  Considered here is a class of Boussinesq systems of Nwogu type. Such systems
describe propagation of nonlinear and dispersive water waves of significant
interest such as solitary and tsunami waves. The initial-boundary value problem
on a finite interval for this family of systems is studied both theoretically
and numerically. First, the linearization of a certain generalized Nwogu system
is solved analytically via the unified transform of Fokas. The corresponding
analysis reveals two types of admissible boundary conditions, thereby
suggesting appropriate boundary conditions for the nonlinear Nwogu system on a
finite interval. Then, well-posedness is established, both in the weak and in
the classical sense, for a regularized Nwogu system in the context of an
initial-boundary value problem that describes the dynamics of water waves in a
basin with wall-boundary conditions. In addition, a new modified Galerkin
method is suggested for the numerical discretization of this regularized system
in time, and its convergence is proved along with optimal error estimates.
Finally, numerical experiments illustrating the effect of the boundary
conditions on the reflection of solitary waves by a vertical wall are also
provided.
