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Data publikacji: 25 maja 2024

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1

Yu, Qiang, Fawang Liu, Ian Turner i Kevin Burrage. "Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, nr 1990 (13.05.2013): 20120150. http://dx.doi.org/10.1098/rsta.2012.0150.

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Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations extend our knowledge of biological materials through a detailed analysis of relaxation time distribution and water diffusion heterogeneity. Here, the dynamic models become more complex as they attempt to correlate new data with a multiplicity of tissue compartments, where processes are often anisotropic. Anomalous diffusion in the human brain using fractional-order calculus has been investigated. Recently, a new diffusion model was proposed by solving the Bloch–Torrey equation using fractional-order calculus with respect to time and space. However, effective numerical methods and supporting error analyses for the fractional Bloch–Torrey equation are still limited. In this paper, the space and time fractional Bloch–Torrey equation (ST-FBTE) in both fractional Laplacian and Riesz derivative form is considered. The time and space derivatives in the ST-FBTE are replaced by the Caputo and the sequential Riesz fractional derivatives, respectively. Firstly, we derive an analytical solution for the ST-FBTE in fractional Laplacian form with initial and boundary conditions on a finite domain. Secondly, we propose an implicit numerical method (INM) for the ST-FBTE based on the Riesz form, and the stability and convergence of the INM are investigated. We prove that the INM for the ST-FBTE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis.
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2

Caubet, Fabien, Houssem Haddar, Jing-Rebecca li i Dang Van Nguyen. "New transmission condition accounting for diffusion anisotropy in thin layers applied to diffusion MRI". ESAIM: Mathematical Modelling and Numerical Analysis 51, nr 4 (30.06.2017): 1279–301. http://dx.doi.org/10.1051/m2an/2016060.

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The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers.
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3

Rotkopf, L. T., E. Wehrse, F. T. Kurz, H. P. Schlemmer i C. H. Ziener. "Efficient discretization scheme for semi-analytical solutions of the Bloch-Torrey equation". Journal of Magnetic Resonance Open 6-7 (czerwiec 2021): 100010. http://dx.doi.org/10.1016/j.jmro.2021.100010.

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4

Seroussi, Inbar, Denis S. Grebenkov, Ofer Pasternak i Nir Sochen. "Microscopic interpretation and generalization of the Bloch-Torrey equation for diffusion magnetic resonance". Journal of Magnetic Resonance 277 (kwiecień 2017): 95–103. http://dx.doi.org/10.1016/j.jmr.2017.01.018.

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5

Magin, Richard L., Osama Abdullah, Dumitru Baleanu i Xiaohong Joe Zhou. "Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation". Journal of Magnetic Resonance 190, nr 2 (luty 2008): 255–70. http://dx.doi.org/10.1016/j.jmr.2007.11.007.

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6

Zhu, Yun, i Zhi-Zhong Sun. "A High-Order Difference Scheme for the Space and Time Fractional Bloch–Torrey Equation". Computational Methods in Applied Mathematics 18, nr 1 (1.01.2018): 147–64. http://dx.doi.org/10.1515/cmam-2017-0034.

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AbstractIn this paper, a high-order difference scheme is proposed for an one-dimensional space and time fractional Bloch–Torrey equation. A third-order accurate formula, based on the weighted and shifted Grünwald–Letnikov difference operators, is used to approximate the Caputo fractional derivative in temporal direction. For the discretization of the spatial Riesz fractional derivative, we approximate the weighed values of the Riesz fractional derivative at three points by the fractional central difference operator. The unique solvability, unconditional stability and convergence of the scheme are rigorously proved by the discrete energy method. The convergence order is 3 in time and 4 in space in {L_{1}(L_{2})}-norm. Two numerical examples are implemented to testify the accuracy of the numerical solution and the efficiency of the difference scheme.
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7

Xu, Tao, Shujuan Lü i Haonan Li. "An implicit numerical method for the space-time variable-order fractional Bloch-Torrey equation". Journal of Physics: Conference Series 1039 (czerwiec 2018): 012008. http://dx.doi.org/10.1088/1742-6596/1039/1/012008.

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8

Barzykin, A. V. "Exact solution of the Torrey-Bloch equation for a spin echo in restricted geometries". Physical Review B 58, nr 21 (1.12.1998): 14171–74. http://dx.doi.org/10.1103/physrevb.58.14171.

