Non-homogeneous boundary value problems for some KdV-type equations on a finite interval: A numerical approach

• Published in: Communications in nonlinear science & numerical simulation 2021-05, Vol.96, p.105669, Article 105669
• Main Author: Grajales, Juan Carlos Muñoz
• Format: Article
• Language: English
• Subjects: Boundary controlling theory; Boundary value problems; Controllability; Finite element analysis; Finite element method; KdV-type equations; Korteweg-Devries equation; Mathematical models; Nonlinear equations; Numerical analysis; Stability
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2020.105669

•This paper introduces numerical schemes for approximation of solutions to some nonhomogeneous boundary value problems associated to 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.•The numerical results show the persistence of the behavior not uniform in time of the control functions computed.•The controllability problems for nonlinear KdV-type systems we consider have been intensively studied for the model equations addressed in this paper, but in the past, emphasis have been placed on theoretical results. In this paper we fill this gap by exploring this problem from the numerical point of view.•Numerical methods are valuable tools for studying dynamics of complicated evolution systems and for exploring open questions on controllability of dispersive-type equations. 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.