Analysis and numerical solution of Benjamin–Bona–Mahony equation with moving boundary

In this work we present the existence, the uniqueness and numerical solutions for a mathematical model associated with equations of Benjamin–Bona–Mahony type in a domain with moving boundary. We apply the Galerkin method, multiplier techniques, energy estimates and compactness results to obtain the existence and uniqueness. For numerical solutions, we shall employ the finite element method together with the Crank–Nicolson method. Some numerical experiments are presented to show the moving boundary for the problem.