A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D

Applied Mathematics and Computation

Volumn 219, Issue 8, 2012, Pages 4082-4095

  Yu, Q ^a   Liu, F ^a   Turner, I ^a   Burrage, K ^a,b  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Alternating direction method; Convergence; Fractional Bloch Torrey equation; Fractional calculus; Implicit numerical method; Stability

Indexed keywords

ALTERNATING DIRECTION METHODS; CONVERGENCE; FRACTIONAL BLOCH-TORREY EQUATION; FRACTIONAL CALCULUS; IMPLICIT NUMERICAL METHOD;

COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL TECHNIQUES;

NUMERICAL METHODS;

EID: 84870058501     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.10.056     Document Type: Article

References (22)

1. R. Magin, O. Abdullah, D. Baleanu, and X. Zhou Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation Journal of Magnetic Resonance 190 2 2008 255 270
2. Q. Yu, F. Liu, I. Turner, K. Burrage, Analytical and numerical solutions of the space and time fractional Bloch-Torrey equation, in: Proceedings of the ASME 2011 International Design Engineering Technical conferences & Computers and Information in Engineering Conference, DECT2011/MESA-47613, 2011.
3. R. Magin, X. Feng, and D. Baleanu Solving the fractional order Bloch equation Concepts in Magnetic Resonance Part A 34A 1 2009 16 23
4. M. Velasco, J. Trujillo, D. Reiter, R. Spencer, W. Li, R. Magin, Anomalous fractional order models of NMR relaxation, in: Proceedings of FDA'10, the 4th IFAC Workshop fractional differentiation and its applications, vol. 1, Paper number FDA10-058, 2010.
5. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin Fractional Bloch equation with delay Computers and Mathematics with Applications 61 5 2011 1355 1365
6. V.V. Anh, and N.N. Leonenko Spectral analysis of fractional kinetic equations with random data J. Stat. Phys. 104 2001 1349 1387
7. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of fractional differential equations 2006 Elsevier Amsterdam
8. J.Jr. Douglas Alternating direction method for three space variables Numerische Mathematik 4 1 1961 41 63
9. J.Jr. Douglas, and J. Gunn A general formulation of alternating direction methods. Part I. Parabolic and hyperbolic problems Numerische Mathematik 6 1 1964 428 453
10. D.W. Peaceman, and H.H. Rachford The numerical solution of parabolic and elliptic differential equations Journal of the Society for Industrial and Applied Mathematics 3 1955 28 41
11. M.M. Meerschaert, H. Scheffler, and C. Tadjeran Finite difference methods for two-dimensional fractional dispersion equation Journal of Computational Physics 211 2006 249 261
12. S. Chen, and F. Liu ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation Journal of Applied Mathematics and Computing 26 2008 295 311
13. Y. Zhang, and Z. Sun Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation Journal of Computational Physics 230 24 2011 8713 8728
14. 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 74 2009 201 229
15. R. Gorenflo, and F. Mainardi Random walk models for space-fractional diffusion processes Fractional Calculus and Applied Analysis 1 2 1998 167 191
16. Q. Yang, F. Liu, and I. Turner Numerical methods for fractional partial differential equations with Riesz space fractional derivatives Applied Mathematical Modelling 34 1 2010 200 218
17. P. Zhuang, F. Liu, V. Anh, and I. Turner Numerical method for the variable-order fractional advection-diffusion equation with a nonlinear source term SIAM Journal of Numerical Analysis 47 3 2009 1760 1781
18. F. Liu, and K. Burrage Novel techniques in parameter estimation for fractional dynamical models arising from biological systems Computer & Mathematics with Application 62 2011 822 833
19. S. Shen, F. Liu, and V. Anh Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation Numerical Algorithms 56 3 2011 383 403
20. 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 91 2007 12 20
21. S. Chen, F. Liu, I. Turner, V. Anh, An implicit numerical method for the two-dimensional fractional percolation equation, Applied Mathematics and Computation (2013), in press.
22. J. Song, Q. Yu, F. Liu, I. Turner, A new implicit numerical method for the space and time fractional Bloch-Torrey equation, in: Proceedings of FDA'12, the 5th IFAC Symposium on fractional differentiation and its applications, vol. 1, Paper number FDA12-018, 2012.