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

TL;DR

This paper uses the similarity method to find exact solutions of the Fokker-Planck equation with time-dependent coefficients, reducing it to an integrable ordinary differential equation and deriving closed-form probability densities.

Contribution

It introduces a novel approach to solve the Fokker-Planck equation with time-dependent parameters using similarity variables, providing new exactly solvable models.

Findings

Derived closed-form solutions for specific Fokker-Planck equations.

Presented new examples of exactly solvable equations.

Analyzed properties of the obtained solutions.

Abstract

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 resulted 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.