High‐order finite difference/spectral‐Galerkin approximations for the nonlinear time–space fractional Ginzburg–Landau equation

Abstract

AbstractThis paper proposes and analyzes a high‐order difference/Galerkin spectral scheme for the time–space fractional Ginzburg–Landau equation. For the time discretization, theL2 ‐ 1σdifferentiation formula is used to approximate the Caputo fractional derivative. While for the space discretization, the Legendre–Galerkin spectral method is used to approximate the Riesz fractional derivative. It is shown that the scheme is efficiently applied with spectral accuracy in space and second‐order in time. The error estimates of the solution are established by applying a fractional Grönwall inequality and its discrete form. In addition, a detailed implementation of the numerical algorithm is provided. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the effect of fractional‐order parameters on the pattern formation of time–space fractional Ginzburg–Landau equation is discussed.