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A numerical method for the fractional Schrödinger type equation of spatial dimension two

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Abstract

This work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

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References

  1. M. Caputo, The Green function of the diffusion of fluids in porous media with memory. Rend. Fis. Acc. Lincei (Ser. 9) 7 (1996), 243–250.

    Article  MATH  Google Scholar 

  2. K. Diethelm, Generalized compound quadrature formulae for finite-part integrals. IMA J. Numer. Anal. 17 (1997), 479–493.

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order. Electronic Transactions on Numerical Analysis 5 (1997), 1–6.

    MathSciNet  MATH  Google Scholar 

  4. D. Elliott, An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals. IMA J. Numer. Anal. 13 (1993), 445–462.

    Article  MathSciNet  MATH  Google Scholar 

  5. N.J. Ford, M.M. Rodrigues, Jingyu Xiao, Yubin Yan, Numerical analysis of a two-parameter fractional telegraph equation. Journal of Computational and Applied Mathematics, To appear.

  6. R. Gorenflo, F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes. Journal of Computational and Applied Mathematics 229 (2009), 400–415.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam (2006).

    Book  MATH  Google Scholar 

  8. N. Laskin, Fractional Schrödinger equation. Phys. Rev. E 66 (2002), 056108.

    Article  MathSciNet  Google Scholar 

  9. N. Laskin, Fractional quantum mechanics. Phy. Rev. E 62, No 3 (2000), 3135–3145.

    Article  MathSciNet  Google Scholar 

  10. N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000), 298–305.

    Article  MathSciNet  MATH  Google Scholar 

  11. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press andWorld Scientific, London — Singapore (2010).

    Book  MATH  Google Scholar 

  12. M. Naber, Time fractional Schrödinger equation. J. Math. Phys. 45, No 8 (2004), 3339–3352.

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Podlubny, Fractional Differential Equations. (An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications). Mathematics in Science and Engineering 198, Academic Press, San Diego (1999).

    MATH  Google Scholar 

  14. M.M. Rodrigues and N. Vieira, Multidimensional fractional Schrödinger equation. In: American Institute of Physics Conf. Proceedings (Special Issue for the ICNPAA 2012 Congress — ICNPAA Congress: Mathematical Problems in Engineering, Aerospace and Sciences), 1493 (2012), 798–804; doi: 10.1063/1.4765579.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Chester, CH1 4BJ, Chester, England, UK

    Neville J. Ford

  2. CIDMA — Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    M. Manuela Rodrigues

  3. CIDMA — Center for Research and Development in Mathematics and Applications and Polytechnical Institute of Leiria, School of Technology and Management, P-2411-901, Leiria, Portugal

    Nelson Vieira

Authors
  1. Neville J. Ford
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  2. M. Manuela Rodrigues
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  3. Nelson Vieira
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Correspondence to Neville J. Ford.

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Dedicated to Professor Francesco Mainardi on the occasion of his 70th anniversary

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Ford, N.J., Manuela Rodrigues, M. & Vieira, N. A numerical method for the fractional Schrödinger type equation of spatial dimension two. fcaa 16, 454–468 (2013). https://doi.org/10.2478/s13540-013-0028-5

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  • DOI: https://doi.org/10.2478/s13540-013-0028-5

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