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9

Beltrachini, Leandro, Zeike A. Taylor i Alejandro F. Frangi. "A parametric finite element solution of the generalised Bloch–Torrey equation for arbitrary domains". Journal of Magnetic Resonance 259 (październik 2015): 126–34. http://dx.doi.org/10.1016/j.jmr.2015.08.008.

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10

Zhao, Yue, Weiping Bu, Xuan Zhao i Yifa Tang. "Galerkin finite element method for two-dimensional space and time fractional Bloch–Torrey equation". Journal of Computational Physics 350 (grudzień 2017): 117–35. http://dx.doi.org/10.1016/j.jcp.2017.08.051.

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11

Bueno-Orovio, Alfonso, i Kevin Burrage. "Exact solutions to the fractional time-space Bloch–Torrey equation for magnetic resonance imaging". Communications in Nonlinear Science and Numerical Simulation 52 (listopad 2017): 91–109. http://dx.doi.org/10.1016/j.cnsns.2017.04.013.

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12

Zhang, Mengchen, Fawang Liu, Ian W. Turner i Vo V. Anh. "Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients". Applied Mathematical Modelling 129 (maj 2024): 169–90. http://dx.doi.org/10.1016/j.apm.2024.01.050.

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13

Nguyen, Dang Van, Jing-Rebecca Li, Denis Grebenkov i Denis Le Bihan. "A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging". Journal of Computational Physics 263 (kwiecień 2014): 283–302. http://dx.doi.org/10.1016/j.jcp.2014.01.009.

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14

Yang, Jiye, Yuqing Li i Zhiyong Liu. "A finite difference/Kansa method for the two-dimensional time and space fractional Bloch-Torrey equation". Computers & Mathematics with Applications 156 (luty 2024): 1–15. http://dx.doi.org/10.1016/j.camwa.2023.12.007.

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15

Sevilla, F. J., i V. M. Kenkre. "Theory of the spin echo signal in NMR microscopy: analytic solutions of a generalized Torrey–Bloch equation". Journal of Physics: Condensed Matter 19, nr 6 (22.01.2007): 065113. http://dx.doi.org/10.1088/0953-8984/19/6/065113.

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16

Song, J., Q. Yu, F. Liu i I. Turner. "A spatially second-order accurate implicit numerical method for the space and time fractional Bloch-Torrey equation". Numerical Algorithms 66, nr 4 (17.09.2013): 911–32. http://dx.doi.org/10.1007/s11075-013-9768-x.

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17

Yu, Q., F. Liu, I. Turner i K. Burrage. "A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D". Applied Mathematics and Computation 219, nr 8 (grudzień 2012): 4082–95. http://dx.doi.org/10.1016/j.amc.2012.10.056.

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18

Liao, Mingzhao, Yu Liu, Yafeng Li, Liangliang Hu, Haihong Niu i Jinzhang Xu. "Simulation of Diffusion Magnetic Resonance Based on Chain Method". Journal of Physics: Conference Series 2607, nr 1 (1.10.2023): 012002. http://dx.doi.org/10.1088/1742-6596/2607/1/012002.

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Abstract Magnetic resonance diffusion imaging is an important tool for pathology research, and the cost of instrument development is extremely high. The construction of simulation models can optimize the parameters of key components of the instrument on the numerical platform, which can greatly reduce the R&D cost and improve the success rate. In this paper, a chain magnetic resonance simulation calculation method is obtained based on equation Bloch-Torrey to simulate the signal change law caused by gradient coding and diffusion characteristics, and MATLAB is used to construct a numerical imitation model and write a simulation program. Based on this calculation method, the diffusion-weighted imaging, diffusion coefficient imaging and diffusion tensor imaging simulation experiments were carried out by using the spin-echo (SE) diffusion-weighted sequence, and the images and results were reconstructed by combining the anti-Fourier transform. Experiments show that the simulated images can accurately reflect the set simulation model, reconstruct the diffusion coefficient and tensor characteristics, and the chain magnetic resonance simulation calculation method deduced in this paper.
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19

Bu, Weiping, Yanmin Zhao i Chen Shen. "Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation". Applied Mathematics and Computation 398 (czerwiec 2021): 125985. http://dx.doi.org/10.1016/j.amc.2021.125985.

