Analytical Solutions of the Fokker–Planck Equation for Generalized Morse and Hulthén Potentials

Abstract

In the present contribution we analytically calculate solutions of the transition probability of the Fokker-Planck equation (FPE) for both the generalized Morse potential and the Hulthén potential. The method is based on the formal analogy of the FPE with the Schrödinger equation using techniques from supersymmetric quantum mechanics.

A new analytical strategy based on a wavelet computing technique for solving Fokker-Plank equation arises in stochastic phenomena

“…This solution of the FPE, which is associated with above pair functions F(x) and D(x), is corresponding to Hulthen potential in the Schrodinger equation [6].The exact solution of this equation has been obtained by Anjos et al using the super symmetric quantum mechanics approach [6], which is in agreement with our result. It is worthwhile to mention that Hulthen is one of the essential short-range potentials with interesting applications in nuclear, solidstate, and atomic physics.…”

“…The study of stochastic events has grown significantly in recent years because to its applications in physics, chemistry, biology, circuit theory, and even economics [1][2][3][4][5][6][7][8][9][10][11][12][13]. One of the most common differential equations, the so-called FPE, naturally occurs in such events across a wide range of natural sciences, including solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory [6]. Historically, it has been used not just to the study of stochastic systems, but also to the Brownian motion of particles [6].…”

“…One of the most common differential equations, the so-called FPE, naturally occurs in such events across a wide range of natural sciences, including solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory [6]. Historically, it has been used not just to the study of stochastic systems, but also to the Brownian motion of particles [6]. FPE is a linear partial differential equation that represents the temporal evolution of the probability density function associated with a random dynamical system.…”

“…The resulting FPE may be exceedingly difficult or fairly simple, depending on the dynamics of drift and diffusion functions in such systems, and numerical techniques or analytical solutions are provided appropriately. Various analytical methods have been proposed, including supersymmetry [6], Lie algebra [11], Laplace transformation [7], Fourier analysis [8], variational methods [9], and the Ornstein-Uhlenbeck process [10].…”

The Dunkl-Fokker-Planck Equation in 1+1 Dimensions

“…Several methods are used to find solutions for the FPE: analytical methods, transformation of the FPE to a Schrödinger-type equation and numerical methods [4]. When the FPE can be transformed to a Schrödinger-type equation, group theory [6,7], supersymmetric quantum mechanics and shape invariance [5,[8][9][10] are quantum mechanical methods that have been used successfully to solve the FPE for the so called confined potentials [4].…”