Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-23T20:00:59.172Z Has data issue: false hasContentIssue false
  • English
  • Français

The time fractional diffusion equation and the advection-dispersion equation

Published online by Cambridge University Press:  17 February 2009

F. Huang  and
F. Liu
F. Huang
Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, China; e-mail: [email protected]. School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China; e-mail: [email protected].
F. Liu
Affiliation:
Department of Mathematics, Xiamen University, Xiamen 361005, China; e-mail: [email protected]. School of Mathematical Sciences, Queensland University of Technology, Qld 4001, Australia; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 3 , January 2005 , pp. 317 - 330
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Agrawal, O. P., “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynam. 29 (2002) 145–155.CrossRefGoogle Scholar
[2]Anh, V. V. and Leonenko, N. N., “Non-Gaussian scenarios for the heat equation with singular initial conditions”, Stochastic Processes Appl. 84 (1999) 91–114.CrossRefGoogle Scholar
[3]Anh, V. V. and Leonenko, N. N., “Scaling laws for fractional diffusion-wave equations with singular data”, Statist. Probab. Let. Vol. 48 (2000) 239–252.CrossRefGoogle Scholar
[4]Anh, V. V. and Leonenko, N. N., “Harmonic analysis of random fractional diffusion-wave equations”, Appl. Math. Comput. 141 (2003) 77–85.Google Scholar
[5]Baeumer, B., Meerschaert, M. M., Benson, D. A. and Wheatcraft, S. W., “Subordinated advection-dispersion equation for contaminant transport”, Water Resources Res. 37 (2001) 1543–1550.CrossRefGoogle Scholar
[6]Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M., “Application of a fractional advection-dispersion equation”, Water Resources Res. 36 (2000) 1403–1412.CrossRefGoogle Scholar
[7]Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M., “The fractional-order governing equation of Lévy motion”, Water Resources Res. 36 (2000) 1413–1424.CrossRefGoogle Scholar
[8]Bouchaud, J. P. and Georges, A., “Anomalous diffusion in disordered media-statistical mechanisms”, Phys. Rep. 195 (1990) 127–293.CrossRefGoogle Scholar
[9]Caputo, M., “Linear model of dissipation whose Q is almost frequency independent II”, Geophys. J. Roy. Astr. Soc. 13 (1967) 529–539.CrossRefGoogle Scholar
[10]Caputo, M., “Vibrations on an infinite viscoelastic layer with a dissipative memory”, J. Acoust. Soc. Amer. 56 (1974) 897–904.CrossRefGoogle Scholar
[11]Caputo, M. and Mainardi, F., “A new dissipation model based on memory mechanism”, Pure Appl. Geophysics 91 (1971) 134–147.CrossRefGoogle Scholar
[12]Chaves, A., “Fractional diffusion equation to describe Lévy flights”, Phys. Lett. A 239 (1998) 13–16.CrossRefGoogle Scholar
[13]Ginoa, M., Cerbelli, S. and Roman, H. E., “Fractional diffusion equation and relaxation in complex viscoelastic materials”, Phys. A 191 (1992) 449–453.CrossRefGoogle Scholar
[14]Gorenflo, R., Luchko, Yu. and Mainardi, F., “Wright function as scale-invariant solutions of the diffusion-wave equation”, J. Comp. Appl. Math. 118 (2000) 175–191.CrossRefGoogle Scholar
[15]Gorenflo, R. and Mainardi, F., “Random walk models for space-fractional diffusion processes”, Fract. Calc. Appl. Anal. 1 (1998) 167–191.Google Scholar
[16]Gorenflo, R., Mainardi, F., Moretti, D. and Paradisi, P., “Time fractional diffusion: a discrete random walk approach”, Nonlinear Dynam. 29 (2002) 129–143.CrossRefGoogle Scholar
[17]Klafter, J., Blumen, A. and Shlesinger, M. F., “Stochastic pathways to anomalous diffusion”, Phys. Rev. A 35 (1987) 3081–3085.CrossRefGoogle ScholarPubMed
[18]Liu, F., Anh, V. V. and Turner, I., “Numerical solution of the space fractional Fokker-Plank equation”, J. Comput. Appl. Math. 166 (2004) 209–219.CrossRefGoogle Scholar
[19]Liu, F., Anh, V. V., Turner, I. and Zhuang, P.. “Time fractional advection-dispersion equation”, J. Appl. Math. Comput. 13 (2003) 223–245.CrossRefGoogle Scholar
[20]Mainardi, F., “Fraction calculus: some basic problems in continuum and statistical mechanics”, in Fractal and Fractional Calin Continuum Mechanics (eds. Carpinteri, A. and Mainardi, F.), (Springer, Wien, 1997) 291–348.CrossRefGoogle Scholar
[21]Mainardi, F., Luchko, Yu. and Pagnini, G., “The fundamental solution of the space-time fractional diffusion equation”, Fract. Calc. Appl. Anal. 4 (2001) 153–192.Google Scholar
[22]Nigmatullin, R. R., “To the theoretical explanation of the universal response”, Physica B 123 (1984) 739–745.Google Scholar
[23]Nigmatullin, R. R., “The realization of the generalized transfer equation in a medium with fractal geometry”, Phys. Stat. Sol. B 133 (1986) 425–430.CrossRefGoogle Scholar
[24]Podlubny, I., Fractional differential equations (Academic Press, San Diego, CA, 1999).Google Scholar
[25]Raberto, M., Scalas, E. and Mainardi, F., “Waiting-time and returns in high-frequency financial data: an empirical study”, Physica A 34 (2002) 749–755.CrossRefGoogle Scholar
[26]Roman, H. E. and Alemany, P. A., “Continuous-time random walks and the fractional diffusion equation”, J. Phys. A 27 (1994) 3407–3410.CrossRefGoogle Scholar
[27]Sabatelli, L., Keating, S., Dudley, J. and Richmond, P., “Waiting time distributions in financial markets”, Eur Phys. J. B 27 (2002) 273–275.CrossRefGoogle Scholar
[28]Saichev, A. I. and Zaslavsky, G. M., “Fractional kinetic equations: Solutions and applications”, Chaos 7 (1997) 753–764.CrossRefGoogle ScholarPubMed
[29]Schneider, W. R. and Wyss, W., “Fractional diffusion and wave equations”, J. Math. Phys. 30 (1989) 134–144.CrossRefGoogle Scholar
[30]Schumer, R., Benson, D. A., Meerschaert, M. M. and Baeumer, B., “Multiscaling fractional advection-dispersion equations and their solutions”, Water Resources Res. 39 (2003) 1022–1032.CrossRefGoogle Scholar
[31]Srivastava, H. M., Gupta, K. C. and Goyal, S. P., The H-functions of one and two variables with applications (South Asian, New Delhi, 1982).Google Scholar
[32]Wyss, W., “The fractional diffusion equation”, J. Math. Phys. 27 (1986) 2782–2785.CrossRefGoogle Scholar
[33]Zaslavsky, G., “Fractional kinetic equation for Hamiltonian chaos, chaotic advection, tracer dynamics and turbulent dispersion”, Phys. D 76 (1994) 110–122.CrossRefGoogle Scholar