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A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space. (English) Zbl 1254.65014

Summary: 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 [B. Cartling, “Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential”, J. Chem. Phys. 87, No. 5, 2638–2648 (1987); W. Deng and C. Li, Numer. Methods Partial Differ. Equations 27, No. 6, 1561-1583 (2011; Zbl 1233.65052)]. 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, loc. cit.] and the five-point scheme in [Deng and Li, loc. cit.]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations

Citations:

Zbl 1233.65052
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References:

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