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Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Author

Listed:
  • Athinoula A. Kosti

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

  • Simon Colreavy-Donnelly

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

  • Fabio Caraffini

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

  • Zacharias A. Anastassi

    (Institute of Artificial Intelligence, School of Computer Science and Informatics, De Montfort University, Leicester LE1 9BH, UK)

Abstract

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations.

Suggested Citation

  • Athinoula A. Kosti & Simon Colreavy-Donnelly & Fabio Caraffini & Zacharias A. Anastassi, 2020. "Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients," Mathematics, MDPI, vol. 8(3), pages 1-12, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:374-:d:329796
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    References listed on IDEAS

    as
    1. Kengne, E., 2014. "Analytical solutions of nonlinear Schrödinger equation with distributed coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 61(C), pages 56-68.
    2. N. A. Ahmad & N. Senu & F. Ismail, 2017. "Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs," Mathematical Problems in Engineering, Hindawi, vol. 2017, pages 1-11, February.
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