AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 291386 10.1155/2013/291386 291386 Research Article Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System Zhao Yang 1,2 Cheng De-Fu 1 Yang Xiao-Jun 3 Băleanu D. 1 College of Instrumentation & Electrical Engineering Jilin University Changchun 130061 China jlu.edu.cn 2 Electronic and Information Technology Department Jiangmen Polytechnic Jiangmen 529090 China jmpt.cn 3 Department of Mathematics and Mechanics China University of Mining and Technology Xuzhou Campus Xuzhou Jiangsu 221008 China cumt.edu.cn 2013 11 9 2013 2013 23 07 2013 07 08 2013 12 08 2013 2013 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.

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  and other references therein.

Recently, the fractional calculus , 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  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 . In present works, the solutions of fractional Schrödinger equations were considered in .

Classical and fractional calculus cannot deal with nondifferentiable functions. However, the local fractional calculus (also called fractal calculus)  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 . The local fractional Fokker-Planck equation was investigated in . The basic theory of elastic problems was considered in . The anomalous diffusion with local fractional derivative was researched in . Newtonian mechanics with local fractional derivative was proposed in . The fractal heat transfer in silk cocoon hierarchy and heat conduction in a semi-infinite fractal bar were presented in  and other references therein.

More recently, the local fractional Schrödinger equation in three-dimensional Cantorian system was considered in  as (1)iαhααψα(x,y,z,t)tα=-hα22m2αψα(x,y,z,t)+Vα(x,y,z)ψα(x,y,z,t), where the local fractional Laplace operator is [39, 40, 42] (2)2α=2αx2α+2αy2α+2αz2α, the wave function ψα(x,y,z,t) is a local fractional continuous function [39, 40], and the local fractional differential operator is given by [39, 40] (3)f(α)(x0)=dαf(x)dxαx=x0=limxx0Δα(f(x)-f(x0))(x-x0)α,

with Δα(f(x)-f(x0))Γ(1+α)Δ(f(x)-f(x0)).

The local fractional Schrödinger equation in two-dimensional Cantorian system can be written as (4)iαhααψα(x,y,t)tα=-hα22m2αψα(x,y,t)+Vα(x,y)ψα(x,y,t), where the local fractional Laplace operator is given by (5)2α=2αx2α+2αy2α. The local fractional Schrödinger equation in one-dimensional Cantorian system is presented as (6)iαhααψα(x,t)tα=-hα22m2αx2αψα(x,t)+Vα(x)ψα(x,t), where the wave function ψα(x,t) is local fractional continuous function.

With the potential energy Vα=0, the local fractional Schrödinger equation in the one-dimensional Cantorian system is (7)iαhααψα(x,t)tα=-hα22m2αx2αψα(x,t).

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 . 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 , we consider the following local fractional differentiable equation: (8)ϕtα=Lαϕ, where Lα is the linear local fractional operator and ϕ is a local fractional continuous function.

In view of (8), the multiterm separated functions with respect to x,t are expressed as follows: (9)ϕ(x,t)=i=0φi(t)ψi(x), where φi(t) and ψi(x) are the local fractional continuous function.

There are nondifferentiable terms, which are written as (10)φi(t)=χitiαΓ(1+iα), where χi is a coefficient.

In view of (10), we get (11)ϕ(x,t)=i=0χitiαΓ(1+iα)ψi(x).

Therefore, (12)ϕtα=i=0χi+1tiαΓ(1+iα)ψi+1(x),Lαϕ=i=0χitiαΓ(1+iα)(Lαψi)(x).

Then, following (12), we have (13)i=0χi+1tiαΓ(1+iα)ψi+1(x)=i=0χitiαΓ(1+iα)(Lαψi)(x).

Let χi+1=χi=1; then (14)i=0tiαΓ(1+iα)ψi+1(x)=i=0tiαΓ(1+iα)(Lαψi)(x).

So, we have (15)ψi+1(x)=Lαψi, where Lα is a linear local fractional operator.

