scholarly journals Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yuxin Zhang
Keyword(s):  
Theoretical Analysis ◽  
Fourier Method ◽  
Planck Equation ◽  
Stability And Convergence ◽  
Numerical Treatment ◽  
Unconditionally Stable ◽  
Fokker Planck Equation ◽  
Temporal And Spatial ◽  
The Difference ◽  
Fokker Planck

A numerical method for the modified time fractional Fokker-Planck equation is proposed. Stability and convergence of the method are rigorously discussed by means of the Fourier method. We prove that the difference scheme is unconditionally stable, and convergence order isO(τ+h4), whereτandhare the temporal and spatial step sizes, respectively. Finally, numerical results are given to confirm the theoretical analysis.

A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing

2019 ◽  
Vol 40 (2) ◽  
pp. 1217-1240 ◽  
Author(s):  
Can Huang ◽  
Kim Ngan Le ◽  
Martin Stynes
Keyword(s):  
Numerical Method ◽  
Blow Up ◽  
Planck Equation ◽  
Stability And Convergence ◽  
Time Stability ◽  
Backward Euler Scheme ◽  
Fokker Planck Equation ◽  
Backward Euler ◽  
Numer Anal ◽  
Fokker Planck

Abstract First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.

scholarly journals Numerical treatment for the fractional Fokker-Planck equation

2007 ◽  
Vol 48 ◽  
pp. 759 ◽  
Author(s):  
Pinghui Zhuang ◽  
Fawang Liu ◽  
Vo Anh ◽  
Ian Turner
Keyword(s):  
Planck Equation ◽  
Numerical Treatment ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Allgemeine 13-Momenten-Näherung zur Fokker-Planck-Gleichung eines Plasmas

1965 ◽  
Vol 20 (10) ◽  
pp. 1243-1255
Author(s):  
Friedrich Hertweck
Keyword(s):  
Heat Flux ◽  
Lower Order ◽  
Planck Equation ◽  
Electrical Current ◽  
Moment Equations ◽  
Fokker Planck Equation ◽  
Electrons And Ions ◽  
The Difference ◽  
Fokker Planck

For the Fokker-Planck-equation for a plasma the system of moment equations is derived. The highest order moments considered are the components of the heat flux. For these the condition must be satisfied that they are small compared with (5/2) p √p/ϱe. All moments of lower order, especially the difference velocity of electrons and ions (i.e. the electrical current) and the anisotropy of pressure are arbitrary in this approximation.

scholarly journals Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1260
Author(s):  
Elsayed I. Mahmoud ◽  
Viktor N. Orlov
Keyword(s):  
Brownian Particle ◽  
Planck Equation ◽  
Variable Coefficients ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Time Space ◽  
Liouville Derivative ◽  
The Stability ◽  
Fokker Planck

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis.

ASYMPTOTIC ANALYSIS OF THE FOKKER-PLANCK EQUATION RELATED TO BROWNIAN MOTION

1994 ◽  
Vol 04 (01) ◽  
pp. 17-33 ◽  
Author(s):  
J. BANASIAK ◽  
J.R. MIKA
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Collision Operator ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Expansion Procedure ◽  
The Difference ◽  
Fokker Planck

In this paper we apply the modified Chapman-Enskog expansion procedure to find the asymptotic solution of the Fokker-Planck equation related to Brownian motion. We prove that the asymptotic solution is defined by the diffusion equation and show that the difference between the exact and asymptotic solutions is of order ε2 where 1/ε is related to the magnitude of the collision operator.

Numerical Method for The Time Fractional Fokker-Planck Equation

2012 ◽  
Vol 4 (06) ◽  
pp. 848-863 ◽  
Author(s):  
Xue-Nian Cao ◽  
Jiang-Li Fu ◽  
Hu Huang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Truncation Error ◽  
Planck Equation ◽  
Local Truncation Error ◽  
Order Of Accuracy ◽  
Fokker Planck Equation ◽  
The Stability ◽  
Fokker Planck

AbstractIn this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.

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
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