This study considers the boundary stabilization for stochastic delayed Cohen–Grossberg neural networks (SDCGNNs) with diffusion terms by the Lyapunov functional method. In the realization of NNs, sometimes time delays and diffusion phenomenon cannot be ignored, so Cohen–Grossberg NNs with time delays and diffusion terms are studied in this article. Moreover, different from the previously distributed control, the boundary control is used to stabilize the system, which can reduce the spatial cost of the controller and is easy to implement. Boundary controllers are presented for system with Neumann boundary and mixed boundary conditions, and criteria are derived such that the controlled system achieves mean-square exponential stabilization. Based on the criterion, the effects of diffusion matrix, coupling strength, coupling matrix, and time delays on exponentially stability are analyzed. In the process of analysis, two difficulties need to be addressed: 1) how to introduce boundary control into system analysis? and 2) how to analyze the influence of system parameters on stability? We deal with these problems by using Poincaré’s inequality and Schur’s complement lemma. Moreover, mean-square exponential synchronization of stochastic delayed Hopfield NNs with diffusion terms, as an application of the theoretical result, is considered under the boundary control. Examples are given to illustrate the effectiveness of the theoretical results.

In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical stability, and the capability in preserving the norm conservation, are verified in the numerical experiments.

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.