  In this article we present a numerical analysis for a third-order
differential equation with non-periodic boundary conditions and time-dependent
coefficients, namely, the linear Korteweg-de Vries Burgers equation. This
numerical analysis is motived due to the dispersive and dissipative phenomena
that government this kind of equations. This work builds on previous methods
for dispersive equations with constant coefficients, expanding the field to
include a new class of equations which until now have eluded the time-evolving
parameters. More precisely, throughout the Legendre-Petrov-Galerkin method we
prove stability and convergence results of the approximation in appropriate
weighted Sobolev spaces. These results allow to show the role and trade off of
these temporal parameters into the model. Afterwards, we numerically
investigate the dispersion-dissipation relation for several profiles, further
provide insights into the implementation method, which allow to exhibit the
accuracy and efficiency of our numerical algorithms.
