SIMILARITY ANALYSIS OF THE ONE-DIMENSIONAL FOKKER-PLANCK EQUATION

Abstract

The goal of this study is to investigate the conditions under which the Fokker Planck equation assumes closed form solutions. The Fokker-Planck equation arises in the study of fluctuations in physical and biological systems. This equation has been subject of several recent studies. We have analyzed the existence of solutions of this equation analytically and have established results using various invariant approaches. For example, we have employed a group of stretching transformations that keeps the original equation invariant under this group actions. This resulted in obtaining a class of closed form solutions. Then we analyzed the conditions under which a Clarkson-Kruskal type similarity functional form would result in closed form solutions. The importance of this type of solutions relies on the fact that they can be used as a test functional tool in checking the accuracy of numerical algorithms. General form of such class of solutions is obtained and several specific examples of this class is discussed. In addition, a general group invariant approach is investigated.