In this paper, we consider a recently proposed hydrodynamical model that represents a quasilinear system of equations in the form of conservation laws. The conservation laws are obtained from a system of moment relations for the Boltzmann transport equation with the maximum entropy principle used in the model for a closure of the system of moments. In this paper, the hydrodynamic model is utilized to find stationary solutions that describe the motion of electrons in a 2D silicon MOSFET (Metal Oxide Semiconductor Field Effect Transistor) transistor with a silicon oxide nanochannel. In the stationary case, the mathematical model is reduced to a system of elliptical quasilinear equations. To find approximate solutions of these equations, we use a computational algorithm based on the method of lines, the stabilization method, and various forms of nonstationary regularization of equations. The computational algorithm is implemented as a software package using Object Pascal in Delphi 6 environment. The results of the obtained solutions are provided.