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.

Numerical Solutions of the Fokker–Planck Equation for Magnetic Nanoparticles

2009 ◽  
Vol 45 (11) ◽  
pp. 5216-5219
Author(s):  
D. P. Ansalone ◽  
C. Ragusa ◽  
M. d'Aquino ◽  
C. Serpico ◽  
G. Bertotti
Keyword(s):  
Magnetic Nanoparticles ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals An interplay between attraction and repulsion in infinite populations

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Yuri Kozitsky
Keyword(s):  
Kinetic Equation ◽  
Short Range ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Infinite Population ◽  
Poisson Law ◽  
Fokker Planck Equation ◽  
Proposed Model ◽  
Fokker Planck

AbstractWe propose and study a model describing an infinite population of point entities arriving in and departing from $$X=\mathbb {R}^d$$ X = R d , $$d\ge 1$$ d ≥ 1 . The already existing entities force each other to leave the population (repulsion) and attract the newcomers. The evolution of the population states is obtained by solving the corresponding Fokker-Planck equation. Without interactions, the evolution preserves states in which the probability $$p(n,\Lambda )$$ p ( n , Λ ) of finding n points in a compact vessel $$\Lambda \subset X$$ Λ ⊂ X obeys the Poisson law. As we show, for pure attraction the decay of $$p(n,\Lambda )$$ p ( n , Λ ) with $$n\rightarrow +\infty $$ n → + ∞ may be essentially slower. The main result is the statement that in the presence of repulsion—even of an arbitrary short range—the evolution preserves states in which the decay of $$p(n,\Lambda )$$ p ( n , Λ ) is at most Poissonian. We also derive the corresponding kinetic equation, the numerical solutions of which can provide more detailed information on the interplay between attraction and repulsion. Further possibilities in studying the proposed model are also discussed.

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