About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2011 (2011), Article ID 231920, 12 pages
http://dx.doi.org/10.1155/2011/231920
Research Article

An Explicit Numerical Method for the Fractional Cable Equation

Departamento de Física, Universidad de Extremadura, 06071 Badajoz, Spain

Received 27 April 2011; Accepted 30 June 2011

Academic Editor: Fawang Liu

Copyright © 2011 J. Quintana-Murillo and S. B. Yuste. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Klages, G. Radons, and I. M. Sokolov, Eds., Anomalous Transport: Foundations and Applications, Elsevier, Amsterdam, The Netherlands, 2008. View at Zentralblatt MATH
  2. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  3. R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, no. 2, pp. 255–270, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
  6. D. Baleanu, “Fractional variational principles in action,” Physica Scripta, vol. 2009, no. T136, Article ID 014006, 4 pages, 2009. View at Publisher · View at Google Scholar
  7. S. B. Yuste and L. Acedo, “Some exact results for the trapping of subdiffusive particles in one dimension,” Physica A, vol. 336, no. 3-4, pp. 334–346, 2004. View at Publisher · View at Google Scholar
  8. S. B. Yuste and K. Lindenberg, “Trapping reactions with subdiffusive traps and particles characterized by different anomalous diffusion exponents,” Physical Review E, vol. 72, no. 6, Article ID 061103, 8 pages, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. B. Yuste and K. Lindenberg, “Subdiffusive target problem: survival probability,” Physical Review E, vol. 76, no. 5, Article ID 051114, 6 pages, 2007. View at Publisher · View at Google Scholar
  10. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, p. 77, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Metzler, E. Barkai, and J. Klafter, “Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach,” Physical Review Letters, vol. 82, no. 18, pp. 3563–3567, 1999. View at Publisher · View at Google Scholar · View at Scopus
  13. S. B. Yuste, E. Abad, and K. Lindenberg, “A reaction-subdiffusion model of morphogen gradient formation,” Physical Review E, vol. 82, no. 6, Article ID 061123, 9 pages, 2010.
  14. J. A. Dix and A. S. Verkman, “Crowding effects on diffusion in solutions and cells,” Annual Review of Biophysics, vol. 37, pp. 247–263, 2008. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
  15. I. Y. Wong, M. L. Gardel, D. R. Reichman, et al., “Anomalous Diffusion Probes Microstructure Dynamics of Entangled F-Actin Networks,” Physical Review Letters, vol. 92, no. 17, Article ID 178101, 4 pages, 2004.
  16. J.-H. Jeon, V. Tejedor, S. Burov, et al., “In Vivo Anomalous Diffusion and Weak Ergodicity Breaking of Lipid Granules,” Physical Review Letters, vol. 106, no. 4, Article ID 048103, 4 pages, 2011.
  17. T. A. M. Langlands, B. I. Henry, and S. L. Wearne, “Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions,” Journal of Mathematical Biology, vol. 59, no. 6, pp. 761–808, 2009. View at Publisher · View at Google Scholar · View at PubMed
  18. B. I. Henry, T. A. M. Langlands, and S. L. Wearne, “Fractional cable models for spiny neuronal dendrites,” Physical Review Letters, vol. 100, no. 12, Article ID 128103, p. 4, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. A. Compte and R. Metzler, “The generalized Cattaneo equation for the description of anomalous transport processes,” Journal of Physics A, vol. 30, no. 21, pp. 7277–7289, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Metzler and T. F. Nonnenmacher, “Fractional diffusion, waiting-time distributions, and Cattaneo-type equations,” Physical Review E, vol. 57, no. 6, pp. 6409–6414, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  21. S. S. Ray, “Exact solutions for time-fractional diffusion-wave equations by decomposition method,” Physica Scripta, vol. 75, no. 1, pp. 53–61, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Momani, A. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379–387, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. H. Jafari and S. Momani, “Solving fractional diffusion and wave equations by modified homotopy perturbation method,” Physics Letters. A, vol. 370, no. 5-6, pp. 388–396, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  25. I. Podlubny, A. V. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara, “Matrix approach to discrete fractional calculus. II. Partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137–3153, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. E. Barkai, “Fractional Fokker-Planck equation, solution, and application,” Physical Review E, vol. 63, no. 4, Article ID 046118, 17 pages, 2001.
  27. M. Enelund and G. A. Lesieutre, “Time domain modeling of damping using anelastic displacement fields and fractional calculus,” International Journal of Solids and Structures, vol. 36, no. 29, pp. 4447–4472, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. G. J. Fix and J. P. Roop, “Least squares finite-element solution of a fractional order two-point boundary value problem,” Computers & Mathematics with Applications. An International Journal, vol. 48, no. 7-8, pp. 1017–1033, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Y. Zheng, C. Li, and Z. Zhao, “A note on the finite element method for the space-fractional advection diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1718–1726, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. Y. Zheng, C. Li, and Z. Zhao, “A fully discrete discontinuous Galerkin method for nonlinear fractional Fokker-Planck equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 279038, p. 26, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, “Time fractional diffusion: a discrete random walk approach,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 129–143, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. S. B. Yuste and J. Quintana-Murillo, “On three explicit difference schemes for fractional diffusion and diffusion-wave equations,” Physica Scripta, vol. 2009, no. T136, Article ID 014025, 6 pages, 2009.
  34. S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,” Journal of Computational Physics, vol. 216, no. 1, pp. 264–274, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. J. Quintana-Murillo and S. B. Yuste, “An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form,” Journal of Computational and Nonlinear Dynamics, vol. 6, no. 2, Article ID 021014, 6 pages, 2011.
  36. C. M. Chen, F. Liu, I. Turner, and V. Anh, “A Fourier method for the fractional diffusion equation describing sub-diffusion,” Journal of Computational Physics, vol. 227, no. 2, pp. 886–897, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. P. Zhuang, F. Liu, V. Anh, and I. Turner, “New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 46, no. 2, pp. 1079–1095, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. F. Liu, Q. Yang, and I. Turner, “Stability and convergence of two new implicit numerical methods for the fractional cable equation,” Journal of Computational and Nonlinear Dynamics, vol. 6, no. 1, Article ID 01109, 7 pages, 2011.
  39. K. W. Morton and D. F. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1994.
  40. Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704–719, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, John Wiley & Sons, New York, NY, USA, 1978.
  42. M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH