Research Article | Open Access

Yang Zhao, De-Fu Cheng, Xiao-Jun Yang, "Approximation Solutions for Local Fractional Schrödinger Equation in the One-Dimensional Cantorian System", *Advances in Mathematical Physics*, vol. 2013, Article ID 291386, 5 pages, 2013. https://doi.org/10.1155/2013/291386

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

**Academic Editor:**D. Băleanu

#### 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).

#### References

- G. Teschi,
*Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators*, vol. 99, American Mathematical Society, Providence, RI, USA, 2009. - R. Shankar,
*Principles of Quantum Mechanics*, vol. 233, Plenum Press, New York, NY, USA, 1994. View at: Zentralblatt MATH | MathSciNet - M. D. Feit, J. A. Fleck, Jr., and A. Steiger, “Solution of the Schrödinger equation by a spectral method,”
*Journal of Computational Physics*, vol. 47, no. 3, pp. 412–433, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Delfour, M. Fortin, and G. Payre, “Finite-difference solutions of a non-linear Schrödinger equation,”
*Journal of Computational Physics*, vol. 44, no. 2, pp. 277–288, 1981. View at: Publisher Site | Google Scholar | MathSciNet - A. Borhanifar and R. Abazari, “Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method,”
*Optics Communications*, vol. 283, no. 10, pp. 2026–2031, 2010. View at: Publisher Site | Google Scholar - A. S. V. R. Kanth and K. Aruna, “Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations,”
*Chaos, Solitons & Fractals*, vol. 41, no. 5, pp. 2277–2281, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. M. Wazwaz, “A study on linear and nonlinear Schrödinger equations by the variational iteration method,”
*Chaos, Solitons & Fractals*, vol. 37, no. 4, pp. 1136–1142, 2008. View at: Publisher Site | Google Scholar | MathSciNet - M. M. Mousa, S. F. Ragab, and Z. Nturforsch, “Application of the homotopy perturbation method to linear and nonlinear Schrödinger equations,”
*Zeitschrift Fur Naturforschung A*, vol. 63, no. 3-4, pp. 140–144, 2008. View at: Google Scholar - N. H. Sweilam and R. F. Al-Bar, “Variational iteration method for coupled nonlinear Schrödinger equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7, pp. 993–999, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Modified variational iteration method for Schrödinger equations,”
*Mathematical and Computational Applications*, vol. 15, no. 3, pp. 309–317, 2010. View at: Google Scholar | MathSciNet - J. Biazar and H. Ghazvini, “Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method,”
*Physics Letters A*, vol. 366, no. 1, pp. 79–84, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Sadighi and D. D. Ganji, “Analytic treatment of linear and nonlinear Schrödinger equations: a study with homotopy-perturbation and Adomian decomposition methods,”
*Physics Letters A*, vol. 372, no. 4, pp. 465–469, 2008. View at: Publisher Site | Google Scholar | MathSciNet - A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, Netherlands, 2006. View at: MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. View at: MathSciNet - K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. View at: MathSciNet - A. Carpinteri and F. Mainardi,
*Fractals Fractional Calculus in Continuum Mechanics*, Springer, New York, NY, USA, 1997. - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at: MathSciNet - F. Mainardi,
*Fractional Calculus and Waves in Linear Viscoelasticity*, Imperial College Press, London, UK, 2010. View at: Publisher Site | MathSciNet - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begerll House, Connecticut, Conn, USA, 2006. - J. Klafter, S. C. Lim, and R. Metzler,
*Fractional Dynamics in Physics: Recent Advances*, World Scientific, Singapore, 2011. View at: MathSciNet - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, Oxford, UK, 2008. View at: MathSciNet - J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. View at: Publisher Site | MathSciNet - V. E. Tarasov,
*Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media*, Springer, Berlin, Germany, 2011. - J. A. T. Machado, A. C. J. Luo, and D. Baleanu,
*Nonlinear Dynamics of Complex Systems: Applications in Physical, Biological and Financial Systems*, Springer, New York, NY, USA, 2011. - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. View at: Publisher Site | MathSciNet - J. T. Machado, A. M. Galhano, and J. J. Trujillo, “Science metrics on fractional calculus development since 1966,”
