L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity

Abstract

Numerical approximation for a linearized time fractional KdV equation with initial singularity using [Formula: see text] scheme on graded mesh is considered. It is proved that the [Formula: see text] scheme can attain order [Formula: see text] convergence rate with appropriate choice of the grading parameter, where [Formula: see text] [Formula: see text] is the order of temporal Caputo fractional derivative. A fully discrete spectral scheme is constructed combing a Petrov–Galerkin spectral method for the spatial discretization, and its stability and convergence are theoretically proved. Some numerical results are provided to verify the theoretical analysis and demonstrated the sharpness of the error analysis.