving the fractional differential equations are developed [9–21], such as the heat balance integral method [9, 10], the homotopy analysis method [11], the variational iteration method [12], the homotopy decomposition method [13, 14], and the Adomian decomposition method [15, 16]. 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 [19–22]. 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. 2. Mathematical Fundamentals In this section, we introduce the basic