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 -

References

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  1. [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. [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. [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. [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 Zbl1167.65419
  5. [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. [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. [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 Zbl1233.65052
  8. [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. [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 Zbl1211.65112
  10. [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 Zbl1057.92012
  11. [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. [12] Kramers H.A., Brownian motion in a field of force and the diffusion model of chemical reactions, Phys., 1940, 7, 284–304 Zbl0061.46405
  13. [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 Zbl1123.60045
  14. [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. [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. [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. [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. [18] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999 
  19. [19] Rice S.O., Mathematical analysis of random noise, Bell System Tech. J., 1945, 24, 46–156 Zbl0063.06487
  20. [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 Zbl0604.76069
  21. [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 Zbl1094.65083
  22. [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. [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. [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. [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 Zbl0941.76579
  26. [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 Zbl1204.01020
  27. [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 Zbl1173.26006

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