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September  2020, 25(9): 3357-3371. doi: 10.3934/dcdsb.2020065

Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations

Hong Lu 1, , Ji Li 2, and  Mingji Zhang 3,4,,

1. 

School of Mathematics and Statistics, Shandong University, Weihai, Shandong 264209, China
2. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China
3. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
4. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

* Corresponding author: Mingji Zhang

Received  May 2019 Revised  November 2019 Published  April 2020

Fund Project: This work was supported the NSF of China (No. 11601278) and MPS Simons Foundation of USA (No. 628308)

In this paper, we consider the numerical approximation of the space and time fractional Bloch-Torrey equations. A fully discrete spectral scheme based on a finite difference method in the time direction and a Galerkin-Legendre spectral method in the space direction is developed. In order to reduce the amount of computation, an alternating direction implicit (ADI) spectral scheme is proposed. Then the stability and convergence analysis are rigorously established. Finally, numerical results are presented to support our theoretical analysis.

Keywords: Space and time fractional Bloch-Torrey equations, Legendre spectral method, alternating direction implicit method, stability and convergence.
Mathematics Subject Classification: Primary: 26A33, 65M70; Secondary: 65M12.
Citation: Hong Lu, Ji Li, Mingji Zhang. Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3357-3371. doi: 10.3934/dcdsb.2020065
References:
[1]

A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424-438.  doi: 10.1016/j.jcp.2014.09.031.  Google Scholar

[2]

C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, 5, North-Holland, Amsterdam, 1997, 209–485. doi: 10.1016/S1570-8659(97)80003-8.  Google Scholar

[3]

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), 264-279.  doi: 10.1016/j.jcp.2014.06.031.  Google Scholar

[4]

A. Bueno-Orovio and K. Burrage, Exact solutions to the fractional time-space Bloch-Torrey equation for magnetic resonance imaging, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 91-109.  doi: 10.1016/j.cnsns.2017.04.013.  Google Scholar

[5]

V. J. Ervinand and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in $\mathbb{R}^d$, Numer. Methods Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.  Google Scholar

[6]

B. I. Henry, T. A. M. Langlands and S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74 (2006), 15pp. doi: 10.1103/PhysRevE.74.031116.  Google Scholar

[7]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[8]

R. L. Magin, O. Abdullah, D. Baleanu and X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magn. Reson., 190 (2008), 255-270.  doi: 10.1016/j.jmr.2007.11.007.  Google Scholar

[9]

R. Magin, X. Feng and D. Baleanu, Solving the fractional order Bloch equation, Concepts Magn. Reson., Part A, 34A (2009), 16-23.  doi: 10.1002/cmr.a.20129.  Google Scholar

[10]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[11]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[12]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev space, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[13] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[14]

J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations, Ph.D. thesis, Clemson University in Clemson, SC, 2004.  Google Scholar

[15]

J. Shen, Efficient spectral-Galerkin method. I. Direct solvers for second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.  Google Scholar

[16]

J. Shen, T. Tang and L. L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[17]

Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 18pp. doi: 10.1098/rsta.2012.0150.  Google Scholar

[18]

Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D, Cent. Eur. J. Phys., 11 (2013), 646-665.  doi: 10.2478/s11534-013-0220-6.  Google Scholar

[19]

Q. Yu, F. Liu, I. Turner and K. Burrage, A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comput., 219 (2012), 4082-4095.  doi: 10.1016/j.amc.2012.10.056.  Google Scholar

[20]

Y. Zhao, W. Bu, X. Zhao and Y. Tang, Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation, J. Comput. Phys., 350 (2017), 117-135.  doi: 10.1016/j.jcp.2017.08.051.  Google Scholar

show all references

References:
[1]

A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280 (2015), 424-438.  doi: 10.1016/j.jcp.2014.09.031.  Google Scholar

[2]

C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, 5, North-Holland, Amsterdam, 1997, 209–485. doi: 10.1016/S1570-8659(97)80003-8.  Google Scholar

[3]

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), 264-279.  doi: 10.1016/j.jcp.2014.06.031.  Google Scholar

[4]

A. Bueno-Orovio and K. Burrage, Exact solutions to the fractional time-space Bloch-Torrey equation for magnetic resonance imaging, Commun. Nonlinear Sci. Numer. Simul., 52 (2017), 91-109.  doi: 10.1016/j.cnsns.2017.04.013.  Google Scholar

[5]

V. J. Ervinand and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in $\mathbb{R}^d$, Numer. Methods Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.  Google Scholar

[6]

B. I. Henry, T. A. M. Langlands and S. L. Wearne, Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations, Phys. Rev. E, 74 (2006), 15pp. doi: 10.1103/PhysRevE.74.031116.  Google Scholar

