A fully spectral collocation approximation formulti-dimensional fractional Schrodinger equations

作者：Bhrawy A H; Abdelkawy M A

来源：Journal of Computational Physics, 2015, 294: 462-483.

DOI：10.1016/j.jcp.2015.03.063

• Fractional Schrodinger equations
• Two-dimensional Schrdinger equations
• Collocation method
• Spectral method
• Gauss-type quadrature

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one-and two-dimensional time fractional Schrodinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre-Gauss-Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss-Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss-Lobatto and Gauss-Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.

• 出版日期2015-8-1