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

Guang-hua Gao; Zhi-zhong Sun

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 101-115
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Guang-hua Gao, and Zhi-zhong Sun. "A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space." Open Mathematics 10.1 (2012): 101-115. <http://eudml.org/doc/269069>.

@article{Guang2012,
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.},
author = {Guang-hua Gao, Zhi-zhong Sun},
journal = {Open Mathematics},
keywords = {Fractional Klein-Kramers equation; Boundary condition; Finite difference scheme; Stability; Convergence; fractional Klein-Kramers equation; boundary condition; finite difference scheme; stability; convergence},
language = {eng},
number = {1},
pages = {101-115},
title = {A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space},
url = {http://eudml.org/doc/269069},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Guang-hua Gao
AU - Zhi-zhong Sun
TI - A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 101
EP - 115
AB - 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.
LA - eng
KW - Fractional Klein-Kramers equation; Boundary condition; Finite difference scheme; Stability; Convergence; fractional Klein-Kramers equation; boundary condition; finite difference scheme; stability; convergence
UR - http://eudml.org/doc/269069
ER -

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