%0 Journal Article
%T Approximation Solutions for Local Fractional Schrˋdinger Equation in the One-Dimensional Cantorian System
%A Yang Zhao
%A De-Fu Cheng
%A Xiao-Jun Yang
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/291386
%X The local fractional Schrˋdinger equations in the one-dimensional Cantorian system are investigated. The approximations solutions are obtained by using the local fractional series expansion method. The obtained solutions show that the present method is an efficient and simple tool for solving the linear partial differentiable equations within the local fractional derivative. 1. Introduction As it is known, in classical mechanics, the equations of motions are described as Newton＊s second law, and the equivalent formulations become the Euler-Lagrange equations and Hamilton＊s equations. In quantum mechanics, Schrˋdinger's equation for a dynamic system like Newton's law plays an important role in Newton's mechanics and conservation of energy. Mathematically, it is a partial differential equation, which is applied to describe how the quantum state of a physical system changes in time [1, 2]. In this work, the solutions of Schrˋdinger equations were investigated within the various methods [3每12] and other references therein. Recently, the fractional calculus [13每30], which is different from the classical calculus, is now applied to practical techniques in many branches of applied sciences and engineering. Fractional Schrˋdinger's equation was proposed by Laskin [31] via the space fractional quantum mechanics, which is based on the Feynman path integrals, and some properties of fractional Schrˋdinger's equation are investigated by Naber [32]. In present works, the solutions of fractional Schrˋdinger equations were considered in [33每38]. Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus) [39每56] is best candidate and has been applied to model the practical problems in engineering, which are nondifferentiable functions. For example, the systems of Navier-Stokes equations on Cantor sets with local fractional derivative were discussed in [42]. The local fractional Fokker-Planck equation was investigated in [43]. The basic theory of elastic problems was considered in [44]. The anomalous diffusion with local fractional derivative was researched in [48每50]. Newtonian mechanics with local fractional derivative was proposed in [51]. The fractal heat transfer in silk cocoon hierarchy and heat conduction in a semi-infinite fractal bar were presented in [53每55] and other references therein. More recently, the local fractional Schrˋdinger equation in three-dimensional Cantorian system was considered in [56] as where the local fractional Laplace operator is [39, 40, 42] the
%U http://www.hindawi.com/journals/amp/2013/291386/