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2012 | 10 | 1 | 101-115
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
1
Strony
101-115
Opis fizyczny
Daty
wydano
2012-02-01
online
2011-12-09
Twórcy
Bibliografia
  • [1] Barkai E., Silbey R.J., Fractional Kramers equation, Journal of Physical Chemistry B, 2000, 104(16), 3866–3874 http://dx.doi.org/10.1021/jp993491m
  • [2] Bicout D.J., Berezhkovskii A.M., Szabo A., Irreversible bimolecular reactions of Langevin particles, J. Chem. Phys., 2001, 114(5), 2293–2303 http://dx.doi.org/10.1063/1.1332807
  • [3] 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 http://dx.doi.org/10.1063/1.453102
  • [4] 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 http://dx.doi.org/10.1016/j.apm.2007.11.005
  • [5] Coffey W.T., Kalmykov Y.P., Titov S.V., Inertial effects in anomalous dielectric relaxation, Journal of Molecular Liquids, 2004, 114, 35–41 http://dx.doi.org/10.1016/j.molliq.2004.02.004
  • [6] 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 http://dx.doi.org/10.1016/j.molliq.2004.02.005
  • [7] 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 http://dx.doi.org/10.1002/num.20596
  • [8] Dieterich P., Klages R., Preuss R., Schwab A., Anomalous dynamics of cell migration, Proc. Natl. Acad. Sci. USA, 2008, 105(2), 459–463 http://dx.doi.org/10.1073/pnas.0707603105
  • [9] 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 http://dx.doi.org/10.1016/j.jcp.2010.10.007
  • [10] 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 http://dx.doi.org/10.1142/S0218202504003726
  • [11] 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 http://dx.doi.org/10.1063/1.2140281
  • [12] Kramers H.A., Brownian motion in a field of force and the diffusion model of chemical reactions, Phys., 1940, 7, 284–304
  • [13] Magdziarz M., Weron A., Numerical approach to the fractional Klein-Kramers equation, Phys. Rev. E, 2007, 76(6), #066708 http://dx.doi.org/10.1103/PhysRevE.76.066708
  • [14] 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 http://dx.doi.org/10.1088/0305-4470/18/18/016
  • [15] 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 http://dx.doi.org/10.1021/jp9934329
  • [16] 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 http://dx.doi.org/10.1103/PhysRevE.61.6308
  • [17] 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 http://dx.doi.org/10.1209/epl/i2002-00421-1
  • [18] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999
  • [19] Rice S.O., Mathematical analysis of random noise, Bell System Tech. J., 1945, 24, 46–156
  • [20] 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 http://dx.doi.org/10.1007/BF01010986
  • [21] Sun Z.-Z., Wu X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 2006, 56(2), 193–209 http://dx.doi.org/10.1016/j.apnum.2005.03.003
  • [22] 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 http://dx.doi.org/10.1016/j.cplett.2003.12.051
  • [23] 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 http://dx.doi.org/10.1063/1.1607315
  • [24] Wang M.C., Uhlenbeck G.E., On the theory of the Brownian motion. II, Rev. Modern Phys., 1945, 17(2–3), 323–342 http://dx.doi.org/10.1103/RevModPhys.17.323
  • [25] 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 http://dx.doi.org/10.1007/BF01049431
  • [26] Zambelli S., Chemical kinetics and diffusion approach: the history of the Klein-Kramers equation, Arch. Hist. Exact Sci., 2010, 64(4), 395–428 http://dx.doi.org/10.1007/s00407-010-0059-9
  • [27] 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 http://dx.doi.org/10.1137/060673114
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0105-0
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