Discrete-ordinate method of solution of Fokker-Planck equations with nonlinear coefficients

Publisher Website
Google Scholar
Abstract
A discrete-ordinate method [J. Comput. Phys. 55, 313 (1984)] based on nonclassical polynomials is applied to the solution of a large class of Fokker-Planck equations with nonlinear coefficients. These Fokker-Planck equations arise in the description of nonequilibrium processes in reactive systems, laser systems, and model systems with bistable potentials. This subject has received considerable attention in recent years in connection with stochastic processes in physics, cooperative phenomena, and synergetics. The present approach is based on an eigenfunction expansion of the time-dependent probability density function. A discrete-ordinate method is employed in a numerical calculation of the eigenvalues and corresponding eigenfunctions of the Fokker-Planck operator. A general procedure for determining the eigenvalue spectrum of such Fokker-Planck operators with the discrete-ordinate method based on nonclassical polynomials, constructed so as to give rapid convergence of the eigenvalues, is described. The method is applied to several systems which include a model problem for which an analytic solution is known, a model with a triple-well potential in the Schrödinger equation equivalent to the Fokker-Planck equation, and to a model for the the trans-gauche isomerization of n-butane in carbon tetrachloride. The present methods for studying eigenvalue and boundary-value problems should be applicable to a wide variety of problems in addition to those presented here.