On separable Fokker-Planck equations with a constant diagonal diffusion matrix

Abstract

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients B 1 ( x), B 2 ( x), B 3 ( x) providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients B( x) must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructed.

Separable Non-Parallel and Unsteady Flow Stability Problems

“…The method has been successfully applied to several equations of mathematical physics (see, e.g., Zhalij, 1999;Zhalij, 1999a, 1999b;Zhalij, 2002).…”

Individual Political Terror During the Civil War in Russia: Problems of Modern Historiography

“…The separation of variables in the Fokker-Planck equations was discussed in the comparatively new papers (e.g. [4,5,6]), but under rather restricted assumptions. The equation of type (2) arises in the theory of 1D localization, where it describes the evolution of the mutual distribution P (ρ, ψ) of the Landauer resistace ρ and the phase variable ψ = θ − ϕ, where θ and ϕ are phases entering the transfer matrix (see Eq.28 in [7] and the comments after it).…”