[7]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[8]

R. L. Magin, O. Abdullah, D. Baleanu and X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magn. Reson., 190 (2008), 255-270.  doi: 10.1016/j.jmr.2007.11.007.  Google Scholar

[9]

R. Magin, X. Feng and D. Baleanu, Solving the fractional order Bloch equation, Concepts Magn. Reson., Part A, 34A (2009), 16-23.  doi: 10.1002/cmr.a.20129.  Google Scholar

[10]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[11]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[12]

E. D. Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev space, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[13] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[14]

J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations, Ph.D. thesis, Clemson University in Clemson, SC, 2004.  Google Scholar

[15]

J. Shen, Efficient spectral-Galerkin method. I. Direct solvers for second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.  Google Scholar

[16]

J. Shen, T. Tang and L. L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[17]

Q. Yu, F. Liu, I. Turner and K. Burrage, Stability and convergence of an implicit numerical method for the space and time fractional Bloch-Torrey equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013), 18pp. doi: 10.1098/rsta.2012.0150.  Google Scholar

[18]

Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D, Cent. Eur. J. Phys., 11 (2013), 646-665.  doi: 10.2478/s11534-013-0220-6.  Google Scholar

[19]

Q. Yu, F. Liu, I. Turner and K. Burrage, A computationally effective alternating direction method for the space and time fractional Bloch-Torrey equation in 3-D, Appl. Math. Comput., 219 (2012), 4082-4095.  doi: 10.1016/j.amc.2012.10.056.  Google Scholar

[20]

Y. Zhao, W. Bu, X. Zhao and Y. Tang, Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation, J. Comput. Phys., 350 (2017), 117-135.  doi: 10.1016/j.jcp.2017.08.051.  Google Scholar

Figure 1.  The left graph with $ \alpha = 0.3,\ \beta = 0.6 $ while the right one with $ \alpha = 0.8,\ \beta = 0.75 $ for Example 6.1
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Figure 2.  $ \alpha = 0.5,\ \beta = 0.8 $ for Example 6.2
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Table 1.  $ L^2 $ errors and convergence rates for Example 6.1
$ \tau $ $ \alpha=0.3\ \beta=0.6 $ Con. rate $ \alpha=0.8\ \beta=0.75 $ Con. rate
1/10 1.4361e-004 2.1596 1.5512e-004 2.0231
1/20 3.2143e-005 2.0832 3.8163e-005 2.0046
1/40 7.5856e-006 2.0480 9.5104e-006 1.9942
1/80 1.8343e-006 1.9814 2.3872e-006 1.9743
1/160 4.6451e-007 1.7299 6.0791e-007 1.8456
1/320 1.4003e-007 - 1.6914e-007 -
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$ \tau $ $ \alpha=0.3\ \beta=0.6 $ Con. rate $ \alpha=0.8\ \beta=0.75 $ Con. rate
1/10 1.4361e-004 2.1596 1.5512e-004 2.0231
1/20 3.2143e-005 2.0832 3.8163e-005 2.0046
1/40 7.5856e-006 2.0480 9.5104e-006 1.9942
1/80 1.8343e-006 1.9814 2.3872e-006 1.9743
1/160 4.6451e-007 1.7299 6.0791e-007 1.8456
1/320 1.4003e-007 - 1.6914e-007 -
Table 2.  $ L^2 $ errors and convergence rates for Example 6.2
$ \tau $ $ \alpha=0.3\ \beta=0.6 $ Con. rate $ \alpha=0.8\ \beta=0.75 $ Con. rate
1/10 1.7946e-003 2.1361 1.6438e-003 2.0271
1/20 4.0827e-004 2.0638 4.0329e-004 2.0098
1/40 9.7653e-005 2.0361 1.0014e-005 1.9949
1/80 2.3810e-005 2.0142 2.5123e-005 1.9935
1/160 5.8943e-006 1.9872 6.3092e-006 1.9763
1/320 1.4867e-006 - 1.6034e-006 -
Table Options

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$ \tau $ $ \alpha=0.3\ \beta=0.6 $ Con. rate $ \alpha=0.8\ \beta=0.75 $ Con. rate
1/10 1.7946e-003 2.1361 1.6438e-003 2.0271
1/20 4.0827e-004 2.0638 4.0329e-004 2.0098
1/40 9.7653e-005 2.0361 1.0014e-005 1.9949
1/80 2.3810e-005 2.0142 2.5123e-005 1.9935
1/160 5.8943e-006 1.9872 6.3092e-006 1.9763
1/320 1.4867e-006 - 1.6034e-006 -
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