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.

We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that the scheme can be unconditionally energy-stable. In particular, the equation becomes the space-fractional Schrödinger equation. Then, the scheme can keep both the discrete mass and energy conserved. Moreover, convergence of the scheme is obtained. Numerical experiments are performed to confirm the theoretical results.

The exact solution to a neutral delay differential equation is generally non-smooth. Some possible loss of accuracy is usually found if certain high-order numerical methods are applied. The discontinuous Galerkin (DG) methods are introduced to numerically solve neutral delay differential equations so as to handle the difficulties. Maximum error estimates of the numerical method is investigated. Theoretical results indicate that the ${p}$-degree DG approximate solution has an accuracy of ${p}$-th order. Numerical experiments are presented to confirm the effectiveness and performance of the DG methods.