Advances in Mathematical Physics

Volume 2013 (2013), Article ID 291386, 5 pages

http://dx.doi.org/10.1155/2013/291386

Research Article

Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System

Yang Zhao,1,2 De-Fu Cheng,¹ and Xiao-Jun Yang³

¹College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130061, China

²Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

³Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China

Received 23 July 2013; Revised 7 August 2013; Accepted 12 August 2013

Academic Editor: D. Băleanu

Copyright © 2013 Yang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The local fractional Schrödinger 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, Schrödinger'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 Schrödinger equations were investigated within the various methods [3–12] and other references therein.

Recently, the fractional calculus [13–30], which is different from the classical calculus, is now applied to practical techniques in many branches of applied sciences and engineering. Fractional Schrödinger'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 Schrödinger's equation are investigated by Naber [32]. In present works, the solutions of fractional Schrödinger equations were considered in [33–38].

Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus) [39–56] 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 [48–50]. 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 [53–55] and other references therein.

More recently, the local fractional Schrödinger equation in three-dimensional Cantorian system was considered in [56] as

where the local fractional Laplace operator is [39, 40, 42]

the wave function is a local fractional continuous function [39, 40], and the local fractional differential operator is given by [39, 40]

with .

The local fractional Schrödinger equation in two-dimensional Cantorian system can be written as

where the local fractional Laplace operator is given by

The local fractional Schrödinger equation in one-dimensional Cantorian system is presented as

where the wave function is local fractional continuous function.

With the potential energy , the local fractional Schrödinger equation in the one-dimensional Cantorian system is

In this paper our aim is to investigate the nondifferentiable solutions for local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method [49]. The organization of the paper is organized as follows. In Section 2, we introduce the local fractional series expansion method. Section 3 is devoted to the solutions for local fractional Schrödinger equations. Finally, conclusions are given in Section 4.

2. The Local Fractional Series Expansion Method

According to local fractional series expansion method [49], we consider the following local fractional differentiable equation:

where is the linear local fractional operator and is a local fractional continuous function.

In view of (8), the multiterm separated functions with respect to are expressed as follows:

where and are the local fractional continuous function.

There are nondifferentiable terms, which are written as

where is a coefficient.

In view of (10), we get

Therefore,

Then, following (12), we have

Let ; then

So, we have

where is a linear local fractional operator.

In [49], the linear local fractional operators are considered as

where is a constant.

Here, we consider the following operator:

where and are two constants.

Using the iterative formula (18), we obtain

which is the solution of (8).

3. Approximation Solutions

Let us change (6) into the formula in the following form:

where the linear local fractional operator is

With , we have

so that

where

Using iteration relation (15), we set up

and an initial value is given by

Therefore, following (25), we get

and so on.

Hence, from (32) we obtain the solution of (23) as

We transform (7) into the following equation:

The initial condition is presented as

Applying (17), we can write the iterative relations as follows:

where

From (36)–(38), we give the local fractional series terms as follows:

and so forth.

Hence, we have the nondifferentiable solution of (34) as follows:

4. Conclusions

In the work, we have obtained the nondifferentiable solutions for the local fractional Schrödinger equations in the one-dimensional Cantorian system by using the local fractional series expansion method. The present method is shown that is an effective method to obtain the local fractional series solutions for the partial differential equations within local fractional differentiable operator.

Acknowledgment

This work was supported by Natural Science Foundation of Hebei Province (no. F2010001322).

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