A fully spectral collocation approximation for multi-dimensional fractional Schrdinger equations
A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrdinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre GaussLobatto 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 LegendreGaussLobatto 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 GaussRadau 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 GaussLobatto and GaussRadau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.
fractional schrdinger equations collocation method two-dimensional schrdinger equations spectral method gauss-type quadrature