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20

Doucette, Jonathan, Luxi Wei, Enedino Hernández-Torres, Christian Kames, Nils D. Forkert, Rasmus Aamand, Torben E. Lund, Brian Hansen i Alexander Rauscher. "Rapid solution of the Bloch-Torrey equation in anisotropic tissue: Application to dynamic susceptibility contrast MRI of cerebral white matter". NeuroImage 185 (styczeń 2019): 198–207. http://dx.doi.org/10.1016/j.neuroimage.2018.10.035.

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21

Magin, Richard L., Hamid Karani, Shuhong Wang i Yingjie Liang. "Fractional Order Complexity Model of the Diffusion Signal Decay in MRI". Mathematics 7, nr 4 (12.04.2019): 348. http://dx.doi.org/10.3390/math7040348.

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Fractional calculus models are steadily being incorporated into descriptions of diffusion in complex, heterogeneous materials. Biological tissues, when viewed using diffusion-weighted, magnetic resonance imaging (MRI), hinder and restrict the diffusion of water at the molecular, sub-cellular, and cellular scales. Thus, tissue features can be encoded in the attenuation of the observed MRI signal through the fractional order of the time- and space-derivatives. Specifically, in solving the Bloch-Torrey equation, fractional order imaging biomarkers are identified that connect the continuous time random walk model of Brownian motion to the structure and composition of cells, cell membranes, proteins, and lipids. In this way, the decay of the induced magnetization is influenced by the micro- and meso-structure of tissues, such as the white and gray matter of the brain or the cortex and medulla of the kidney. Fractional calculus provides new functions (Mittag-Leffler and Kilbas-Saigo) that characterize tissue in a concise way. In this paper, we describe the exponential, stretched exponential, and fractional order models that have been proposed and applied in MRI, examine the connection between the model parameters and the underlying tissue structure, and explore the potential for using diffusion-weighted MRI to extract biomarkers associated with normal growth, aging, and the onset of disease.
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22

Zhang, Mengchen, Fawang Liu, Ian W. Turner, Vo V. Anh i Libo Feng. "A finite volume method for the two-dimensional time and space variable-order fractional Bloch-Torrey equation with variable coefficients on irregular domains". Computers & Mathematics with Applications 98 (wrzesień 2021): 81–98. http://dx.doi.org/10.1016/j.camwa.2021.06.013.

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23

Akgul, Esra. "A novel method for the space and time fractional Bloch-Torrey equations". Thermal Science 22, Suppl. 1 (2018): 253–58. http://dx.doi.org/10.2298/tsci170715293a.

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Reproducing kernel technique was implemented to solve the fractional Bloch-Torrey equations. This efficient technique was used via some useful reproducing kernel functions, to obtain approximations to the exact solution in form of series solutions. A numerical example has been presented to prove efficiency of developed technique.
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24

Lu, Hong, Ji Li i Mingji Zhang. "Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations". Discrete & Continuous Dynamical Systems - B 25, nr 9 (2020): 3357–71. http://dx.doi.org/10.3934/dcdsb.2020065.

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25

Qin, Shanlin, Fawang Liu, Ian W. Turner, Qianqian Yang i Qiang Yu. "Modelling anomalous diffusion using fractional Bloch–Torrey equations on approximate irregular domains". Computers & Mathematics with Applications 75, nr 1 (styczeń 2018): 7–21. http://dx.doi.org/10.1016/j.camwa.2017.08.032.

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26

Choquet, Catherine, i Marie-Christine Néel. "Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium". Methodology and Computing in Applied Probability 22, nr 1 (5.12.2018): 49–74. http://dx.doi.org/10.1007/s11009-018-9688-2.

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27

Ding, Hengfei, i Changpin Li. "Numerical algorithms for the time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations". Numerical Methods for Partial Differential Equations 36, nr 4 (10.12.2019): 772–99. http://dx.doi.org/10.1002/num.22451.

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28

Sun, Hong, Zhi-zhong Sun i Guang-hua Gao. "Some high order difference schemes for the space and time fractional Bloch–Torrey equations". Applied Mathematics and Computation 281 (kwiecień 2016): 356–80. http://dx.doi.org/10.1016/j.amc.2016.01.044.

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29

Kenkre, V. M., Eiichi Fukushima i D. Sheltraw. "Simple Solutions of the Torrey–Bloch Equations in the NMR Study of Molecular Diffusion". Journal of Magnetic Resonance 128, nr 1 (wrzesień 1997): 62–69. http://dx.doi.org/10.1006/jmre.1997.1216.

