The Generalized Fokker-Planck Equation in terms of Dunkl-type Derivatives

Abstract

In this work we introduce two different generalizations of the Fokker-Planck equation in (1+1) dimensions by replacing the spatial derivatives in terms of generalized Dunkl-type derivatives involving reflection operators. As applications of these results, we solve exactly the generalized Fokker-Planck equations for the harmonic oscillator and the centrifugal-type potentials.

Figures (2)

• Figure 1: Plots of Bessel eigenfunctions with $a=2$, $\lambda=4$. Figure (a) shows the even parity eigenfunctions $x^{0}J_2(2x)$ (dot), $x^{-1}J_7(2x)$ (dash), $x^{-2}J_6(2x)$ (solid). Figure (b) shows the odd parity eigenfunctions $x^{0}J_1(2x)$ (dot), $x^{-1}J_4(2x)$ (dash), $x^{-2}J_3(2x)$ (solid).
• Figure 2: Eigenfunctions of the harmonic oscillator plus a centrifugal-type potential with $a=4.3$, $\mu=0.6$. Figure (a) shows the even parity eigenfunctions for $\alpha_e=3.5$, $\gamma=0.43$ for $n_e=0$ (dot), $n_e=1$ (dash), $n_e=2$ (solid). Figure (b) shows the odd parity eigenfunctions for $\alpha_o=2$, $\gamma=-0.48$ for $n_o=0$ (dot), $n_o=1$ (dash), $n_o=2$ (solid).