Numerical solution of the Fokker-Planck equation via chebyschev polynomial approximations with reference to first passage time probability density functions
Chebyschev approximations are employed to solve the one-dimensional, time-dependent Fokker-Planck (forward Kolmogrov) equation in the presence of two barriers a finite "distance" apart. Solutions are presented for the fundamental intervals (-1, +1) and (0, +1). In order to speed up the calculations, sparse matrix routines are utilized. The first passage time probability density function is also evaluated. Illustrative numerical results are presented for the Wiener process with drift, and the Ornstein-Uhlenbeck process for a variety of combinations of boundary conditions.