Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.
@article{bwmeta1.element.doi-10_2478_s11533-011-0105-0,
author = {Guang-hua Gao and Zhi-zhong Sun},
title = {A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {101-115},
zbl = {1254.65014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0105-0}
}
Guang-hua Gao; Zhi-zhong Sun. A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space. Open Mathematics, Tome 10 (2012) pp. 101-115. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0105-0/
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