Numerical investigation of the Carleman system

In the article the Cauchy problem of the Carleman equation is considered. The Carleman system of equations is a model problem of the kinetic theory of gases. It is a discrete kinetic model of one-dimensional gas consisting of identical monatomic molecules. The molecules can have one of two speeds, which have equal values and opposite directions. This system of the equations is quasi-linear hyperbolic system of partial differential equations. There is no analytic solution for this problem in general case. So, the numerical investigation of the Cauchy problem of the Carleman system solution is very important.The paper presents and discusses the results of the numerical investigation of the Cauchy problem for the studied system solution with periodic initial conditions. The dependence of the stabilization time of the solution and the time dependence of energy exchange from small parameter are obtained.The second point of the paper is numerical investigation of the solution of the Cauchy problem with non-periodic initial conditions. The solution stabilization to the equilibrium state is obtained. The solution stabilization time is compared with stabilization time in periodic case.The final point of the paper is numerical investigation of the Cauchy problem with stationary normal processes as initial conditions. The solution to this problem is two stationary stochastic processes for any fixed value of time variable. As a rule, the practical interest is not a stochastic solution but its statistical characteristics. The stochastic solution realization is presented and discussed. The dependence of the mathematical expectation of the solution deviation modulus from equilibrium state is obtained. It demonstrates the process of the solution stabilization.