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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 293706, 10 pages
http://dx.doi.org/10.1155/2013/293706
Research Article

Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

1Department of Mathematics, East China Normal University, Shanghai 200241, China
2Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China

Received 25 June 2013; Accepted 8 July 2013

Academic Editor: Ming Li

Copyright © 2013 Yuan-Ming Wang. 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. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  2. R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, and P. Paradisi, “Discrete random walk models for space-time fractional diffusion,” Chemical Physics, vol. 284, pp. 521–541, 2002.
  3. B. I. Henry and S. L. Wearne, “Fractional reaction-diffusion,” Physica A, vol. 276, no. 3-4, pp. 448–455, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. Giona and H. E. Roman, “Fractional diffusion equation for transport phenomenna in random media,” Physica A, vol. 185, pp. 87–97, 1992.
  5. T. Kosztolowicz, “Subdiffusion in a system with a thick membrane,” Journal of Membrane Science, vol. 320, pp. 492–499, 2008.
  6. 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, pp. 3563–3567, 1999.
  7. 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
  8. R. Metzler and J. Klafter, “Boundary value problems for fractional diffusion equations,” Physica A, vol. 278, no. 1-2, pp. 107–125, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M. Li, “Approximating ideal filters by systems of fractional order,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 365054, 6 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. Li and W. Zhao, “Essay on fractional Riemann-Liouville integral operator versus Mikusinski’s,” Mathematical Problems in Engineering, vol. 2013, Article ID 635412, 3 pages, 2013. View at Publisher · View at Google Scholar
  12. M. Li and W. Zhao, “Solving the Abel's type integral equation with the Mikusinski's operator of fractional order,” Advances in Mathematical Physics, vol. 2013, Article ID 806984, 4 pages, 2013. View at Publisher · View at Google Scholar
  13. J. A. Tenreiro MacHado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. View at Publisher · View at Google Scholar
  14. H. Sun, Y. Chen, and W. Chen, “Random-order fractional differential equation models,” Signal Processing, vol. 91, no. 3, pp. 525–530, 2011.
  15. C. Cattani, “Fractional calculus and Shannon wavelet,” Mathematical Problems in Engineering, vol. 2012, Article ID 502812, 26 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  16. C. Cattani, G. Pierro, and G. Altieri, “Entropy and multifractality for the myeloma multiple TET 2 gene,” Mathematical Problems in Engineering, vol. 2012, Article ID 193761, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. B. Baeumer, M. Kovács, and M. M. Meerschaert, “Numerical solutions for fractional reaction-diffusion equations,” Computers & Mathematics with Applications, vol. 55, no. 10, pp. 2212–2226, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. C.-M. Chen and F. Liu, “A numerical approximation method for solving a three-dimensional space Galilei invariant fractional advection-diffusion equation,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 219–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. 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
  20. C.-M. Chen, F. Liu, V. Anh, and I. Turner, “Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation,” SIAM Journal on Scientific Computing, vol. 32, no. 4, pp. 1740–1760, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. C.-m. Chen, F. Liu, and K. Burrage, “Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 754–769, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term,” Journal of Computational Physics, vol. 240, pp. 36–48, 2013.
  23. G.-H. Gao, Z.-Z. Sun, and Y.-N. Zhang, “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions,” Journal of Computational Physics, vol. 231, no. 7, pp. 2865–2879, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. T. A. M. Langlands and B. I. Henry, “The accuracy and stability of an implicit solution method for the fractional diffusion equation,” Journal of Computational Physics, vol. 205, no. 2, pp. 719–736, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,” Journal of Computational Physics, vol. 225, no. 2, pp. 1533–1552, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. F. Liu, C. Yang, and K. Burrage, “Numerical method and analytic technique of the modified anomalous subdiffusion equation with a nonlinear source term,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 160–176, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  27. F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C.-M. Chen, F. Liu, V. Anh, and I. Turner, “Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation,” Mathematics of Computation, vol. 81, no. 277, pp. 345–366, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Q. Liu, F. Liu, I. Turner, and V. Anh, “Numerical simulation for the 3D seepage flow with fractional derivatives in porous media,” IMA Journal of Applied Mathematics, vol. 74, no. 2, pp. 201–229, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation,” Journal of Computational Physics, vol. 213, no. 1, pp. 205–213, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. 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
