Analysis of Fractal Wave Equations by Local Fractional Fourier Series Method

The fractal wave equations with local fractional derivatives are investigated in this paper. The analytical solutions are obtained by using local fractional Fourier series method. The present method is very efficient and accurate to process a class of local fractional differential equations. 1. Introduction Fractional calculus deals with derivative and integrals of arbitrary orders [1]. During the last four decades, fractional calculus has been applied to almost every field of science and engineering [2–6]. In recent years, there has been a great deal of interest in fractional differential equations [7]. As a result, various kinds of analytical methods were developed [8–18]. For example, there are the exp-function method [8], the variational iteration method [9, 10], the homotopy perturbation method [11], the homotopy analysis method [12], the heat-balance integral method [13], the fractional variational iteration method [14, 15], the fractional difference method [16], the finite element method [17], the fractional Fourier and Laplace transforms [18], and so on. Recently, local fractional calculus was applied to deal with problems for nondifferentiable functions; see [19–26] and the references therein. There are also analytical methods for solving the local fractional differential equations, which are referred to in [27–34]. The local fractional series method [32–34] was applied to process the local fractional wave equation in fractal vibrating [32] and local fractional heat-conduction equation [33]. More recently, the wave equation on the Cantor sets was considered as [21, 28] Local damped wave equation was written in the form [30] and local fractional dissipative wave equation in fractal strings was [31] In this paper, we investigate the application of local fractional series method for solving the following local fractional wave equation: where initial and boundary conditions are presented as The organization of the paper is as follows. In Section 2, the basic concepts of local fractional calculus and local fractional Fourier series are introduced. In Section 3, we present a local fractional Fourier series solution of wave equation with local fractional derivative. Two examples are shown in Section 4. Finally, Section 5 is devoted to our conclusions. 2. Mathematical Tools In this section, we present some concepts of local fractional continuity, local fractional derivative, and local fractional Fourier series. Definition 1 (see [21, 28, 30–32]). Suppose that there is with , for , and , . Then is called local fractional continuous at , where with .