Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation

[1]  Yifa Tang,et al.  Space-time finite element method for the distributed-order time fractional reaction diffusion equations , 2020 .

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

[3]  Jiwei Zhang,et al.  A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems , 2018, SIAM J. Numer. Anal..

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

[5]  X. Zhu,et al.  Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains , 2016, J. Comput. Phys..

[6]  Meng Li,et al.  Nonconforming Virtual Element Method for the Time Fractional Reaction–Subdiffusion Equation with Non-smooth Data , 2019, Journal of Scientific Computing.

[7]  Hong-lin Liao,et al.  Adaptive second-order Crank-Nicolson time-stepping schemes for time fractional molecular beam epitaxial growth models , 2019, SIAM J. Sci. Comput..

[8]  YangQuan Chen,et al.  A new collection of real world applications of fractional calculus in science and engineering , 2018, Commun. Nonlinear Sci. Numer. Simul..

[9]  Fawang Liu,et al.  A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation , 2014, Numerical Algorithms.

[10]  Changpin Li,et al.  Numerical methods for fractional partial differential equations , 2018, Int. J. Comput. Math..

[11]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[12]  Jiwei Zhang,et al.  Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations , 2018, Journal of Scientific Computing.

[13]  Fawang Liu,et al.  Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains , 2019, Comput. Math. Appl..

[14]  Max D. Gunzburger,et al.  Θ Schemes for Finite Element Discretization of the Space-time Fractional Diffusion Equations , 2015, J. Comput. Appl. Math..

[15]  I. Turner,et al.  Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains , 2018, Applied Mathematical Modelling.

[16]  Anatoly A. Alikhanov,et al.  A new difference scheme for the time fractional diffusion equation , 2014, J. Comput. Phys..

[17]  Zhimin Zhang,et al.  Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations , 2015, 1511.03453.

[18]  Alfonso Bueno-Orovio,et al.  Exact solutions to the fractional time-space Bloch-Torrey equation for magnetic resonance imaging , 2017, Commun. Nonlinear Sci. Numer. Simul..

[19]  Hong Sun,et al.  Some high order difference schemes for the space and time fractional Bloch-Torrey equations , 2016, Appl. Math. Comput..

[20]  Xin Li,et al.  A second-order fast compact scheme with unequal time-steps for subdiffusion problems , 2020, Numerical Algorithms.

[21]  Aiguo Xiao,et al.  Space-time finite element method for the multi-term time-space fractional diffusion equation on a two-dimensional domain , 2019, Comput. Math. Appl..

[22]  Hong-lin Liao,et al.  Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation , 2019, Advances in Computational Mathematics.

[23]  Yue Zhao,et al.  Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation , 2017, J. Comput. Phys..

[24]  Chuanju Xu,et al.  Müntz Spectral Methods for the Time-Fractional Diffusion Equation , 2017, Comput. Methods Appl. Math..

[25]  Mehdi Dehghan,et al.  An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations , 2018, Applied Numerical Mathematics.

[26]  I. Podlubny Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[27]  Viktor Vegh,et al.  Tissue microstructure features derived from anomalous diffusion measurements in magnetic resonance imaging , 2017, Human brain mapping.

[28]  Ali H. Bhrawy,et al.  A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations , 2015, J. Comput. Phys..

[29]  Fawang Liu,et al.  A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements , 2018, Commun. Nonlinear Sci. Numer. Simul..

[30]  Jiye Yang,et al.  Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations , 2015, J. Comput. Phys..

[31]  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..

[32]  Zhi-Zhong Sun,et al.  A High-Order Difference Scheme for the Space and Time Fractional Bloch–Torrey Equation , 2018, Comput. Methods Appl. Math..

[33]  M. Stynes,et al.  Optimal $$L^\infty (L^2)$$L∞(L2) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem , 2018 .

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

[35]  Jiwei Zhang,et al.  Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations , 2018, SIAM J. Numer. Anal..

[36]  Fawang Liu,et al.  A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain , 2017 .

[37]  Kassem Mustapha,et al.  A Discontinuous Petrov-Galerkin Method for Time-Fractional Diffusion Equations , 2014, SIAM J. Numer. Anal..

[38]  Han Zhou,et al.  A class of second order difference approximations for solving space fractional diffusion equations , 2012, Math. Comput..