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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

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.

Linked References

  1. G. Teschi, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, vol. 99, American Mathematical Society, Providence, RI, USA, 2009.
  2. R. Shankar, Principles of Quantum Mechanics, vol. 233, Plenum Press, New York, NY, USA, 1994. View at Zentralblatt MATH · View at MathSciNet
  3. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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 · View at Google Scholar · View at MathSciNet
  5. 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 · View at Google Scholar
  6. 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 · View at Google Scholar · View at Zentralblatt MATH
  7. 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 · View at Google Scholar · View at MathSciNet
  8. 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.
  9. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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 MathSciNet
  11. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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 · View at Google Scholar · View at MathSciNet
  13. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006. View at MathSciNet
  14. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  15. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  16. A. Carpinteri and F. Mainardi, Fractals Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
  17. 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
  18. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  19. R. L. Magin, Fractional Calculus in Bioengineering, Begerll House, Connecticut, Conn, USA, 2006.
  20. J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics in Physics: Recent Advances, World Scientific, Singapore, 2011. View at MathSciNet
  21. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008. View at MathSciNet
  22. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  23. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, Germany, 2011.
  24. 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.
  25. 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 · View at Google Scholar · View at MathSciNet
  26. 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 · View at Google Scholar · View at MathSciNet
  27. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. 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 · View at Google Scholar
  29. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. 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 · View at Google Scholar · View at MathSciNet
  31. N. Laskin, “Fractional Schrödinger equation,” Physical Review E, vol. 66, no. 5, Article ID 056108, 7 pages, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. Naber, “Time fractional Schrödinger equation,” Journal of Mathematical Physics, vol. 45, no. 8, article 3339, 14 pages, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. 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 · View at Google Scholar · View at Zentralblatt MATH
  34. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. 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 · View at Google Scholar · View at MathSciNet
  37. 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 · View at Google Scholar · View at Zentralblatt MATH
  38. 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.
  39. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  40. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, China, 2011.
  41. 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 · View at Google Scholar
  42. 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 · View at Google Scholar
  43. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. 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 · View at Google Scholar
  45. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. W. Chen, “Time-space fabric underlying anomalous diffusion,” Chaos Solitons Fractals, vol. 28, pp. 923–929, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  49. 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 · View at Google Scholar
  50. 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.
  51. 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.
  52. 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. 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 · View at Google Scholar
  54. 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.
  55. 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 · View at Google Scholar
  56. 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 · View at Google Scholar