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A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

  • Ali BaşhanEmail author
  • Yusuf Uçar
  • N. Murat Yağmurlu
  • Alaattin Esen
Regular Article

Abstract.

In the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrödinger (NLS) equation. For this purpose, first of all, the Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, \(L_{2}\) and \(L_{\infty}\), as well as the two lowest invariants, \(I_{1}\) and \(I_{2}\), have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.

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Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ali Başhan
    • 1
    Email author
  • Yusuf Uçar
    • 2
  • N. Murat Yağmurlu
    • 2
  • Alaattin Esen
    • 2
  1. 1.Department of Mathematics, Faculty of Science and ArtBulent Ecevit UniversityZonguldakTurkey
  2. 2.Department of Mathematics, Faculty of Science and ArtInonu UniversityMalatyaTurkey

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