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30

Jochimsen, Thies H., Andreas Schäfer, Roland Bammer i Michael E. Moseley. "Efficient simulation of magnetic resonance imaging with Bloch–Torrey equations using intra-voxel magnetization gradients". Journal of Magnetic Resonance 180, nr 1 (maj 2006): 29–38. http://dx.doi.org/10.1016/j.jmr.2006.01.001.

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31

Bu, Weiping, Yifa Tang, Yingchuan Wu i Jiye Yang. "Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations". Journal of Computational Physics 293 (lipiec 2015): 264–79. http://dx.doi.org/10.1016/j.jcp.2014.06.031.

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32

Xu, Tao, Fawang Liu, Shujuan Lü i Vo V. Anh. "Numerical approximation of 2D multi-term time and space fractional Bloch–Torrey equations involving the fractional Laplacian". Journal of Computational and Applied Mathematics 393 (wrzesień 2021): 113519. http://dx.doi.org/10.1016/j.cam.2021.113519.

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33

Chen, Ruige, Fawang Liu i Vo Anh. "A fractional alternating-direction implicit method for a multi-term time–space fractional Bloch–Torrey equations in three dimensions". Computers & Mathematics with Applications 78, nr 5 (wrzesień 2019): 1261–73. http://dx.doi.org/10.1016/j.camwa.2018.11.035.

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34

Yang, Zongze, Fawang Liu, Yufeng Nie i Ian Turner. "An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains". Journal of Computational Physics 408 (maj 2020): 109284. http://dx.doi.org/10.1016/j.jcp.2020.109284.

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35

Xu, Tao, Fawang Liu, Shujuan Lü i Vo V. Anh. "Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains". Computers & Mathematics with Applications 80, nr 12 (grudzień 2020): 3173–92. http://dx.doi.org/10.1016/j.camwa.2020.11.007.

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36

Dehghan, Mehdi, i Mostafa Abbaszadeh. "An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations". Applied Numerical Mathematics 131 (wrzesień 2018): 190–206. http://dx.doi.org/10.1016/j.apnum.2018.04.009.

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37

Liu, Fawang, Libo Feng, Vo Anh i Jing Li. "Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch–Torrey equations on irregular convex domains". Computers & Mathematics with Applications 78, nr 5 (wrzesień 2019): 1637–50. http://dx.doi.org/10.1016/j.camwa.2019.01.007.

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38

Sayevand, K., N. Ghanbari i I. Masti. "A robust computational framework for analyzing the Bloch–Torrey equation of fractional order". Computational and Applied Mathematics 40, nr 4 (3.05.2021). http://dx.doi.org/10.1007/s40314-021-01513-7.

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39

Mesgarani, H., Y. Esmaeelzade Aghdam i H. Tavakoli. "Numerical Simulation to Solve Two-Dimensional Temporal-Space Fractional Bloch–Torrey Equation Taken of the Spin Magnetic Moment Diffusion". International Journal of Applied and Computational Mathematics 7, nr 3 (14.05.2021). http://dx.doi.org/10.1007/s40819-021-01024-3.

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40

Feng, Libo, Fawang Liu i Vo V. Anh. "Galerkin finite element method for a two-dimensional tempered time-space fractional diffusion equation with application to a Bloch–Torrey equation retaining Larmor precession". Mathematics and Computers in Simulation, grudzień 2022. http://dx.doi.org/10.1016/j.matcom.2022.11.024.

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41

Zhang, Mengchen, i Fawang Liu. "Fractional diffusion model generalised by the distributed-order operator involving variable diffusion coefficients". ANZIAM Journal 64 (23.10.2023). http://dx.doi.org/10.21914/anziamj.v64.17959.

