On the solution of the model kinetic equation ES

The article describes a method for finding a solution to a linearized ellipsoidal-statistical kinetic equation (ES) with a homogeneous boundary condition based on the Chebyshev polynomial approximation in the framework of the problem of modeling the axial flow of a rarefied gas in a long channel. The channel is formed from two cylinders having a common central axis. Diffuse Maxwell reflection is used as a model for the reflection of gas molecules from cylinders. The gas flow is due to a small absolute value of the pressure gradient directed along the axis of the cylinders. The calculation of the mass flow of gas in the channel is carried out depending on the rarefaction parameter and the ratio of the radii of the cylinders. The unknown function approximating the solution of the linearized ES equation is represented as a partial sum of the expansion in Chebyshev polynomials of the first kind. By choosing interpolation nodes and applying the properties of finite sums of Chebyshev polynomials, the problem is reduced to a system of linear algebraic equations with respect to the values of the desired function at these nodes. The expressions for the gas mass velocity in the channel and the gas mass flow are obtained in terms of the partial sums of the series of Chebyshev polynomials.