scholarly journals On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

2012 ◽  
Vol 50 (3) ◽  
pp. 1136-1161 ◽  
Author(s):  
Jie Shen ◽  
Haijun Yu
Keyword(s):  
Planck Equation ◽  
Two Dimensions ◽  
Dumbbell Model ◽  
Fokker Planck Equation ◽  
Nonlinear Elastic ◽  
Elastic Dumbbell Model ◽  
Fokker Planck

A Numerical Solution Based on the Fokker-Planck Equation for Dilute Polymer Solutions Using High Order RBF Methods

2014 ◽  
Vol 553 ◽  
pp. 187-192
Author(s):  
H.Q. Nguyen ◽  
C.D. Tran ◽  
N. Pham-Sy ◽  
T. Tran-Cong
Keyword(s):  
Numerical Solutions ◽  
Polymer Solutions ◽  
Planck Equation ◽  
High Order ◽  
Collocation Methods ◽  
Dilute Polymer Solutions ◽  
Fokker Planck Equation ◽  
Start Up ◽  
Nonlinear Elastic ◽  
Fokker Planck

This paper presents a numerical method for the Fokker-Planck Equation (FPE) based on mesoscopic modelling of dilute polymer solutions using Radial Basis Function (RBF) approaches. The stress is determined by the solution of a FPE while the velocity field is locally calculated via the solution of conservation Differential Equations (DEs) [1,2]. The FPE and PDEs are approximated separately by two different Integrated RBF methods. The time implicit discretisation of both FPE and PDEs is carried out using collocation methods where the high order RBF approximants improve significantly the accuracy of the numerical solutions and the convergence rate. As an illustration of the method, the time evolution of a start-up flow is studied for the Finitely Extensible Nonlinear Elastic (FENE) dumbbell model.

scholarly journals Semi-Dilute Dumbbells: Solutions of the Fokker–Planck Equation

J ◽  
2021 ◽  
Vol 4 (3) ◽  
pp. 341-355
Author(s):  
Stephen Chaffin ◽  
Julia Rees
Keyword(s):  
Constitutive Equation ◽  
Mean Field ◽  
Planck Equation ◽  
Intermolecular Forces ◽  
Concentration Effects ◽  
Fokker Planck Equation ◽  
Dynamics Simulations ◽  
Nonlinear Elastic ◽  
Background Shear ◽  
Fokker Planck

Spring bead models are commonly used in the constitutive equations for polymer melts. One such model based on kinetic theory—the finitely extensible nonlinear elastic dumbbell model incorporating a Peterlin closure approximation (FENE-P)—has previously been applied to study concentration-dependent anisotropy with the inclusion of a mean-field term to account for intermolecular forces in dilute polymer solutions for background profiles of weak shear and elongation. These investigations involved the solution of the Fokker–Planck equation incorporating a constitutive equation for the second moment. In this paper, we extend this analysis to include the effects of large background shear and elongation beyond the Hookean regime. Further, the constitutive equation is solved for the probability density function which permits the computation of any macroscopic variable, allowing direct comparison of the model predictions with molecular dynamics simulations. It was found that if the concentration effects at equilibrium are taken into account, the FENE-P model gives qualitatively the correct predictions, although the over-shoot in extension in comparison to the infinitely dilute case is significantly underpredicted.

AN APPLICATION OF FOKKER–PLANCK EQUATION TO JOSEPHSON JUNCTION

1989 ◽  
Vol 9 (1) ◽  
pp. 109-120
Author(s):  
G. Liao ◽  
A.F. Lawrence ◽  
A.T. Abawi
Keyword(s):  
Josephson Junction ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Time-changed fractional Ornstein-Uhlenbeck process

2020 ◽  
Vol 23 (2) ◽  
pp. 450-483 ◽  
Author(s):  
Giacomo Ascione ◽  
Yuliya Mishura ◽  
Enrica Pirozzi
Keyword(s):  
Planck Equation ◽  
Uhlenbeck Process ◽  
Fokker Planck Equation ◽  
Ornstein Uhlenbeck Process ◽  
Fokker Planck

AbstractWe define a time-changed fractional Ornstein-Uhlenbeck process by composing a fractional Ornstein-Uhlenbeck process with the inverse of a subordinator. Properties of the moments of such process are investigated and the existence of the density is shown. We also provide a generalized Fokker-Planck equation for the density of the process.

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