On an exact solution of the non-linear Fokker-Planck equation with sink term

Abstract

The Fokker-Planck equation with a sink term is resolved exactly, using the Fujita diffusivity and Parlange relationship between conductivity and diffusivity. In order to obtain the solution, the Kirchhoff potential and the Fujita-Storm mobile coordinate are introduced. The differential equation takes the form of the Burgers equation, which is linear in the diffusive term. The convective coefficient of the latter is replaced by the Hopf-Cole transformation for the purpose of deriving the classical linear heat equation. During the transformation, the sink term is defined functionally, so that the end result is precisely the heat equation without sink term. The exact solution of the Hopf-Cole potential is deduced by using the classical Laplace transform for certain initial and boundary conditions of interest. The solution of the Fokker-Planck equation in the physical space is obtained through the inverse transformations. The solution includes as particular cases both the Sanders et al. and Broadbridge and White solutions. The exact solution can be used to validate numerical solutions of the Fokker-Planck equation and in studies on water extraction by plant roots.