Preferences help
enabled [disable] Abstract
Number of results
2013 | 11 | 6 | 646-665
Article title

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

Title variants
Languages of publication
Recently, the fractional Bloch-Torrey model has been used to study anomalous diffusion in the human brain. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions: Model-1 with the Riesz fractional derivative; Model-2 with the one-dimensional fractional Laplacian operator; and Model-3 with the two-dimensional fractional Laplacian operator.Firstly, we propose a spatially second-order accurate implicit numerical method for Model-1 whereby we discretize the Riesz fractional derivative using a fractional centered difference. We consider a finite domain where the time and space derivatives are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Secondly, we utilize the matrix transfer technique for solving Model-2 and Model-3. Finally, some numerical results are given to show the behaviours of these three models especially on varying domain sizes with zero Dirichlet boundary conditions.
Physical description
1 - 6 - 2013
9 - 10 - 2013
  • [1] I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)
  • [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006)
  • [3] R. Hilfer, Applications of fractional calculus in physics (World Scientific Publishing, New York, 2000)[Crossref]
  • [4] R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)[Crossref]
  • [5] G. Zaslavsky, Phys. Rep. 371, 461 (2002)[Crossref]
  • [6] R. Gorenflo, F. Mainardi, D. Moretti, P. Paradisi, Nonlinear Dynam. 29, 129 (2002)[Crossref]
  • [7] M. M. Meerschaert, C. Tadjeran, J. Comput. Appl. Math. 172, 65 (2004)[Crossref]
  • [8] Q. Yu, F. Liu, V. Anh, I. Turner, Internat. J. Numer. Meth. Eng. 74, 138 (2008)[Crossref]
  • [9] I. Podlubny, A. Chechkin, T. Skovranek, Y. Q. Chen, B. M. V. Jara, J. Comput. Phys. 228, 3137 (2009)[Crossref]
  • [10] K. Diethelm, The Analysis of Fractional Differential Equations: An Application Oriented Exposition Using Differential Operators of Caputo Type (Springer, Berlin, 2010)[Crossref]
  • [11] A. Abragam, Principles of nuclear magnetism (Oxford University Press, New York, 2002)
  • [12] H. C. Torrey, Phys. Rev. 104, 563 (1956)[Crossref]
  • [13] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. Magin, Comput. Math. Appl. 64, 3367 (2012)[Crossref]
  • [14] R. Magin, X. Feng, D. Baleanu, Concept. Magnetic Res. A 34, 16 (2009)[Crossref]
  • [15] R. Magin, X. Feng, D. Baleanu, Magn. Reson. Engr. 34, 16 (2009)[Crossref]
  • [16] I. Petráš, Comput. Math. Appl. 61, 341 (2011)[Crossref]
  • [17] R. Magin, W. Li, M. Velasco, J. Trujillo, D. Reiter, A. Morgenstern, R. Spencer, J. Magn. Reson. 210, 184 (2011)[Crossref]
  • [18] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, R. Magin, Comput. Math. Appl. 61, 1355 (2011)[Crossref]
  • [19] V. M. Kenkre, E. Fukushima, D. Sheltraw, J. Magn. Reson. 128, 62 (1997)[Crossref]
  • [20] A. V. Barzykin, Phys. Rev. B 58, 14171 (1998)[Crossref]
  • [21] T. H. Jochimsena, A. Schäferb, R. Bammera, M. E. Moseleya, J. Magn. Reson. 180, 29 (2006)[Crossref]
  • [22] Q. Yu, F. Liu, I. Turner, K. Burrage, Appl. Math. Comput. 219, 4082 (2012)[Crossref]
  • [23] Q. Yu, F. Liu, I. Turner, K. Burrage, Philos. T. Roy. Soc. A 371, 20120150 (2013)[Crossref]
  • [24] R. Magin, O. Abdullah, D. Baleanu, X. Zhou, J. Magn. Reson. 190, 255 (2008)[Crossref]
  • [25] X. Zhou, Q. gao, O. Abdullah, and R. Magin, Magnet. Reson. Med. 63, 562 (2010)[Crossref]
  • [26] M. Ilic, F. Liu, I. Turner, V. Anh, Fract. Calc. Appl. Anal. 9, 333 (2006)
  • [27] Q. Yang, I. Turner, F. Liu, M. Ilic, SIAM J. Sci. Comput. 33, 1159 (2011)[Crossref]
  • [28] K. Burrage, N. Hale, D. Kay, SIAM J. SIAM J. Sci. Comput. 34, A2145 (2012)[Crossref]
  • [29] A. Bueno-Orovio, D. Kay, K. Burrage, J. Comput. Phys. (2013) (in press)
  • [30] P. Zhuang, F. Liu, V. Anh, I. Turner, SIAM J. Numer. Anal. 47, 1760 (2009)[Crossref]
  • [31] Q. Yang, F. Liu, I. Turner, Appl. Math. Model. 34, 200 (2010)[Crossref]
  • [32] S. Shen, F. Liu, V. Anh, Numer. Algorithms 56, 383 (2011)[Crossref]
  • [33] M. D. Ortigueira, Int. J. Math. Stat. Sci. 2006, 48391 (2006)
  • [34] C. Celik, M. Duman, J. Comput. Phys. 231, 1743 (2012)[Crossref]
  • [35] F. Liu, K. Burrage, Comput. Math. Appl. 62, 822 (2011)[Crossref]
  • [36] F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Appl. Math. Comput. 191, 12 (2007)[Crossref]
Document Type
Publication order reference
YADDA identifier
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.