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Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations

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Abstract

A distributed-order time- and Riesz space-fractional Schrödinger equation (DOT–RSFSE) is considered. Distributed-order derivatives indicate fractional derivatives that are integrated over the order of the differentiation within a given range. That is to say, the order of the time derivative ranges from zero to one. The space-fractional derivative is defined in the Riesz sense. In this paper, a new numerical approach is developed for simulating DOT–RSFSE. The main characteristic behind this approach is to investigate a space-time spectral approximation for spatial and temporal discretizations. Firstly, the given problem in one and two dimensions is transformed into a system of distributed-order fractional differential equations by using Jacobi–Gauss–Lobatto (J–G–L) collocation approach. Then, an efficient spectral method based on Jacobi–Gauss–Radau (J–G–R) collocation approach is applied to solve this system. Furthermore, the error of the approximate solution is theoretically estimated and numerically confirmed in both temporal and spatial discretizations. In order to highlight the effectiveness of our approaches, several numerical examples are given and compared with those reported in the literature.

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Acknowledgements

This paper is dedicated to the memory of Professor Ali H. Bhrawy, who passed away shortly after its completion. The authors are very grateful to the editor and reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    A. H. Bhrawy

  2. Department of Applied Mathematics, National Research Centre, Dokki, Giza, 12622, Egypt

    M. A. Zaky

  3. Zewail City of Science and Technology, University of Science and Technology, Zayed City, Giza, 12588, Egypt

    M. A. Zaky

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  1. A. H. Bhrawy
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  2. M. A. Zaky
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Correspondence to M. A. Zaky.

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Bhrawy, A.H., Zaky, M.A. Numerical simulation of multi-dimensional distributed-order generalized Schrödinger equations. Nonlinear Dyn 89, 1415–1432 (2017). https://doi.org/10.1007/s11071-017-3525-y

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