Numerical solution for Fokker-Planck equations in accelerators

Abstract

A finite difference scheme is presented to solve the Fokker-Planck equation in (2+1) variables numerically. This scheme is applied to study stochastic beam dynamics in two-dimensional phase space.

Finding stability domains and escape rates in kicked Hamiltonians

“…We will model the noise phenomenologically with uncorrelated Gaussian noise and linear damping. This assumption has been used earlier to model noise in accelerators [32,33]. A more realistic treatment of the noise would be multiplicative and could be added in principle though some parts of the calculation will then have to be done numerically.…”

An efficient algorithm based on Haar wavelets for numerical simulation of Fokker-Planck equations with constants and variable coefficients

“…The Equation ( 2 The nonlinear Fokker-Planck Equation ( 6) has important applications in various areas such as plasma physics, surface physics, population dynamics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, polymer physics laser physics, and pattern formation, psychology, and marketing (see (Tatari et al, 2007) and references therein). Fokker-Planck equation provides a very useful tool for modeling a wide variety of stochastic phenomena arising in physics (Zorzano et al, 1998), chemistry, biology, finance (Choe et al, 2013;Kopp et al, 2012), etc. The large numbers of applications of the Fokker-Planck equation, a lot of analytical methods such as Adomain decomposition method (Tatari et al, 2007), He's variational iteration method (Dehghan and Tatari, 2006), modify path integration method (Narayanana and Kumar, 2012), discrete eigenvalue spectrum method (Brics et al, 2013), Chebyshev spectral collocation method (Zarebnia and Jalili, 2011), differential transform method (Hesam et al, 2012), Tau method (Vanani and Aminataei, 2012), homotopy perturbation method (Jafari and Aminataei, 2009), etc.…”

Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions

“…For some other investigations on this model or some other similar models the interested readers can see references [6][7][8][9][10]. Authors of [11] developed a finite difference technique [12][13][14][15] to solve the type of Fokker-Planck equations describing the stochastic dynamics of a particle in a storage ring. One important problem in accelerator physics is to study the dynamics of charged particles under the influence of electromagnetic fields and noise.…”

“…One important problem in accelerator physics is to study the dynamics of charged particles under the influence of electromagnetic fields and noise. These models lead to stochastic differential equations in six-dimensional phase space or equivalently to the Fokker-Planck equation [11]. In [16] a finite difference procedure is given for solving the Fokker-Planck equation in two dimensions.…”