The Exact Solution of the Cauchy Problem for Two Generalized Fokker-Planck Equations — Algebraic Approach

Abstract

By employing algebraic techniques we find the exact solutions of the Cauchy problem for two equations, which may be considered as n-dimensional generalization of the famous Fokker–Planck equation. Our approach is a combination of the disentangling techniques of R. Feynman with operational method developed in modern functional analysis in particular in the theory of partial differential equations. Our method may be considered as a generalization of the M. Suzuki method of solving the Fokker–Planck equation.

Exact solution to the cauchy problem for a generalized “linear” vectorial Fokker-Planck equation: Algebraic approach

“…• When â4 (t) is scalar: â4 (t) = a 4 (t) 1 ( in this case â4 : ∇∇ = a 4 ∆ ) the "anisotropic" problem (3) turns to the "isotropic" one, with the exact solution found in [20]. It is easy to check that the solution (21) turns to the solution obtained in [20] (there is an error in [20]: the sign before a 2 in the Eqs. ( 17) and (34) there, should be (+)).…”

“…Our method may be regarded as a combination of the disentangling techniques of R. Feynman [22] with the operational methods developed in the functional analysis and in particular in the theory of pseudodifferential equations with partial derivatives [23]− [27]. As we have emphasized in [20] and [21] this approach is an extension and generalization of the M. Suzuki's method [18] for solving the one-dimensional linear FPE (1).…”