*Fractional Calculus and Applied Analysis*, vol. 16, no. 2, pp. 479–500, 2013. View at: Publisher Site | Google Scholar | MathSciNet - H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2006–2012, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at: Publisher Site | Google Scholar - S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 910–919, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 587179, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - N. Laskin, “Fractional Schrödinger equation,”
*Physical Review E*, vol. 66, no. 5, Article ID 056108, 7 pages, 2002. View at: Publisher Site | Google Scholar | MathSciNet - M. Naber, “Time fractional Schrödinger equation,”
*Journal of Mathematical Physics*, vol. 45, no. 8, article 3339, 14 pages, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Ara, “Approximate solutions to time-fractional Schrödinger equation via homotopy analysis method,”
*ISRN Mathematical Physics*, vol. 2012, Article ID 197068, 11 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. I. Muslih, O. P. Agrawal, and D. Baleanu, “A fractional Schrödinger equation and its solution,”
*International Journal of Theoretical Physics*, vol. 49, no. 8, pp. 1746–1752, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,”
*Physics Letters A*, vol. 372, no. 5, pp. 553–558, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Felmer, A. Quaas, and J. Tan, “Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 142, no. 6, pp. 1237–1262, 2012. View at: Publisher Site | Google Scholar | MathSciNet - J. P. Dong and M. Y. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,”
*Journal of Mathematical Physics*, vol. 48, no. 7, Article ID 072105, 14 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Yildirim, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 4, pp. 445–450, 2009. View at: Google Scholar - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, China, 2011. - X. J. Ma, H. M. Srivastava, D. Baleanu, and X. J. Yang, “A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 325121, 6 pages, 2013. View at: Publisher Site | Google Scholar - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Systems of Navier-Stokes equations on Cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at: Publisher Site | Google Scholar - K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,”
*Physical Review Letters*, vol. 80, no. 2, pp. 214–217, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,”
*Computers & Structures*, vol. 82, no. 6, pp. 499–508, 2004. View at: Publisher Site | Google Scholar - A. Babakhani and V. D. Gejji, “On calculus of local fractional derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 66–79, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. Ben Adda and J. Cresson, “About non-differentiable functions,”
*Journal of Mathematical Analysis and Applications*, vol. 263, no. 2, pp. 721–737, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,”
*Journal of Mathematical Analysis and Applications*, vol. 362, no. 1, pp. 17–33, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Chen, “Time-space fabric underlying anomalous diffusion,”
*Chaos Solitons Fractals*, vol. 28, pp. 923–929, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. M. Yang, X. J. Yang, and Z. B. Li, “Local fractional series expansion method for solving wave and diffusion equations on cantor sets,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 351057, 5 pages, 2013. View at: Publisher Site | Google Scholar - X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximate solutions for diffusion equations on cantor space-time,”
*Proceedings of the Romanian Academy A*, vol. 14, no. 2, pp. 127–133, 2013. View at: Google Scholar - A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,”
*Romania Reports in Physics*, vol. 65, pp. 84–93, 2013. View at: Google Scholar - G. Jumarie, “Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions,”
*Chaos, Solitons & Fractals*, vol. 40, no. 3, pp. 1428–1448, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C.-F. Liu, S.-S. Kong, and S.-J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, Article ID 120826, p. 75, 2013. View at: Publisher Site | Google Scholar - J. H. He, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,”
*Nonlinear Science Letters A*, vol. 4, no. 1, pp. 15–20, 2013. View at: Google Scholar - A. M. Yang, Y. Z. Zhang, and Y. Long, “The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar,”
*Thermal Science*, p. 74, 2013. View at: Publisher Site | Google Scholar - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, no. 1, pp. 131–146, 2013. View at: Publisher Site | Google Scholar

#### Copyright

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.