A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space |
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Authors: | Guang-hua Gao Zhi-zhong Sun |
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Institution: | 1.Department of Mathematics,Southeast University,Nanjing,China |
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Abstract: | 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. |
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