Numerical Analysis For Fokker-Planck Equations With Discontinuous Drift

Stochastic dynamical systems with discontinuities are used as various models of physical and biological systems.Uncertainty of noise disturbance and discontinuous property make the dynamical systems display more abundant phenomena and have more widely uses which are worth further study.For this reason,we take as an analysis the model of Brownian motion with dry friction,which is a simple model of stochastic dynamical system with discontinuity.At present,the study of transition probability distribution and first-passage time problem to Brownian motion with dry friction for the one-variable case has many conclusions.However,for the two-variable case,the relevant theoretical or numerical results are rarely found in the literature so far.If we solve the transition probability distribution and furthermore the displacement distribution for the two-variable case,one can compare the displacement distribution obtained from experiments with the numerical results to see whether there is a dry friction effect between a solid object and a solid surface(e.g.,two materials).As for the first-passage time problem,it is always important in mathematical finance,biology,physical chemistry,engineering,and others.It will provide an important reference in practical applications if the first-passage time distribution is obtained.In this paper,the transition probability distribution and the first-passage time problem of Brownian motion with dry friction are solved numerically by finite difference methods.The main work includes:The problem of transition probability distribution is solved.We derive appropriate explicit and implicit difference schemes and prove their stability for the one-variable Fokker-Planck equation.The numerical results are compared with the analytical solutions to test the reliability of the schemes.For the two-variable Fokker-Planck equation,since analytical expressions of the joint probability distribution are unknown,we turn to use the alternating direction implicit method,and consider the marginal probability distributions of the velocity and the displacement.The marginal probability distribution of the velocity is compared with the analytical solution as the one-variable case.For the marginal probability distribution of the displacement,since the distribution is Gaussian in the long time limit,the first two moments are compared with the exact solutions.The comparative results indicate that the scheme is effective and the numerical results are accurate.We solve the first-passage time problem.Numerical results of the problem for the one-variable case by using explicit difference scheme are compared with the available analytical solutions.Then we derive a correcting explicit difference scheme for the boundary conditions of two-variable case by the same method and obtain the first-passage time distribution as well as the mean first-passage time which are analyzed respectively.At last the first moment of first-passage time distribution and mean first-passage time obtained directly are compared with each other to illustrate the correctness of the numerical results.