A fast second-order predictor-corrector method for a nonlinear time-fractional Benjamin-Bona-Mahony-Burgers equation

A second-order predictor-corrector method of Nguyen and Jang (Fract. Calc. Appl. Anal. 20(2), 447-476, 2017) is generalised to graded meshes to solve nonlinear fractional initial-value problems whose typical solutions have a weak singularity at the initial time. In comparison with existing predictor-corrector methods in the literature, this new method significantly improves the numerical accuracy while reducing the computational cost. Moreover, to increase its computational efficiency still further, a corresponding fast algorithm based on the sum-of-exponentials approximation to the kernel of the scheme is described. An error analysis is given for problems whose right-hand sides satisfy a Lipschitz condition. The method (and its fast variant) is then extended to solve the nonlinear time-fractional Benjamin-Bona-Mahony-Burgers (BBMB) initial-boundary value problem, combined with a standard discretisation of the spatial derivatives on a uniform mesh. Estimates are derived for the discrete H-1-norm errors in the computed solution for the BBMB problem; to enable this analysis, a new Gronwall inequality is proved. Finally, several numerical experiments show the sharpness of our theoretical error bounds for both problems.