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

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.

[1]  Fawang Liu,et al.  Novel techniques in parameter estimation for fractional dynamical models arising from biological systems , 2011, Comput. Math. Appl..

[2]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[3]  A. Barzykin.  EXACT SOLUTION OF THE TORREY-BLOCH EQUATION FOR A SPIN ECHO IN RESTRICTED GEOMETRIES , 1998 .

[4]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[5]  Fawang Liu,et al.  Solving linear and non‐linear space–time fractional reaction–diffusion equations by the Adomian decomposition method , 2008 .

[6]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[7]  J. Koenderink.  Q… , 2014, Les noms officiels des communes de Wallonie, de Bruxelles-Capitale et de la communaute germanophone.

[8]  Dumitru Baleanu,et al.  Fractional Bloch equation with delay , 2011, Comput. Math. Appl..

[9]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[10]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[11]  Fawang Liu,et al.  Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation , 2011, Numerical Algorithms.

[12]  Eiichi Fukushima,et al.  Simple Solutions of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion , 1997 .

[13]  Richard L. Magin,et al.  Transient chaos in fractional Bloch equations , 2012, Comput. Math. Appl..

[14]  Xiaohong Joe Zhou,et al.  Studies of anomalous diffusion in the human brain using fractional order calculus , 2010, Magnetic resonance in medicine.

[15]  Andrew G. Glen,et al.  APPL , 2001 .

[16]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[17]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[18]  R. Gorenflo,et al.  Time Fractional Diffusion: A Discrete Random Walk Approach , 2002 .

[19]  M. Moseley,et al.  Efficient simulation of magnetic resonance imaging with Bloch-Torrey equations using intra-voxel magnetization gradients. , 2006, Journal of magnetic resonance.

[20]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[21]  Ivo Petrás,et al.  Modeling and numerical analysis of fractional-order Bloch equations , 2011, Comput. Math. Appl..

[22]  H. C. Torrey.  Bloch Equations with Diffusion Terms , 1956 .

[23]  I. Turner,et al.  Numerical Approximation of a Fractional-In-Space Diffusion Equation, I , 2005 .

[24]  Juan Trujillo,et al.  Anomalous NMR relaxation in cartilage matrix components and native cartilage: fractional-order models. , 2011, Journal of magnetic resonance.

[25]  Fawang Liu,et al.  Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions , 2011, SIAM J. Sci. Comput..

[26]  Xiaohong Joe Zhou,et al.  Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. , 2008, Journal of magnetic resonance.

[27]  H. Carr,et al.  The Principles of Nuclear Magnetism , 1961 .

[28]  I. Podlubny.  Fractional differential equations , 1998 .

[29]  Yangquan Chen,et al.  Matrix approach to discrete fractional calculus II: Partial fractional differential equations , 2008, J. Comput. Phys..

[30]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

[31]  Nicholas Hale,et al.  An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations , 2012, SIAM J. Sci. Comput..

[32]  Fawang Liu,et al.  Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[33]  Fawang Liu,et al.  A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D , 2012, Appl. Math. Comput..

[34]  Richard L. Magin,et al.  Solving the fractional order Bloch equation , 2009 .

[35]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[36]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[37]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..