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Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations

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Abstract

In the present paper, we construct the numerical solution for time fractional (1 + 1)- and (1 + 2)-dimensional Schrödinger equations (TFSEs) subject to initial boundary. The solution is expanded in a series of shifted Jacobi polynomials in time and space. A collocation method in two steps is developed and applied. First step depends mainly on application of shifted Jacobi Gauss-Lobatto-collocation method for spatial discretization on the approximate solution and its spatial derivatives occurring in the TFSE and substitution in the boundary conditions or treatment of the non-local conservation conditions by the Jacobi Gauss-Lobatto quadrature rule. As a result, a system of fractional differential equation for the expansion coefficients is obtained. The second step is to use a shifted Jacobi Gauss-Radau- collocation scheme, for temporal discretization, to reduce such system into a system of nonlinear Newton iterative method. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithms demonstrating superiority over other methods.

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Authors and Affiliations

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

    Ali H. Bhrawy & Mohamed A. Abdelkawy

  2. Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

    Ali H. Bhrawy & Mohamed A. Abdelkawy

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Jameel F. Alzaidy & Anjan Biswas

  4. Department of Mathematical Sciences, Delaware State University, Dover, DE, 19901-2277, USA

    Anjan Biswas

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  1. Ali H. Bhrawy
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  2. Jameel F. Alzaidy
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  3. Mohamed A. Abdelkawy
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Correspondence to Anjan Biswas.

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Bhrawy, A.H., Alzaidy, J.F., Abdelkawy, M.A. et al. Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrödinger equations. Nonlinear Dyn 84, 1553–1567 (2016). https://doi.org/10.1007/s11071-015-2588-x

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