In , the linear local fractional operators are considered as (16)Lα=2αx2α,Lα=x2αΓ(1+2α)2αx2α,(17)Lα=μ2αx2α, where μ is a constant.

Here, we consider the following operator: (18)Lα=η2αx2α+γ, where η and γ are two constants.

Using the iterative formula (18), we obtain (19)ϕ(x,t)=i=0tiαΓ(1+iα)ψi(x), which is the solution of (8).

3. Approximation Solutions

Let us change (6) into the formula in the following form: (20)αψα(x,t)tα=  -hα2iαm2αx2αψα(x,t)+Vα(x)iαhαψα(x,t), where the linear local fractional operator is (21)Lα=-hα2iαm2αx2α+Vα(x)iαhα.

With Vα(x)=1, we have (22)Lα=-hα2iαm2αx2α+1iαhα,

so that (23)αψα(x,t)tα=Lαψα(x,t), where (24)ψα(x,0)=x2αΓ(1+2α).

Using iteration relation (15), we set up (25)ψα,n+1(x,t)=Lαψa,n(x,t),

and an initial value is given by (26)ψa,0(x,t)=x2αΓ(1+2α).

Therefore, following (25), we get (27)ψa,0(x,t)=x2αΓ(1+2α),(28)ψa,1(x,t)=(-hα2iαm2αx2α+1iαhα)ψa,0(x,t)=-hα2iαm+1iαhαx2αΓ(1+2α),(29)ψa,2(x,t)=(-hα2iαm2αx2α+1iαhα)ψa,1(x,t)=(-hα2iαm2αx2α+1iαhα)×(-hα2iαm+1iαhαx2αΓ(1+2α))=-22i2αm+1i2αhα2x2αΓ(1+2α),(30)ψa,3(x,t)=(-hα2iαm2αx2α+1iαhα)ψa,2(x,t)=(-hα2iαm2αx2α+1iαhα)×(-1i2αm+1i2αhα2x2αΓ(1+2α))=-32i3αmhα+1i3αhα3x2αΓ(1+2α),(31)ψa,4(x,t)=(-hα2iαm2αx2α+1iαhα)ψa,3(x,t)=(-hα2iαm2αx2α+1iαhα)×(-32i3αmhα+1i3αhα3x2αΓ(1+2α))=-42i4αmhα2+1i4αhα4x2αΓ(1+2α),(32)ψa,n(x,t)=(-hα2iαm2αx2α+1iαhα)ψa,n-1(x,t)=-hαn-2n2inαm+1inαhαnx2αΓ(1+2α),

and so on.

Hence, from (32) we obtain the solution of (23) as (33)ψα(x,t)=n=0ψa,n(x,t)=n=0tnαΓ(1+nα)(-hαn-2n2inαm+1inαhαnx2αΓ(1+2α)).

We transform (7) into the following equation: (34)αψα(x,t)tα=-hα2miα2αx2αψα(x,t).

The initial condition is presented as (35)ψα(x,0)=x2αΓ(1+2α).

Applying (17), we can write the iterative relations as follows: (36)ψα,n+1(x,t)=Lαψa,n(x,t),ψa,0(x,t)=x2αΓ(1+2α), where (37)Lα=-hα2miα2αx2α. From (36)–(38), we give the local fractional series terms as follows: (38)ψa,0(x,t)=x2αΓ(1+2α),(39)ψa,1(x,t)=-hα2miα2αx2αψa,0(x,t)=-hα2iαm,(40)ψa,2(x,t)=-hα2miα2αx2αψa,1(x,t)=0,(41)ψa,3(x,t)=-hα2miα2αx2αψa,2(x,t)=0,(42)ψa,n(x,t)=-hα2miα2αx2αψa,n-1(x,t)=0,

and so forth.

Hence, we have the nondifferentiable solution of (34) as follows: (43)ψα(x,t)=n=0ψa,n(x,t)=x2αΓ(1+2α)+hα2miα.

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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