Similarity solutions of Fokker-Planck equation with time-dependent coefficients and fixed/moving boundaries

TL;DR

This paper develops a method to find exact solutions to the Fokker-Planck equation with time-dependent coefficients and boundaries by reducing it to an integrable ordinary differential equation using similarity variables.

Contribution

It introduces a similarity method to solve the Fokker-Planck equation with time-dependent parameters, providing new exactly solvable examples.

Findings

Derived closed-form solutions for specific Fokker-Planck equations

Demonstrated the method's applicability to fixed and moving boundaries

Presented new classes of exactly solvable models

Abstract

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.