%0 Journal Article
%T The Yang-Laplace Transform for Solving the IVPs with Local Fractional Derivative
%A Chun-Guang Zhao
%A Ai-Min Yang
%A Hossein Jafari
%A Ahmad Haghbin
%J Abstract and Applied Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/386459
%X The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform. 1. Introduction In recent years, the ordinary and partial differential equations have found applications in many problems in mathematical physics [1, 2]. Initial value problems (IVPs) for ordinary and partial differential equations have been developed by some authors in [3¨C6]. There are analytical methods and numerical methods for solving the differential equations, such as the finite element method [6], the harmonic wavelet method [7¨C9], the Adomian decomposition method [10¨C12], the homotopy analysis method [13, 14], the homotopy decomposition method [15, 16], the heat balance integral method [17, 18], the homotopy perturbation method [19], the variational iteration method [20], and other methods [21]. Recently, owing to limit of classical and fractional differential equations, the local fractional differential equations have been applied to describe nondifferentiable problems for the heat and wave in fractal media [22, 23], the structure relation in fractal elasticity [24], and Fokker-Planck equation in fractal media [25]. Some methods were utilized to solve the local fractional differential equations. For example, the local fractional variation iteration method was used to solve the heat conduction in fractal media [26, 27]. The local fractional decomposition method for solving the local fractional diffusion and heat-conduction equations was considered in [28, 29]. The local fractional series expansion method for solving the Schr£żdinger equation with the local fractional derivative was presented [30]. The Yang-Laplace transform structured in 2011 [22] was suggested to deal with local fractional differential equations [31, 32]. The coupling method for variational iteration method within Yang-Laplace transform for solving the heat conduction in fractal media was proposed in [33]. In this paper, our aim is to use the Yang-Laplace transform to solve IVPs with local fractional derivative. The structure of the paper is as follows. In Section 2, some definitions and properties for the Yang-Laplace transform are given. Section 3 is devoted to the solutions for the homogeneous and nonhomogeneous IVPs with local fractional derivative. Finally, conclusions are presented in Section 4. 2. Yang-Laplace Transform In this section we show some definitions and properties for the Yang-Laplace transform. The local fractional integral operator is
%U http://www.hindawi.com/journals/aaa/2014/386459/