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The diffusion process plays a crucial role in various fields, such as fluid dynamics, microorganisms, heat conduction and food processing. Since molecular diffusion usually takes place in complex materials and disordered media, there still exist many challenges in describing the diffusion process in the real world. Fractional calculus is a powerful tool for modelling complex physical processes due to its non-local property. This research generalises a fractional diffusion model by using the distributed-order operator in time and the Riesz fractional derivative in space. Moreover, variable diffusion coefficients are introduced to better capture the diffusion complexity. The fractional diffusion model is discretised by the finite element method in space. The approximation of the distributed-order operator is implemented by Simpson’s rule and the L2-1σ formula. A numerical example is provided to verify the effectiveness of the proposed numerical methods. This generalised fractional diffusion model may offer more insights into characterising diffusion behaviours in complex and disordered media. References A. A. Alikhanov. A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280 (2015), pp. 424–438. doi: 10.1016/j.jcp.2014.09.031 R. L. Burden, J. D. Faires, and A. M. Burden. Numerical Analysis. Cengage Learning, 2015. url: https://au.cengage.com/c/isbn/9781305253667/ W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti. Applications of distributed-order fractional operators: A review. Entropy 23.1 (2021), p. 110. doi: 10.3390/e23010110 G. Gao, A. A. Alikhanov, and Z. Sun. The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73.1 (2017), pp. 93–121. doi: 10.1007/s10915-017-0407-x P. Gouze, Y. Melean, T. Le Borgne, M. Dentz, and J. Carrera. Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res. 44.11 (2016), pp. 2276–2283. doi: 10.1029/2007WR006690 S. E. Maier, Y. Sun, and R. V. Mulkern. Diffusion imaging of brain tumors. NMR Biomed. 23.7 (2010), pp. 849–864. doi: 10.1002/nbm.1544 M. M. Meerschaert. Fractional calculus, anomalous diffusion, and probability. Fractional Dynamics: Recent Advances. Ed. by J. Klafter, S. C. Lim, and R. Metzler. World Sci., 2011, pp. 265–284. doi: 10.1142/9789814340595_0011 I. Podlubny. Fractional differential equations. New York: Academic Press, 1999. url: https://shop.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9 S. Qin, F. Liu, and I. W. Turner. A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements. Commun. Nonlin. Sci. Numer. Sim. 56 (2018), pp. 270–286. doi: 10.1016/j.cnsns.2017.08.014 J. N. Reddy. An introduction to the finite element method. Vol. 1221. McGraw-Hill New York, 2004. url: https://www.accessengineeringlibrary.com/content/book/9781259861901 J. P. Roop. Variational solution of the fractional advection dispersion equation. PhD thesis. Clemson University, 2004. url: https://tigerprints.clemson.edu/arv_dissertations/1466/ J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, and A. F. Galhano. Some applications of fractional calculus in engineering. Math. Prob. Eng. 2010, 639801 (2010). doi: 10.1155/2010/639801 T. Xu, F. Liu, S. Lü, and V. V. Anh. Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains. In: Comput. Math. App. 80.12 (2020), pp. 3173–3192. doi: 10.1016/j.camwa.2020.11.007 O. C. Zienkiewicz, R. L. Taylor, P. Nithiarasu, and J. Zhu. The finite element method. Vol. 3. McGraw-Hill London, 1977
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42

Zhang, Mengchen, Fawang Liu, Ian W. Turner i Vo V. Anh. "A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch-Torrey equation with fractional Laplacian". Communications in Nonlinear Science and Numerical Simulation, lipiec 2022, 106666. http://dx.doi.org/10.1016/j.cnsns.2022.106666.

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43

Zhang, Mengchen, Fawang Liu, Ian Turner i Vo Anh. "A Vertex-Centred Finite Volume Method for the 3d Multi-Term Time and Space Fractional Bloch-Torrey Equation with Fractional Laplacian". SSRN Electronic Journal, 2022. http://dx.doi.org/10.2139/ssrn.4010730.

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44

Yang, Zheyi, Chengran Fang i Jing-Rebecca Li. "Incorporating interface permeability into the diffusion MRI signal representationwhile using impermeable Laplace eigenfunctions". Physics in Medicine & Biology, 14.08.2023. http://dx.doi.org/10.1088/1361-6560/acf022.

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Abstract {\bf Objective}\\ 
The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients 
acting on a permeable medium can be modeled \soutnew{}{by} the Bloch-Torrey partial differential equation. 
The diffusion MRI signal has a representation in the basis of the Laplace eigenfunctions of the medium. 
However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability. 
\\{\bf Approach}\\
In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable.
\\{\bf Main results}\\
We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method.
\\{\bf Significance}\\ 
Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future.
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Karaca, Yeliz. "Fractional Calculus Operators - Bloch-Torrey Partial Differential Equation - Artificial Neural Networks-Computational Complexity Modeling of the Micro-Macrostructural Brain Tissues with Diffusion MRI Signal Processing and Neuronal Multicomponents". Fractals, 8.09.2023. http://dx.doi.org/10.1142/s0218348x23402041.

