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The time fractional diffusion equation and the advection-dispersion equation

  • F. Huang (a1) (a2) and F. Liu (a1) (a3)
Abstract
Abstract

The time fractional diffusion equation with appropriate initial and boundary conditions in an n-dimensional whole-space and half-space is considered. Its solution has been obtained in terms of Green functions by Schneider and Wyss. For the problem in whole-space, an explicit representation of the Green functions can also be obtained. However, an explicit representation of the Green functions for the problem in half-space is difficult to determine, except in the special cases α = 1 with arbitrary n, or n = 1 with arbitrary α. In this paper, we solve these problems. By investigating the explicit relationship between the Green functions of the problem with initial conditions in whole-space and that of the same problem with initial and boundary conditions in half-space, an explicit expression for the Green functions corresponding to the latter can be derived in terms of Fox functions. We also extend some results of Liu, Anh, Turner and Zhuang concerning the advection-dispersion equation and obtain its solution in half-space and in a bounded space domain.

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[1]O. P. Agrawal , “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynam. 29 (2002) 145155.

[2]V. V. Anh and N. N. Leonenko , “Non-Gaussian scenarios for the heat equation with singular initial conditions”, Stochastic Processes Appl. 84 (1999) 91114.

[3]V. V. Anh and N. N. Leonenko , “Scaling laws for fractional diffusion-wave equations with singular data”, Statist. Probab. Let. Vol. 48 (2000) 239252.

[5]B. Baeumer , M. M. Meerschaert , D. A. Benson and S. W. Wheatcraft , “Subordinated advection-dispersion equation for contaminant transport”, Water Resources Res. 37 (2001) 15431550.

[6]D. A. Benson , S. W. Wheatcraft and M. M. Meerschaert , “Application of a fractional advection-dispersion equation”, Water Resources Res. 36 (2000) 14031412.

[7]D. A. Benson , S. W. Wheatcraft and M. M. Meerschaert , “The fractional-order governing equation of Lévy motion”, Water Resources Res. 36 (2000) 14131424.

[8]J. P. Bouchaud and A. Georges , “Anomalous diffusion in disordered media-statistical mechanisms”, Phys. Rep. 195 (1990) 127293.

[9]M. Caputo , “Linear model of dissipation whose Q is almost frequency independent II”, Geophys. J. Roy. Astr. Soc. 13 (1967) 529539.

[10]M. Caputo , “Vibrations on an infinite viscoelastic layer with a dissipative memory”, J. Acoust. Soc. Amer. 56 (1974) 897904.

[11]M. Caputo and F. Mainardi , “A new dissipation model based on memory mechanism”, Pure Appl. Geophysics 91 (1971) 134147.

[12]A. Chaves , “Fractional diffusion equation to describe Lévy flights”, Phys. Lett. A 239 (1998) 1316.

[13]M. Ginoa , S. Cerbelli and H. E. Roman , “Fractional diffusion equation and relaxation in complex viscoelastic materials”, Phys. A 191 (1992) 449453.

[14]R. Gorenflo , Yu. Luchko and F. Mainardi , “Wright function as scale-invariant solutions of the diffusion-wave equation”, J. Comp. Appl. Math. 118 (2000) 175191.

[16]R. Gorenflo , F. Mainardi , D. Moretti and P. Paradisi , “Time fractional diffusion: a discrete random walk approach”, Nonlinear Dynam. 29 (2002) 129143.

[17]J. Klafter , A. Blumen and M. F. Shlesinger , “Stochastic pathways to anomalous diffusion”, Phys. Rev. A 35 (1987) 30813085.

[18]F. Liu , V. V. Anh and I. Turner , “Numerical solution of the space fractional Fokker-Plank equation”, J. Comput. Appl. Math. 166 (2004) 209219.

[19]F. Liu , V. V. Anh , I. Turner and P. Zhuang . “Time fractional advection-dispersion equation”, J. Appl. Math. Comput. 13 (2003) 223245.

[20]F. Mainardi , “Fraction calculus: some basic problems in continuum and statistical mechanics”, in Fractal and Fractional Calin Continuum Mechanics (eds. A. Carpinteri and F. Mainardi ), (Springer, Wien, 1997) 291348.

[23]R. R. Nigmatullin , “The realization of the generalized transfer equation in a medium with fractal geometry”, Phys. Stat. Sol. B 133 (1986) 425430.

[26]H. E. Roman and P. A. Alemany , “Continuous-time random walks and the fractional diffusion equation”, J. Phys. A 27 (1994) 34073410.

[27]L. Sabatelli , S. Keating , J. Dudley and P. Richmond , “Waiting time distributions in financial markets”, Eur Phys. J. B 27 (2002) 273275.

[28]A. I. Saichev and G. M. Zaslavsky , “Fractional kinetic equations: Solutions and applications”, Chaos 7 (1997) 753764.

[29]W. R. Schneider and W. Wyss , “Fractional diffusion and wave equations”, J. Math. Phys. 30 (1989) 134144.

[30]R. Schumer , D. A. Benson , M. M. Meerschaert and B. Baeumer , “Multiscaling fractional advection-dispersion equations and their solutions”, Water Resources Res. 39 (2003) 10221032.

[32]W. Wyss , “The fractional diffusion equation”, J. Math. Phys. 27 (1986) 27822785.

[33]G. Zaslavsky , “Fractional kinetic equation for Hamiltonian chaos, chaotic advection, tracer dynamics and turbulent dispersion”, Phys. D 76 (1994) 110122.

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