  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. 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
  34. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 1760–1781, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. J.-P. Bouchaud and A. Georges, “Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications,” Physics Reports, vol. 195, no. 4-5, pp. 127–293, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  36. P. Zhuang, F. Liu, V. Anh, and I. Turner, “Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process,” IMA Journal of Applied Mathematics, vol. 74, no. 5, pp. 645–667, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  37. Y. T. Gu, P. Zhuang, and F. Liu, “An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation,” Computer Modeling in Engineering & Sciences, vol. 56, no. 3, pp. 303–333, 2010. View at Zentralblatt MATH · View at MathSciNet
  38. 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 · View at MathSciNet
  39. Y.-N. Zhang, Z.-Z. Sun, and H.-W. Wu, “Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation,” SIAM Journal on Numerical Analysis, vol. 49, no. 6, pp. 2302–2322, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. G.-h. Gao and Z.-z. Sun, “A compact finite difference scheme for the fractional sub-diffusion equations,” Journal of Computational Physics, vol. 230, no. 3, pp. 586–595, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. X. Li and C. Xu, “A space-time spectral method for the time fractional diffusion equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2108–2131, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. X. Zhao and Z.-z. Sun, “A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions,” Journal of Computational Physics, vol. 230, no. 15, pp. 6061–6074, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. C.-M. Chen, F. Liu, I. Turner, and V. Anh, “Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation,” Numerical Algorithms, vol. 54, no. 1, pp. 1–21, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. P. Zhuang and F. Liu, “Finite difference approximation for two-dimensional time fractional diffusion equation,” Journal of Algorithms & Computational Technology, vol. 1, no. 1, pp. 1–15, 2007.
  45. Q. Liu, Y. T. Gu, P. Zhuang, F. Liu, and Y. F. Nie, “An implicit RBF meshless approach for time fractional diffusion equations,” Computational Mechanics, vol. 48, no. 1, pp. 1–12, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  46. H. Brunner, L. Ling, and M. Yamamoto, “Numerical simulations of 2D fractional subdiffusion problems,” Journal of Computational Physics, vol. 229, no. 18, pp. 6613–6622, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu, “Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions,” ANZIAM Journal. Electronic Supplement, vol. 52, pp. C395–C409, 2010. View at MathSciNet
  48. Q. Yang, I. Turner, F. Liu, and M. Ilić, “Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions,” SIAM Journal on Scientific Computing, vol. 33, no. 3, pp. 1159–1180, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. Q. X. Liu and F. W. Liu, “Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives,” Mathematica Numerica Sinica, vol. 31, no. 2, pp. 179–194, 2009. View at Zentralblatt MATH · View at MathSciNet
  50. S. Chen and F. Liu, “ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation,” Journal of Applied Mathematics and Computing, vol. 26, no. 1-2, pp. 295–311, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. M. M. Meerschaert, H.-P. Scheffler, and C. Tadjeran, “Finite difference methods for two-dimensional fractional dispersion equation,” Journal of Computational Physics, vol. 211, no. 1, pp. 249–261, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. H. Wang and K. Wang, “An O(Nlog2N) alternating-direction finite difference method for two-dimensional fractional diffusion equations,” Journal of Computational Physics, vol. 230, no. 21, pp. 7830–7839, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. C. Tadjeran and M. M. Meerschaert, “A second-order accurate numerical method for the two-dimensional fractional diffusion equation,” Journal of Computational Physics, vol. 220, no. 2, pp. 813–823, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. Y.-N. Zhang, Z.-Z. Sun, and X. Zhao, “Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation,” SIAM Journal on Numerical Analysis, vol. 50, no. 3, pp. 1535–1555, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. M. Cui, “Compact alternating direction implicit method for two-dimensional time fractional diffusion equation,” Journal of Computational Physics, vol. 231, no. 6, pp. 2621–2633, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  56. S. Chen, F. Liu, P. Zhuang, and V. Anh, “Finite difference approximations for the fractional Fokker-Planck equation,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 256–273, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  57. W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008/09. View at Publisher · View at Google Scholar · View at MathSciNet
  58. Z.-Z. Sun and X. Wu, “A fully discrete difference scheme for a diffusion-wave system,” Applied Numerical Mathematics, vol. 56, no. 2, pp. 193–209, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  59. Y.-N. Zhang and Z.-Z. Sun, “Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation,” Journal of Computational Physics, vol. 230, no. 24, pp. 8713–8728, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  60. A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet