Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed.

► Scaling form of the Fokker–Planck equation with time-dependent drift and diffusion coefficients is derived.
► Exact similarity solution of the Fokker–Planck equation is given in closed forms.
► New examples of Fokker–Planck equations exactly solvable by similarity methods are discussed.