Two-level difference scheme for the two-dimensional Fokker–Planck equation

2021 ◽  
Vol 180 ◽  
pp. 276-288
Author(s):  
Muhammad Munir Butt
Keyword(s):  
Difference Scheme ◽  
Planck Equation ◽  
Two Dimensional ◽  
Level Difference ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nisrine Maarouf ◽  
Khalid Hilal
Keyword(s):  
Power Series Expansion ◽  
Planck Equation ◽  
Expansion Method ◽  
Analytic Solutions ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Liouville Fractional Derivative ◽  
Detailed Derivation ◽  
Conservation Theorem ◽  
Fokker Planck

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

Kinetic theory of a two-dimensional magnetized plasma. Part 2. Balescu-Lenard limit

1972 ◽  
Vol 8 (3) ◽  
pp. 357-374 ◽  
Author(s):  
George Vahala
Keyword(s):  
Kinetic Theory ◽  
Plasma Parameter ◽  
Planck Equation ◽  
Large Integer ◽  
Collision Term ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Long Time ◽  
Time Average ◽  
Fokker Planck

The kinetic theory of a two-dimensional one-species plasma in a uniform d.c. magnetic field is investigated in the small plasma parameter limit. The plasma consists of charged rods interacting through the logarithmic Coulomb potential. Vahala & Montgomery earlier derived a Fokker –;Planck equation for this system, but it contained a divergent integral, which had to be cut-off on physical grounds. This cut-off is compared to the standard cut-off introduced in the two-dimensional unmagnetized Fokker –;Planck equation. In the small plasma parameter limit, it is shown (under the assumption that for large integer n, γn/γn+1 = O(np), with p < 2, where γn = ωn −nΩ. with ωn the nth. Bernstein mode and Q the electron gyro frequency) that the Balescu-Lenard collision term is zero in the long time average limit if one considers only two-body interactions. The energy transfer from a test particle to an equilibrium plasma is discussed and also shown to be zero in the long time average limit. This supports the unexpected result of zero Balescu-Lenard collision term.

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