Exact Solutions of the Time Derivative Fokker-Planck Equation: A Novel Approach

TL;DR

This paper presents a novel method to find exact solutions of the fractional Fokker-Planck equation by transforming it into classical equations, revealing insights into how fractional derivatives influence probability distributions and system dynamics.

Contribution

It introduces a new approach combining Hopf transformation and extended unified method to solve fractional Fokker-Planck equations exactly, reducing them to classical forms.

Findings

Solutions are bi-Gaussian in form.

High friction reduces standard deviation.

Fractionality impacts the distribution more than fractality.

Abstract

In the present article, an approach to find the exact solution of the fractional Fokker-Planck equation is presented. It is based on transforming it to a system of first-order partial differential equation via Hopf transformation, together with implementing the extended unified method. On the other hand, reduction of the fractional derivatives to non autonomous ordinary derivative. Thus the fractional Fokker-Planck equation is reduced to non autonomous classical ones. Some explicit solutions of the classical, fractional time derivative Fokker-Planck equation, are obtained . It is shown that the solution of the Fokker-Planck equation is bi-Gaussian's. It is found that high friction coefficient plays a significant role in lowering the standard deviation. Further, it is found the fractionality has stronger effect than fractality. It is worthy to mention that the mixture of Gaussian's is a powerful tool in machine learning. Further, when varying the order of the fractional time derivatives, results to slight effects in the probability distribution function. Also, it is shown that the mean and mean square of the velocity vary slowly.