An L1 Legendre–Galerkin spectral method with fast algorithm for the two-dimensional nonlinear coupled time fractional Schrödinger equation and its parameter estimation

Abstract In this paper, we derive an L1 Legendre–Galerkin spectral method with fast algorithm based on an efficient sum-of-exponentials (SOE) approximation for the kernel t − 1 − α to solve the two-dimensional nonlinear coupled time fractional Schrodinger equations. The numerical method is stable without the Courant–Friedrichs–Lewy (CFL) conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. At the same time, we use the fully implicit method to deal with the nonlinear terms. For the non-smooth solution, we employ the graded mesh method. Moreover, we estimate the parameters including fractional derivative index and the coefficients of the nonlinear term in the equations by applying the Cuckoo Search algorithm related to Levy flights. Numerical examples are given to verify the theoretical analysis and confirm the effectiveness of the fast algorithm and the estimation method.