A fast method based on variable time steps for 2D nonlinear time-fractional generalized Benjamin–Bona–Mahony–Burgers equation: Error and stability analysis

Abstract

This paper presents the stability and convergence analysis of a hybrid scheme that combines the Galerkin-spectral method with a nonuniform time-stepping temporal discretization. The scheme is used to solve the two-dimensional nonlinear time-fractional generalized Benjamin–Bona–Mahony–Burgers equation with admissible regularities. It is widely accepted that fractional differential equations often exhibit singularity near the initial time, which makes uniform time-stepping methods unsuitable to approximate their solution. We convert the problem to an equivalent integral form and then solve it using a reliable time-stepping method with nonuniform time steps. We have developed an unconditionally stable discretization technique that approximates Riemann–Liouville fractional integrals with third-order convergence. Furthermore, we have used the spectral Galerkin method based on the Legendre polynomials for spatial discretization of the mentioned nonlinear problem in two dimensions. Also, we have proved the unconditional stability and convergence of the full-discretization scheme. Using the proposed method, we have conducted several numerical simulations which fully confirmed the theoretical results.