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Numerical Method for The Time Fractional Fokker-Planck Equation

Published online by Cambridge University Press:  03 June 2015

Xue-Nian Cao ,
Jiang-Li Fu  and
Hu Huang
Xue-Nian Cao*
Affiliation:
School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China
Jiang-Li Fu*
Affiliation:
School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China
Hu Huang*
Affiliation:
School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan 411105, China
*
Corresponding author. Email: [email protected]
Email: [email protected]
Email: [email protected]
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Abstract

In this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.

Keywords

34A0826A3365L1265L20Fractional Fokker-Planck equationRiemann-Liouville fractional derivativetruncation errorstability
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012 , pp. 848 - 863
Copyright
Copyright © Global-Science Press 2012

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References

[1]Barkai, E., Metzler, R., Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61(2000), 132–138.CrossRefGoogle Scholar
[2]Barkai, E., Fractional Fokker-Planck equation, solution and application, Phys. Rev. E, 63(2001), 046118.Google Scholar
[3]Benson, D. A., Wheatcraft, S. W., Meerschaert, M. M., Application of a fractional advection-despersion equation, Water Resour. Res., 36(6)(2000), 1403–1412.Google Scholar
[4]Benson, D. A, Wheatcraft, S. W., Meerschaert, M. M., The fractional-order Governing equation of Levy motion, Water Resour. Res., 36(6)(2000), 1423–1423.CrossRefGoogle Scholar
[5]Chen, S., Liu, F., Zhuang, P., Anh, V., Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33(2009), 256–273.Google Scholar
[6]Deng, W. H., Numerical algorithm for the time fractional Fokker-Planck equation, J. Com-put. Phys., 227(2007), 1510–1522.Google Scholar
[7]Diethelm, K., Ford, N. J., Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29(2002), 3–22.Google Scholar
[8]Diethelm, K., Ford, N. J., Freed, A. D., Details error anaysis for franctional adams method, Numer. Agorithms, 2004(36), 31–52.Google Scholar
[9]Heinsalu, E., Patriarca, M., Goychuk, I., Schmid, G., Haünggi, P., Fractional Fokker-Planck dynamics: numerical algorithms and simulations, Phys. Rev. E, 73(2006), 046133.CrossRefGoogle Scholar
[10]Huang, J., Tang, Y.-F., Vazquez, L., Convergence analysis of a block-by-block method for fractional differential equations Numer. Math. Theor. Meth. Appl., 5 (2012), 229–241.Google Scholar
[11]Ji, X., Tang, H.-Z., High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations, Numer. Math. Theor. Meth. Appl., 5 (2012), 333–358.Google Scholar
[12]Jumarie, G., A Fokker-Planck equation of fractional order with respect to time, J. Math. Phys., 33(1992), 3536–3542.Google Scholar
[13]Koeller, R. C., Application of franctional calculus to the theory of viscoelasticity, Appl. Mech., 5(1984), 299–307.Google Scholar
[14]Kusnezov, D., Bulgac, A., Dang, G. D., Quantum levy processes and fractional kinetics, Phys. Rev. Lett., 82(1999), 1136–1139.Google Scholar
[15]Lenzi, E. K., Mendes, R. S., Fa, K. S., Malacarne, L. C., Anomalous diffusion: fractional Fokker-Planck equation and its solution, J. Math. Phys., 44 (2003), 2179–2185.CrossRefGoogle Scholar
[16]Li, X. and Xu, C., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016–1051.Google Scholar
[17]Liu, F., Anh, V., Turner, I., Zhuang, P., Numerical simulation for solute transport in franctal porous media, ANZIAM, 45(E)(2004), 461–473.Google Scholar
[18]Mark, R. J., Hall, M. W., Different egralinter polation from a bandlimited signal’s samples, IEEE Trans. Acoust. Speech Signal Processing, 29(1981), 872–877.Google Scholar
[19]Meerschaert, M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equation, Comp. Appl. Math., 172(2004), 65–77.Google Scholar
[20]Metzler, R., Barkai, E., Klafter, J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82(18)(1999), 3563–3567.Google Scholar
[21]Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, 1974.Google Scholar
[22]Podlubny, I., Franctional Differential Equations, Academic Press, 1999.Google Scholar
[23]Shaj, A. II, Smith, D., Simulation of plasma based on Fokker-Planck equation, MKS Instru-862 ments, 90 Industrial Way, Wilmington, MA, USA, 01887.Google Scholar
[24]Schneider, W. R., Wyss, W., Fractional diffusion and wave equation, J. Math. Phys., 30(1989), 134–144.Google Scholar
[25]Singh, M. and Gupta, P. K., Homotopy perturbation method for time-fractional shock wave equation, Adv. Appl. Math. Mech., 3 (2011), 774–783.Google Scholar
[26]Yang, S., Xiao, A., Pan, X., Dependence analysis of the solutions on the parameters of fractional delay differential equations, Adv. Appl. Math. Mech., 3 (2011), 586–597.Google Scholar
[27]Zhang, N., Deng, W.-H., Wu, Y.-J., Finite difference/element method for a two-dimensional modified fractional diffusion equation, Adv. Appl. Math. Mech., 4 (2012), 496–518.Google Scholar
[28]Zhuang, P., Liu, F., Implicit difference approximation for the time fractional diffusion equation, Appl. Math. Comput., 22(3)(2006), 87–99.Google Scholar