
-
Previous Article
Dynamical behavior of a rotavirus disease model with two strains and homotypic protection
- DCDS-B Home
- This Issue
-
Next Article
Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms
Spectral methods for two-dimensional space and time fractional Bloch-Torrey equations
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, 1993. |
[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. |
[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.
![]() |
[14] |
J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations, Ph.D. thesis, Clemson University in Clemson, SC, 2004. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, Inc., New York, 1993. |
[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. |
[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.
![]() |
[14] |
J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations, Ph.D. thesis, Clemson University in Clemson, SC, 2004. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |


Con. rate | 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 | - |
Con. rate | 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 | - |
Con. rate | 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 | - |
Con. rate | 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 | - |
[1] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[2] |
Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 |
[3] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[4] |
Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 |
[5] |
Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021068 |
[6] |
Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 |
[7] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[8] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[9] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[10] |
María J. Garrido-Atienza, Bohdan Maslowski, Jana Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 |
[11] |
Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021005 |
[12] |
Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 |
[13] |
Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017 |
[14] |
Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002 |
[15] |
Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263 |
[16] |
Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149 |
[17] |
Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021014 |
[18] |
Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200 |
[19] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[20] |
Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]