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Yu, Qiang, Fawang Liu, Ian Turner i Kevin Burrage. "Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D". Open Physics 11, nr 6 (1.01.2013). http://dx.doi.org/10.2478/s11534-013-0220-6.

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AbstractRecently, 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.
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Liu, Yi, Xiaoyun Jiang i Fawang Liu. "The finite element method for the space fractional magnetohydrodynamic flow and heat transfer on an irregular domain". ANZIAM Journal 64 (1.11.2023). http://dx.doi.org/10.21914/anziamj.v64.17912.

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We consider the magnetohydrodynamic flow and heat transfer of a classical Newtonian fluid in a straight channel with fixed irregular cross section. A spatial fractional operator is introduced to modify the classical Fourier's law of thermal conduction, and we obtain the space fractional coupled model. With the help of the finite element method, the coupled model is solved numerically. Finally, a special numerical example is proposed to verify the stability and efficiency of the presented method. References S. Aman, Q. Al-Mdallal, and I. Khan. Heat transfer and second order slip effect on MHD flow of fractional Maxwell fluid in a porous medium. J. King Saud Uni. Sci. 32.1 (2020), pp. 450–458. doi: 10.1016/j.jksus.2018.07.007 W. Bu, Y. Tang, Y. Wu, and J. Yang. Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations. J. Comput. Phys. 293 (2015), pp. 264–279. doi: 10.1016/j.jcp.2014.06.031 X. Chi and H. Zhang. Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with Hall effects. App. Math. Lett. 120, 107312 (2021). doi: 10.1016/j.aml.2021.107312 T. G. Cowling. Magnetohydrodynamics. New York: Interscience, 1957 W. Fan, F. Liu, X. Jiang, and I. Turner. A novel unstructured mesh finite element method for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. Frac. Calc. Appl. Anal. 20.2 (2017), pp. 352–383. doi: 10.1515/fca-2017-0019 L. Feng, F Liu, I. Turner, Q. Yang, and P. Zhuang. Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains. Appl. Math. Model. 59 (2018), pp. 441–463. doi: 10.1016/j.apm.2018.01.044 L. Feng, F. Liu, I. Turner, and L. Zheng. Novel numerical analysis of multi-term time fractional viscoelastic non-Newtonian fluid models for simulating unsteady MHD Couette flow of a generalized Oldroyd-B fluid. Frac. Calc. Appl. Anal. 21.4 (2018), pp. 1073–1103. doi: 10.1515/fca-2018-0058 C. Li and A. Chen. Numerical methods for fractional partial differential equations. Int. J. Comp. Math. 95.6–7 (2018), pp. 1048–1099. doi: 10.1080/00207160.2017.1343941 C. Li and F. Zeng. Finite difference methods for fractional differential equations. Int. J. Bifur. Chaos 22.4, 1230014 (2012). doi: 10.1142/S0218127412300145 Y. Liu, X. Chi, H. Xu, and X. Jiang. Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid. App. Math. Comput. 430, 127255 (2022). doi: 10.1016/j.amc.2022.127255 H. Zhang, F. Liu, and V. Anh. Galerkin finite element approximation of symmetric space-fractional partial differential equations. App. Math. Comput. 217.6 (2010), pp. 2534–2545. doi: 10.1016/j.amc.2010.07.066 H. Zhang, F. Zeng, X. Jiang, and G. E. Karniadakis. Convergence analysis of the time-stepping numerical methods for time-fractional nonlinear subdiffusion equations. Frac. Calc. Appl. Anal. 25.2 (2022), pp. 453–487. doi: 10.1007/s13540-022-00022-6 M. Zhang, M. Shen, F. Liu, and H. Zhang. A new time and spatial fractional heat conduction model for Maxwell nanofluid in porous medium. Comput. Math. Appl. 78.5 (2019), pp. 1621–1636. doi: 10.1016/j.camwa.2019.01.006 L. Zheng, Y. Liu, and X. Zhang. Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlin. Anal.: Real World Appl. 13.2 (2012), pp. 513–523. doi: 10.1016/j.nonrwa.2011.02.016
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Liu, Fawang, Libo Feng i Vo Anh. "Numerical Approximation of the Multi-term Time-space Fractional Bloch-Torrey Equations on Irregular Convex Domains". SSRN Electronic Journal, 2018. http://dx.doi.org/10.2139/ssrn.3286005.

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