References
- J. Cao and C. Li, Finite difference scheme for the time–space fractional diffusion equations, Cent. Eur. J. Phys. 11(10) (2013), pp. 1440–1456.
- C. Celik and M. Duman, Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys. 231 (2012), pp. 1743–1750. doi: 10.1016/j.jcp.2011.11.008
- H. Chen, D. Xu, and Y. Peng, An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation, Int. J. Comput. Math. 92(10) (2015), pp. 2178–2197. doi: 10.1080/00207160.2014.975694
- H. Chen, S. Gan, and D. Xu, A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation, J. Fract. Calc. Appl. 7(1) (2016), pp. 133–146.
- S. Chen, F. Liu, X. Jiang, I. Turner, and K. Burrage, Fast finite difference approximation for identifying parameters in a two-dimensional space-fractional nonlocal model with variable diffusivity coefficients, SIAM J. Numer. Anal. 54(2) (2016), pp. 606–624. doi: 10.1137/15M1019301
- K. Diethelm, N.J. Ford, and A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algor. 36 (2004), pp. 31–52. doi: 10.1023/B:NUMA.0000027736.85078.be
- J. Dixon and S. McKee, Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech. 66 (1986), pp. 535–544. doi: 10.1002/zamm.19860661107
- 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, Fract. Calc. Appl. Anal. 672 (2017), pp. 352–383.
- L.B. Feng, P. Zhuang, F. Liu, I. Turner, and J. Li, High–order numerical methods for the Riesz space fractional advection–dispersion equations, Comput. Math. Appl. (2016). Available at https://doi.org/10.1016/j.camwa.2016.01.015.
- R. Gorenflo, F. Mainardi, D. Moretti, and G. Pagnini, Discrete random walk models for space-time fractional diffusion, Chem. Phys. 284 (2002), pp. 521–541. doi: 10.1016/S0301-0104(02)00714-0
- E. Hanert and C. Piret, A Chebyshev pseudo-spectral method to solve the space-time tempered fractional diffusion equation, SIAM J. Sci. Comput. 36 (2014), pp. A1797–A1812. doi: 10.1137/130927292
- J. Huang and D. Yang, A unified difference-spectral method for time-space fractional diffusion equations, Int. J. Comput. Math. 94(6) (2017), pp. 1172–1184. doi: 10.1080/00207160.2016.1184262
- J.F. Huang, Y.F. Tang, and L. Väzquez, Convergence analysis of a block-by-block method for fractional differential equations, Numer. Math. Theor. Meth. Appl. 5 (2012), pp. 229–241. doi: 10.4208/nmtma.2012.m1038
- R. Kutner, A. Pekalski, and K. Sznajd-Weron, Anomalous Diffusion, Basics to Application, Springer, Berlin, 1999.
- C.P. Li and F.H. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim. 34 (2013), pp. 149–179. doi: 10.1080/01630563.2012.706673
- F. Liu, P. Zhuang, V. Anh, and I. Turner, A fractional-order implicit difference approximation for the space-time fractional diffusion equation, ANZIAM J. 47 (2006), pp. C48–C68. doi: 10.21914/anziamj.v47i0.1030
- F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burra, Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation, Appl. Math. Comput. 191 (2007), pp. 12–20.
- F. Liu, P. Zhuang, and Q. Liu, Numerical Methods of Fractional Partial Differential Equations and Applications, Science Press, China, 2015. (in Chinese), November 2015, ISBN 978-7-03-046335-7.
- F. Liu, P. Zhuang, I. Turner, V. Anh, and K. Burrage, A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys. 293 (2015), pp. 252–263. doi: 10.1016/j.jcp.2014.06.001
- F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Frac. Calc. Appl. Anal. 4(2) (2001), pp. 153–192.
- M.D. Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci. (2006), pp. 1–12. doi: 10.1155/IJMMS/2006/48391
- I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B.M.V. Jara, Matrix approach to discrete fractional calculus II: partial fractional differential equations, J. Comput. Phys. 228 (2009), pp. 3137–3153. doi: 10.1016/j.jcp.2009.01.014
- J. Sabatier, O.P. Agrawal, and J.A. Tenreiro, Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
- T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11 (1993), pp. 309–319. doi: 10.1016/0168-9274(93)90012-G
- Q. Yang, F. Liu, and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model. 34(1) (2010), pp. 200–218. doi: 10.1016/j.apm.2009.04.006
- B. Yu, X. Jiang, and H. Xu, A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation, Numer. Algor. 68(4) (2015), pp. 923–950. doi: 10.1007/s11075-014-9877-1
- M. Zheng, F. Liu, I. Turner, and V. Anh, A novel high order space-time spectral method for the time-fractional Fokker-Planck equation, SIAM J. Sci. Comput. 37(2) (2015), pp. A701–A724. doi: 10.1137/140980545
- P. Zhuang, F. Liu, V. Anh, and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal. 46 (2008), pp. 1079–1095. doi: 10.1137/060673114