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Fractional Calculus and Applied Analysis

, Volume 16, Issue 2, pp 454–468 | Cite as

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

  • Neville J. FordEmail author
  • M. Manuela Rodrigues
  • Nelson Vieira
Research Paper

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.

Key Words and Phrases

fractional partial differential equation fractional Schrödinger equation finite difference method stability Mittag-Leffler function 

MSC 2010

Primary 35R11 Secondary 42A38, 33E12, 65M06, 47H10 

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Copyright information

© Versita Warsaw and Springer-Verlag Wien 2013

Authors and Affiliations

  • Neville J. Ford
    • 1
    Email author
  • M. Manuela Rodrigues
    • 2
  • Nelson Vieira
    • 3
  1. 1.Department of MathematicsUniversity of ChesterChester, EnglandUK
  2. 2.CIDMA — Center for Research and Development in Mathematics and Applications Department of MathematicsUniversity of Aveiro Campus Universitário de SantiagoAveiroPortugal
  3. 3.CIDMA — Center for Research and Development in Mathematics and Applications and Polytechnical Institute of LeiriaSchool of Technology and ManagementLeiriaPortugal

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