This paper addresses the approximation of solutions to some non-homogeneous boundary value problems associated with the nonlinear Korteweg-de Vries equation (KdV) and a system of two coupled KdV-type equations derived by Gear and Grimshaw posed on a bounded interval. An efficient Galerkin scheme that combines a finite element strategy for space discretization with a second-order implicit scheme for time-stepping is employed to approximate time dynamics of model equations studied. Several numerical experiments, including boundary controllability problems for nonlinear KdV and GG equations, are presented for different final states to show the performance of the numerical strategies proposed. The numerical results with nonlinear models agree with previous analytic theory and show the persistence of the behavior not uniform in time of the control functions computed already observed by Rosier [22] in the case of the linear KdV equation.

本文讨论了与非线性Korteweg-de Vries方程（KdV）和由Gear和Grimshaw导出的两个耦合KdV型方程组相关的一些非齐次边值问题的解的逼近，并将它们置于有界区间上。一种有效的Galerkin方案，将空间离散化的有限元策略与用于时间步长的二阶隐式方案相结合，用于近似研究模型方程的时间动态。针对不同的最终状态，进行了一些数值实验，包括非线性KdV和GG方程的边界可控性问题，以证明所提出的数值策略的性能。