Analytical study of the mathematical model of wave propagation in shallow water by the Galerkin method

Of concern is an initial-boundary value problem for the modified Boussinesq equation (IMBq equation) is considered. The equation is often used to describe the propagation of waves in shallow water under the condition of mass conservation in the layer and taking into account capillary effects. In addition, it is used in the study of shock waves. The modified Boussinesq equation belongs to the Sobolev type equations. Earlier, using the theory of relatively $p$-bounded operators, the theorem of existence and uniqueness of the solution to the initial-boundary value problem was proved. In this paper, we will prove that the solution constructed by the Galerkin method using the system orthornormal eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator converges $^*$-weakly to an precise solution. Based on the compactness method and Gronwall's inequality, the existence and uniqueness of solutions to the Cauchy–Dirichlet and the Showalter–Sidorov–Dirichlet problems for the modified Boussinesq equation are proved.