Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 53
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Chen, Hu Lü, Shujuan and Chen, Wenping 2016. Spectral and pseudospectral approximations for the time fractional diffusion equation on an unbounded domain. Journal of Computational and Applied Mathematics, Vol. 304, p. 43.


    Du, Rui Hao, Zhao-peng and Sun, Zhi-zhong 2016. Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions. East Asian Journal on Applied Mathematics, Vol. 6, Issue. 02, p. 131.


    Elsaid, A. Abdel Latif, M. S. and Maneea, M. 2016. Similarity Solutions for Multiterm Time-Fractional Diffusion Equation. Advances in Mathematical Physics, Vol. 2016, p. 1.


    Gao, Guang-hua and Sun, Zhi-zhong 2016. Two unconditionally stable and convergent difference schemes with the extrapolation method for the one-dimensional distributed-order differential equations. Numerical Methods for Partial Differential Equations, Vol. 32, Issue. 2, p. 591.


    Ivanova, Elena Malti, Rachid and Moreau, Xavier 2016. Time-domain simulation of MIMO fractional systems. Nonlinear Dynamics, Vol. 84, Issue. 4, p. 2057.


    Liu, Lin Zheng, Liancun and Zhang, Xinxin 2016. Fractional anomalous diffusion with Cattaneo–Christov flux effects in a comb-like structure. Applied Mathematical Modelling, Vol. 40, Issue. 13-14, p. 6663.


    Povstenko, Yuriy and Klekot, Joanna 2016. The Dirichlet problem for the time-fractional advection-diffusion equation in a line segment. Boundary Value Problems, Vol. 2016, Issue. 1,


    Zhi-Yun, Wang and Pei-Jie, Chen 2016. Stochastic Oscillations of General Relativistic Disks Described by a Fractional Langevin Equation with Fractional Gaussian Noise. Journal of Astrophysics and Astronomy, Vol. 37, Issue. 2,


    Saxena, Ram Tomovski, Zivorad and Sandev, Trifce 2015. Analytical Solution of Generalized Space-Time Fractional Cable Equation. Mathematics, Vol. 3, Issue. 2, p. 153.


    Wu, Guo-Cheng Baleanu, Dumitru Zeng, Sheng-Da and Deng, Zhen-Guo 2015. Discrete fractional diffusion equation. Nonlinear Dynamics, Vol. 80, Issue. 1-2, p. 281.


    Žecová, Monika and Terpák, Ján 2015. Fractional Heat Conduction Models and Thermal Diffusivity Determination. Mathematical Problems in Engineering, Vol. 2015, p. 1.


    Zhang, Jianying and Yan, Guangwu 2015. Lattice Boltzmann method for the fractional sub-diffusion equation. International Journal for Numerical Methods in Fluids, p. n/a.


    Brunner, Hermann Han, Houde and Yin, Dongsheng 2014. Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain. Journal of Computational Physics, Vol. 276, p. 541.


    Chen, J. Liu, F. Liu, Q. Chen, X. Anh, V. Turner, I. and Burrage, K. 2014. Numerical simulation for the three-dimension fractional sub-diffusion equation. Applied Mathematical Modelling, Vol. 38, Issue. 15-16, p. 3695.


    Ghaffari, Rezvan and Hosseini, S. Mohammad 2014. Obtaining artificial boundary conditions for fractional sub-diffusion equation on space two-dimensional unbounded domains. Computers & Mathematics with Applications, Vol. 68, Issue. 1-2, p. 13.


    Gong, Chunye Bao, Weimin Tang, Guojian Yang, Bo and Liu, Jie 2014. An efficient parallel solution for Caputo fractional reaction–diffusion equation. The Journal of Supercomputing, Vol. 68, Issue. 3, p. 1521.


    Liu, Jie Gong, Chunye Bao, Weimin Tang, Guojian and Jiang, Yuewen 2014. Solving the Caputo Fractional Reaction-Diffusion Equation on GPU. Discrete Dynamics in Nature and Society, Vol. 2014, p. 1.


    Povstenko, Y. Z. 2014. Fundamental Solutions to Time-Fractional Advection Diffusion Equation in a Case of Two Space Variables. Mathematical Problems in Engineering, Vol. 2014, p. 1.


    Saxena, R. K. Mathai, A. M. and Haubold, H. J. 2014. Distributed order reaction-diffusion systems associated with Caputo derivatives. Journal of Mathematical Physics, Vol. 55, Issue. 8, p. 083519.


    Yang, Yin Chen, Yanping and Huang, Yunqing 2014. SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS. Journal of the Korean Mathematical Society, Vol. 51, Issue. 1, p. 203.


    ×

The time fractional diffusion equation and the advection-dispersion equation

  • F. Huang (a1) (a2) and F. Liu (a1) (a3)
  • DOI: http://dx.doi.org/10.1017/S1446181100008282
  • Published online: 01 February 2009
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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The time fractional diffusion equation and the advection-dispersion equation
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      The time fractional diffusion equation and the advection-dispersion equation
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      The time fractional diffusion equation and the advection-dispersion equation
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax