An approximate orthogonal decomposition method for the solution of generalized Liouville equations

An analytical-numerical integration method for the generalized Liouville equation is proposed and analyzed. Taking into account a Cauchy condition f(q,p,t)_|t=0=f₀(q,p) for the phase space distribution function, we constructed the problem solution as series expansion in time variable t using orthogonal polynomials and Hermite function. Also we proved the corresponding convergence theorems under certain boundedness conditions upon a Liouville operator.