A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results.

A linearized numerical scheme is proposed to solve the nonlinear time-fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson method, and extrapolation methods in the temporal direction. A novel discrete fractional Grönwall inequality is established. Thanks to the inequality, the error estimate of a fully discrete scheme is obtained. Several numerical examples are provided to verify the effectiveness of the fully discrete numerical method.

In the article titled “Fractional Crank-Nicolson-Galerkin Finite Element Methods for Nonlinear Time Fractional Parabolic Problems with Time Delay” [1], there are a number of minor typographical errors introduced to the equations during the typesetting of the article. The corrected article is as follows.

A linearized transformed Fourier spectral-Galerkin method is presented for numerically solving the multi-dimensional time fractional Allen-Cahn equations. Optimal error estimates of the fully discrete scheme are given. Such error estimates are obtained without any time-step restrictions dependent on the spatial mesh sizes by combining the discrete fractional Grönwall inequality,some Sobolev embedding inequalities and inverse inequalities. Furthermore, the variational energy dissipation law is given. Numerical examples are presented to confirm the theoretical results