Skip to main content
SpringerLink
Log in

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

  • Research Article
  • Published:
Central European Journal of Physics

Abstract

Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. M. Pismen, Patterns and Interfaces in Dissipative Systems (Springer Verlag, Berlin, 2006) 163–169

    Google Scholar 

  2. S. B. Yuste, K. Lindenberg, Phys. Rev. Lett. 87, 118301 (2001)

    Article  ADS  Google Scholar 

  3. S. B. Yuste, L. Acedo, K. Lindenberg, Phys. Rev. E 69, 036126 (2004)

    Article  ADS  Google Scholar 

  4. R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. S. Picozzi, B. West, Phys. Rev. E 66, 046118 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  6. F. Liu, V. Anh, I. Turner, J. Comp. Appl. Math. 166, 209 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Appl. Math. Comp. 191, 12 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. I. Podlubny, Fractional Differential Equations (Academic Press, London and New York, 1999)

    MATH  Google Scholar 

  9. D. Baleanu, Rep. Math. Phys. 61, 199 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. E. M. Rabei, I. M. Rawashdeh, S. Muslih, D. Baleanu, Int. J. Theor. Phys. 50, 1569 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Jarad, T. Abdeljawad, D. Baleanu, Abstr. App. Anal. 2012, 8903976 (2012)

    Google Scholar 

  12. D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Com plexity, Nonlinearity and Chaos (World Scientific, Singapore, 2012)

    Google Scholar 

  13. J. A. Ochoa-Tapia, F. J. Valdes-Parada, J. Alvarez-Ramirez, Physica A 374, 1 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. F. J. Valdes-Parada, J. A. Ochoa-Tapia, J. Alvarez-Ramirez, Physica A 373, 339 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  15. J. H. Cushman, B. X. Hu, T. R. Ginn, J. Stat. Phy. 75, 859 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. D. Del-Castillo-Negrete, Phys. Rev. E 79, 031120 (2009)

    Article  ADS  Google Scholar 

  17. V. B. L. Chaurasia, J. Singh, Appl. Math. Sci. 60, 2989 (2011)

    MathSciNet  Google Scholar 

  18. R. Gorenflo, F. Mainardi, Fractional Calc. App. Anal. 1, 167 (1998)

    MATH  MathSciNet  Google Scholar 

  19. M. M. Meerschaert, H. P. Scheffler, C. Tadjeran, J. Comput. Phys. 211, 249 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. R. FitzHugh, Biophys. J. 1, 445 (1961)

    Article  ADS  Google Scholar 

  21. J. Nagumo, S. Animoto, S. Yoshizawa, Proc. Inst. Radio Eng. 50, 2061 (1962)

    Google Scholar 

  22. A. Bueno-Orovio, D. Kay, K. Burrage, J. Comp. Phys. (in press)

  23. Y. Zhang, D. A. Benson, D. M. Reeves, Adv. Water Resour. 32, 561 (2009)

    Article  ADS  Google Scholar 

  24. S. Chen, F. Liu, I. Turner, V. Anh, App. Math. Comput. 219, 4322 (2013)

    Article  MathSciNet  Google Scholar 

  25. Q. Yu, F. Liu, I. Turner, K. Burrage, Appl. Math. Comp. 219, 4082 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld., 4001, Australia

    Fawang Liu, Ian Turner, Kevin Burrage & Vo Anh

  2. Department of Mathematics, Quanzhou Normal University, Quanzhou, Fujian, China

    Shiping Chen

  3. Department of Computer Science and OCISB, University of Oxford, Oxford, OXI 3QD, UK

    Kevin Burrage

Authors
  1. Fawang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ian Turner
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kevin Burrage
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Vo Anh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fawang Liu.

About this article

Cite this article

Liu, F., Chen, S., Turner, I. et al. Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term. centr.eur.j.phys. 11, 1221–1232 (2013). https://doi.org/10.2478/s11534-013-0296-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11534-013-0296-z

Keywords

Search

Navigation