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.
Similar content being viewed by others
References
Barkai E., Silbey R.J., Fractional Kramers equation, Journal of Physical Chemistry B, 2000, 104(16), 3866–3874
Bicout D.J., Berezhkovskii A.M., Szabo A., Irreversible bimolecular reactions of Langevin particles, J. Chem. Phys., 2001, 114(5), 2293–2303
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
Chen S., Liu F., Zhuang P., Anh V., Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 2009, 33(1), 256–273
Coffey W.T., Kalmykov Y.P., Titov S.V., Inertial effects in anomalous dielectric relaxation, Journal of Molecular Liquids, 2004, 114, 35–41
Coffey W.T., Kalmykov Y.P., Titov S.V., Anomalous dielectric relaxation in a double-well potential, Journal of Molecular Liquids, 2004, 114, 43–49
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
Dieterich P., Klages R., Preuss R., Schwab A., Anomalous dynamics of cell migration, Proc. Natl. Acad. Sci. USA, 2008, 105(2), 459–463
Gao G.-H., Sun Z.-Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 2011, 230(3), 586–595
Hadeler K.P., Hillen T., Lutscher F., The Langevin or Kramers approach to biological modeling, Math. Models Methods Appl. Sci., 2004, 14(10), 1561–1583
Kalmykov Y.P., Coffey W.T., Titov S.V., Thermally activated escape rate for a Brownian particle in a double-well potential for all values of the dissipation, J. Chem. Phys., 2006, 124(2), #024107
Kramers H.A., Brownian motion in a field of force and the diffusion model of chemical reactions, Phys., 1940, 7, 284–304
Magdziarz M., Weron A., Numerical approach to the fractional Klein-Kramers equation, Phys. Rev. E, 2007, 76(6), #066708
Marshall T.W., Watson E.J., A drop of ink falls from my pen... it comes to earth, I know not when, J. Phys. A, 1985, 18(18), 3531–3559
Metzler R., Klafter J., From a generalized Chapman-Kolmogorov equation to the fractional Klein-Kramers equation, Journal of Physical Chemistry B, 2000, 104(16), 3851–3857
Metzler R., Klafter J., Subdiffusive transport close to thermal equilibrium: from the Langevin equation to fractional diffusion, Phys. Rev. E, 2000, 61(6), 6308–6311
Metzler R., Sokolov I.M., Superdiffusive Klein-Kramers equation: normal and anomalous time evolution and Lévy walk moments, Europhys. Lett., 2002, 58(4), 482–488
Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999
Rice S.O., Mathematical analysis of random noise, Bell System Tech. J., 1945, 24, 46–156
Selinger J.V., Titulaer U.M., The kinetic boundary layer for the Klein-Kramers equation; a new numerical approach, J. Statist. Phys., 1984, 36(3–4), 293–319
Sun Z.-Z., Wu X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 2006, 56(2), 193–209
Trahan C.J., Wyatt R.E., Classical and quantum phase space evolution: fixed-lattice and trajectory solutions, Chem. Phys. Lett., 2004, 385(3–4), 280–285
Trahan C.J., Wyatt R.E., Evolution of classical and quantum phase-space distributions: A new trajectory approach for phase space hydrodynamics, J. Chem. Phys., 2003, 119(14), 7017–7029
Wang M.C., Uhlenbeck G.E., On the theory of the Brownian motion. II, Rev. Modern Phys., 1945, 17(2–3), 323–342
Widder M.E., Titulaer U.M., Kinetic boundary layers in gas mixtures: systems described by nonlinearly coupled kinetic and hydrodynamic equations and applications to droplet condensation and evaporation, J. Stat. Phys., 1993, 70(5–6), 1255–1279
Zambelli S., Chemical kinetics and diffusion approach: the history of the Klein-Kramers equation, Arch. Hist. Exact Sci., 2010, 64(4), 395–428
Zhuang P., Liu F., Anh V., Turner I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 2008, 46(2), 1079–1095
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Gao, Gh., Sun, Zz. A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space. centr.eur.j.math. 10, 101–115 (2012). https://doi.org/10.2478/s11533-011-0105-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-011-0105-0