Unsteady solutions of kinetic models with velocity-dependent collision frequency

Abstract

The method of elementary solutions recently extended to treat steady problems with kinetic models with velocity-dependent collision frequency is now extended to cover also time-dependent problems. The theory is somewhat different from the previous one holding for the Bhatnagar, Gross, and Krook model, since the continuous spectrum of space transients covers now a two-dimensional region of the complex plane. Consequently, in order to solve explicitly half-space problems we are not faced with one-dimensional singular integral equations but two-dimensional equations with integrable kernel. This circumstance does not make it possible to use the Muskhelishvili techniques which were used in previous papers. However, a connection of the present theory with the theory of generalized analytic functions is shown, and, on this basis, an analytic method of solving the relevant equations is constructed. The present theory would be easily applied to typical time-dependent problems, such as the Rayleigh problem, the oscillating wall, and the evolution of an initial discontinuity.