<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.</p></abstract>

The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross–Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system.

A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space–Fractional Gross–Pitaevskii Equation

Department of Computational Mathematics and Computer Science

Department of Mathematics, Faculty of Science

Department of Mathematics and Physics Autonomous

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

The present work departs from an extended form of the classical multi-dimensional Gross–Pitaevskii equation, which considers fractional derivatives of the...

A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space–Fractional Gross–Pitaevskii Equation

Published Online: Dec 31, 2019

Page range: 713 - 723

Received: Jan 15, 2019

Accepted: May 28, 2019

DOI: https://doi.org/10.2478/amcs-2019-0053

Keywords
generalized Gross–Pitaevskii system, Riesz fractional diffusion, discrete uniform Sobolev inequality, conservative method, optimal error bounds

© 2019 Ahmed S. Hendy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space–Fractional Gross–Pitaevskii Equation

Published Online: Dec 31, 2019

Page range: 713 - 723

Received: Jan 15, 2019

Accepted: May 28, 2019

DOI: https://doi.org/10.2478/amcs-2019-0053

Keywordsgeneralized Gross–Pitaevskii system, Riesz fractional diffusion, discrete uniform Sobolev inequality, conservative method, optimal error bounds

© 2019 Ahmed S. Hendy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
generalized Gross–Pitaevskii system, Riesz fractional diffusion, discrete uniform Sobolev inequality, conservative method, optimal error bounds