A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

• Autorzy: Guang-hua Gao , Zhi-zhong Sun
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Fractional Klein-Kramers equation; Boundary condition; Finite difference scheme; Stability; Convergence

Kategorie tematyczne

• 65M06: Finite difference methods
• 35R11: Fractional partial differential equations
• 65M12: Stability and convergence of numerical methods

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 1
• Strony: 101-115

Daty

wydano

2012-02-01

online

2011-12-09

Twórcy

autor

Guang-hua Gao

• Southeast University

autor

Zhi-zhong Sun

• Southeast University

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Identyfikatory

DOI

10.2478/s11533-011-0105-0