Boltayeva G, Kurbonova K, Kodirova D

Numerical Resolution of Unsteady Fokker-Planck Type Kinetic Equation
Boltayeva G , Kurbonova K , Kodirova D
Urgench State University, Urgench, Uzbekistan;
[e-mail]


The Fokker-Planck equation is often used to approximate the description of particle transport processes with highly forward-peaked scattering. Pomraning has shown that if the physical scattering kernel is sufficiently dominated by small-angle scattering,then the Fokker-Planck equation is an asymptotic approximation to the linear Boltzmann transport equation (BTE).
From practical point of view, it is important to consider a problem for the time
dependent FPE whose solution is used to compute the absorbed dose of radiation. The problem is given as follows:
for (1)
(2)
for (3)
for (4)
where
For solving the problem (1)-(4), first of all, we rewrite it in the following form:
(5)
(6)
where . Since the problem above is an initial value problem, one can employ explicit or implicit Euler method in order to get a discretization with respect to time variable. The solution allows to compute the angular flux density of particles for each time fixed. We can easily compute for fixed value of t since we have already computed the solution of steady FPE when angular flux density of particles depends on only µ, z with a direct and iterative methods in [1, 2].Numerical solution of similar problem can be seen in [3].
ЛИТЕРАТУРА
1. L´opez Pouso, Oscar and Jumaniyazov, Nizomjon. Numerical experiments with the Fokker-Planck equation in 1D slab geometry//Journal of Computational and Theoretical Transport. 2016. 45, № 3 pp. 184-201.
2. L´opez Pouso, Oscar and Jumaniyazov, Nizomjon. Direct versus iterative methods for forward - backward diffusion equations. Numerical comparisons on a particular transport kinetic model//SeMA Journal. 2021. 78, pp. 271-286.
3. L´opez Pouso, Oscar and Jumaniyazov, Nizomjon. Numerical solution of the azimuth-dependent Fokker-Planck equation in 1D slab geometry//Journal of Computational and Theoretical Transport. 2021. pp. 102-133.