Fokker-Planck equation for Coulomb relaxation and wave-particle diffusion: Spectral solution and the stability of the Kappa distribution to Coulomb collisions

The Fokker-Planck equation has been used by a number of authors (Jokipii 1966, 1971; Hall and Sturrock 1967; Hasselmann and Wibberentz 1968; Roelof 1968) to deduce the diffusion coefficients of cosmic-ray particles in the interplanetary magnetic field. However, these calculations suggest that the diffusion of particles perpendicular to the mean magnetic field is implausibly large; so large that the validity of a Fokker-Planck approach as applied to the interplanetary medium must be doubted.

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

Mathematics ◽

10.3390/math9111260 ◽

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.

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Numerical Method for The Time Fractional Fokker-Planck Equation

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.12-12s13 ◽

2012 ◽

Vol 4 (06) ◽

pp. 848-863 ◽

Cited By ~ 13

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.

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Two-dimensional bounce-averaged Fokker–Planck modelling of an electron cyclotron resonance plasma source

Journal of Plasma Physics ◽

10.1017/s0022377897006259 ◽

1998 ◽

Vol 59 (2) ◽

pp. 243-257 ◽

Cited By ~ 1

Author(s):

A. P. SMIRNOV ◽

N. V. SUETIN ◽

A. B. SHMELEV

Keyword(s):

Cyclotron Resonance ◽

Electron Cyclotron Resonance ◽

Planck Equation ◽

Electron Cyclotron ◽

Electron Cyclotron Resonance Heating ◽

Two Dimensional ◽

Breakdown Threshold ◽

Coulomb Collisions ◽

Fokker Planck Equation ◽

Fokker Planck

The electron energy distribution function in a microwave discharge at the electron cyclotron resonance (ECR) condition is studied using a two-dimensional Fokker–Planck equation in the bounce-averaged approach. Our model takes into account the effects of linearized Coulomb collisions, electron cyclotron resonance heating in the quasilinear approximation, the effects of ionization and excitation of atoms, elastic scattering of electrons on atoms, and the self-consistent ambipolar potential. The plasma is considered to be slightly collisional, so that bounce averaging is valid. We perform a numerical investigation of the Fokker–Planck equation and obtain the dependences of the discharge characteristics on the parameters of the model, such as breakdown threshold values of neutral density, and the dependence of the electron density, temperature and ambipolar potential on the parameters of the ECR wave and gas density. Some results are compared with experimental data.

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