Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T22:34:03.496Z Has data issue: false hasContentIssue false
  • English
  • Français

THE FUNDAMENTAL AND NUMERICAL SOLUTIONS OF THE RIESZ SPACE-FRACTIONAL REACTION–DISPERSION EQUATION

Part of: Real functions Markov processes Stochastic processes Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 July 2008

J. CHEN ,
F. LIU ,
I. TURNER  and
V. ANH
J. CHEN
Affiliation:
School of Science, Jimei University, Xiamen 361021, China
F. LIU*
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au) School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China
I. TURNER
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au)
V. ANH
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia (email: f.liu@qut.edu.au)
*
✉For correspondence; e-mail: f.liu@qut.edu.au
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.

A Riesz space-fractional reaction–dispersion equation (RSFRDE) is obtained from the classical reaction–dispersion equation (RDE) by replacing the second-order space derivative with a Riesz derivative of order β∈(1,2]. In this paper, using Laplace and Fourier transforms, we obtain the fundamental solution for a RSFRDE. We propose an explicit finite-difference approximation for a RSFRDE in a bounded spatial domain, and analyse its stability and convergence. Some numerical examples are presented.

Keywords

random walkfractional derivativesprobability densitystabilityconvergence

MSC classification

Secondary: 60J60: Diffusion processes 60G52: Stable processes 26A33: Fractional derivatives and integrals 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 45 - 57
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Benson, D. A., “The fractional advection–dispersion equation”, Ph. D. thesis, University of Nevada, Reno, NV, 1998.Google Scholar
[2]Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M., “Application of a fractional advection–despersion equation”, Water Resources Res. 36 (2000a) 1403–1412.CrossRefGoogle Scholar
[3]Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M., “The fractional-order governing equation of Lévy motion”, Water Resources Res. 36 (2000b) 1413–1423.CrossRefGoogle Scholar
[4]Feller, W., “On a generalization of Marcel Riesz potentials and the semi-groups generated by them” Meddelanden Lunds Universities Matematiska Seminarium (Comn. Sem. Mathem. Universite de Lund), Lund, 1952, 73–81.Google Scholar
[5]Gorenflo, R., Luchko, Yu. and Mainardi, F., “Wright function as scale-invariant solutions of the diffusion-wave equation”, J. Comput. Appl. Math. 118 (2000) 175–191.Google Scholar
[6]Gorenflo, R. and Mainardi, F., “Random walk models for space-fractional diffusion processes”, Fract. Calc. Appl. Anal. 1 (1998) 167–191.Google Scholar
[7]Gorenflo, R. and Mainardi, F., “Approximation of Lévy–Feller diffusion by random walk”, Z. Anal. Anwendungen 18 (1999) 231–246.CrossRefGoogle Scholar
[8]Henry, B. I. and Wearne, S. L., “Existence of Turing instabilities in a two-species fractional reaction–diffusion system”, SIAM J. Appl. Math. 62 (2002) 870–887.Google Scholar
[9]Hilfer, R., Application of fractional calculus in physics (World Scientific, Singapore, 2000).Google Scholar
[10]Lin, R. and Liu, F., “Fractional high order methods for the nonlinear fractional ordinary differential equation”, Nonlinear Anal. 66 (2007) 856–869.Google Scholar
[11]Liu, F., Anh, V. and Turner, I., “Numerical solution of the fractional-order advection–dispersion equation”, Proc. Int. Conf. on Boundary and Interior Layers – Computational and Asymptotic Methods, Perth, Australia, 2002, 159–164.Google Scholar
[12]Liu, F., Anh, V. and Turner, I., “Numerical solution of space-fractional Fokker–Planck equation”, J. Comput. Appl. Math. 166 (2004) 209–219.Google Scholar
[13]Liu, F., Anh, V., Turner, I. and Zhuang, P., “Time-fractional advection–dispersion equation”, J. Appl. Math. Comput. 13 (2003) 233–246.CrossRefGoogle Scholar
[14]Liu, F., Anh, V., Turner, I. and Zhuang, P., “Numerical simulation for solute transport in fractal porous media”, ANZIAM J. 45(E) (2004) 461–473.CrossRefGoogle Scholar
[15]Lynch, V. E., Carreras, B. A., del Castillo-Negrete, D., Ferreira-Mejias, K. M. and Hicks, H. R., “Numerical methods for the solution of partial differential equations of fractional order”, J. Comput. Phys. 192 (2003) 406–421.Google Scholar
[16]Mainardi, F., “The fundamental solutions for the fractional diffusion-wave equation”, Appl. Math. 9 (1996) 23–28.Google Scholar
[17]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
[18]Meerschaert, M. M. and Tadjeran, C., “Finite difference approximations for two-sided space-fractional partial differential equations”, Appl. Numer. Math. 56 (2006) 80–90.Google Scholar
[19]Podlubny, I., Fractional differential equations (Academic Press, New York, 1999).Google Scholar
[20]Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional integrals and derivatives: theory and applications (Gordon and Breach, Newark, NJ, 1993).Google Scholar
[21]Scalas, E., Gorenflo, R. and Mainardi, F., “Fractional calculus and continuous-time finance”, Physica A 284 (2000) 376–384.CrossRefGoogle Scholar
[22]Schumer, R. and Benson, D. A., “Eulerian derivative of the fractional advection–dispersion equation”, J. Contaminant 48 (2001) 69–88.CrossRefGoogle ScholarPubMed
[23]Shen, S. and Liu, F., “Error analysis of an explicit finite difference approximation for the space-fractional diffusion equation with insulated ends”, ANZIAM J. 46 (2005) 871–887.CrossRefGoogle Scholar
[24]Smith, G. D., Numerical solution of partial differential equations: Finite difference methods (Clarendon Press, Oxford, 1985).Google Scholar
[25]Wyss, W., “The fractional Black–Scholes equation”, Fract. Calc. Appl. Anal. 3 (2000) 51–61.Google Scholar