In this work, we consider the numerical approximation of a linearized time fractional KdV equation in a bounded domain. A fully discrete scheme combining a Petrov–Galerkin spectral method for the spatial discretization and L1-approximation for the Caputo temporal derivative is proposed. Stability and convergence of the fully discrete scheme are rigourously established. Numerical results are presented to confirm the theoretical results.

We consider the numerical approximation for a time fractional Schrödinger equation whose solution exhibits an initial weak singularity. A fully discrete scheme is constructed using $L1$ scheme on graded mesh for the discretiaztion of temporal Caputo derivative and spectral method for spatial discretization. It is shown that with appropriate choice of the grading parameter, the scheme can attain order $2-\alpha$ convergence in temporal direction, where α $(0<\alpha <1)$ is the order of time Caputo fractional derivative, and spectral accuracy in spatial direction if the solution is sufficiently smooth in its spatial part. Numerical results confirm the sharpness of the error analysis.

In this paper, we propose a new Galerkin spectral element method for one-dimensional fourth-order boundary value problems. We first introduce some quasi-orthogonal approximations in one dimension, and establish a series of results on these approximations, which serve as powerful tools in the spectral element method. By applying these results to the fourth-order boundary value problems, we establish sharp $H^2$ and $L^2$ error bounds of the Galerkin spectral element method. The efficient algorithm is implemented in detail. Numerical results demonstrate its high accuracy, and confirm the theoretical analysis well.