Finite difference method for two-dimensional nonlinear time-fractional subdiffusion equation

References

[1] D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000), 1403–1412.10.1029/2000WR900031Search in Google Scholar

[2] J.P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195 (1990), 127–293.10.1016/0370-1573(90)90099-NSearch in Google Scholar

[3] B.L. Buzbee, G.H. Golub and C.W. Nielson, On direct methods for solving Poisson’s equations. SIAM J. Numer. Anal. 7 (1970), 627–656.10.1137/0707049Search in Google Scholar

[4] W.R. Cao, F.H. Zeng, Z.Q. Zhang and G.E. Karniadakis, Implicit-explicit difference schemes for nonlinear fractional differential equations with non-smooth solutions. SIAM J. Sci. Comput. 38 (2016), A3070–A3093.10.1137/16M1070323Search in Google Scholar

[5] A. Chen and C.P. Li, A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions. Int. J. Comput. Math. 93 (2016), 889–914.10.1080/00207160.2015.1009905Search in Google Scholar

[6] M. Cui, Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J. Comput. Phys. 231 (2012), 2621–2633.10.1016/j.jcp.2011.12.010Search in Google Scholar

[7] K. Diethelm, N.J. Ford and A.D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algor. 36 (2004), 31–52.10.1023/B:NUMA.0000027736.85078.beSearch in Google Scholar

[8] H.F. Ding and C.P. Li, High-order algorithms for Riesz derivative and their applications (III). Fract. Calc. Appl. Anal. 19, No 1 (2016), 19–55; 10.1515/fca-2016-0003; https://www.degruyter.com/view/j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml.Search in Google Scholar

[9] H.F. Ding and C.P. Li, Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations. Fract. Calc. Appl. Anal. 20, No 3 (2017), 722–764; 10.1515/fca-2017-0038; https://www.degruyter.com/view/j/fca.2017.20.issue-3/issue-files/fca.2017.20.issue-3.xml.Search in Google Scholar

[10] N.J. Ford and Y. Yan, An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data. Fract. Calc. Appl. Anal. 20, No 5 (2017), 1076–1105; 10.1515/fca-2017-0058; https://www.degruyter.com/view/j/fca.2017.20.issue-5/issue-files/fca.2017.20.issue-5.xml.Search in Google Scholar

[11] R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects. Math. Comput. Simulation110 (2015), 96–112.10.1016/j.matcom.2013.09.012Search in Google Scholar

[12] B.T. Jin, R. Lazarov, Y.K. Liuand Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281 (2015), 825–843.10.1016/j.jcp.2014.10.051Search in Google Scholar

[13] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Application of Fractional Differential Equations. Elsevier, Amersterdam (2006).Search in Google Scholar

[14] M. Klimek, M. Ciesielski and T. Blaszczyk, Exact and numerical solutions of the fractional Sturm-Liouville problem. Fract. Calc. Appl. Anal. 21, No 1 (2018), 45–71; 10.1515/fca-2018-0004; https://www.degruyter.com/view/j/fca.2018.21.issue-1/issue-files/fca.2018.21.issue-1.xml.Search in Google Scholar

[15] R. Koeller, Applications of fractional calculus to the theory of viscoelasticity. ASME J. Appl. Mech. 51 (1984), 299–307.10.1115/1.3167616Search in Google Scholar

[16] D. Kulkarni, D. Schmidt and S.-K. Tsui, Eigenvalues of tridiagonal pseudo-Toeplitz matrices. Linear Alg. Appl. 297 (1999), 63–80.10.1016/S0024-3795(99)00114-7Search in Google Scholar

[17] C.P. Li and A. Chen, Numerical methods for fractional partial differential equations. Int. J. Comput. Math. 95 (2018), 1048–1099.10.1080/00207160.2017.1343941Search in Google Scholar

[18] C.P. Li, Q. Yi and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations. J. Comput. Phys. 316 (2016), 614–631.10.1016/j.jcp.2016.04.039Search in Google Scholar

[19] C.P. Li and F.H. Zeng, The finite difference methods for fractional ordinary differential equations. Numer. Funct. Anal. Optim. 34 (2013), 149–179.10.1080/01630563.2012.706673Search in Google Scholar

[20] C.P. Li and F.H. Zeng, Numerical Methods for Fractional Calculus. Chapman and Hall/CRC Press, Boca Raton, USA (2015).10.1201/b18503Search in Google Scholar

[21] C.P. Li, F.H. Zeng and F.W. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383–406; 10.2478/s13540-012-0028-x; https://www.degruyter.com/view/j/fca.2012.15.issue-3/issue-files/fca.2012.15.issue-3.xml.Search in Google Scholar

[22] M. Li and C.M. Huang, Galerkin finite element method for nonlinear fractional Schrodinger equations. Numer. Algorithms74 (2017), 499–525.10.1007/s11075-016-0160-5Search in Google Scholar

[23] Z.Q. Li, Y.B. Yan and N.J. Ford, Error estimates of a high order numerical method for solving linear fractional differential equations. Appl. Numer. Math. 114 (2017), 201–220.10.1016/j.apnum.2016.04.010Search in Google Scholar

[24] C. Lubich, Discretized fractional calculus. SIAM J. Math. Anal. 17 (1986), 704–719.10.1137/0517050Search in Google Scholar

[25] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar

[26] P. Perdikaris and G.E. Karniadakis, Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42 (2014), 1012–1023.10.1007/s10439-014-0970-3Search in Google Scholar PubMed

[27] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Search in Google Scholar

[28] 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), 3137–3153.10.1016/j.jcp.2009.01.014Search in Google Scholar

[29] Y.A. Rossikhin and M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63 (2010), Art. # 010801.10.1115/1.4000563Search in Google Scholar

[30] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, New York (1993).Search in Google Scholar

[31] H. Sheng, Y. Chen and T. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer-Verlag, London (2011).10.1007/978-1-4471-2233-3Search in Google Scholar

[32] C. Tadjeran and M.M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007), 813–823.10.1016/j.jcp.2006.05.030Search in Google Scholar

[33] H. Wang and N. Du, Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. J. Comput. Phys. 258 (2014), 305–318.10.1016/j.jcp.2013.10.040Search in Google Scholar

[34] Z. Yang, Z. Yuan, Y. Nie, J. Wang, X. Zhu and F. Liu, Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains. J. Comput. Phys. 330 (2017), 863–883.10.1016/j.jcp.2016.10.053Search in Google Scholar

[35] S.B. Yuste and J. Quintana-Murillo, A finite difference method with non-uniform timesteps for fractional diffusion equations. Comput. Phys. Commun. 183 (2012), 2594–2600.10.1016/j.cpc.2012.07.011Search in Google Scholar

[36] F.H. Zeng, C.P. Li, F. Liu and I. Turner, Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J. Sci. Comput. 37 (2015), A55–A78.10.1137/14096390XSearch in Google Scholar

[37] F.H. Zeng, Z.Q. Zhang and G.E. Karniadakis, Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations. J. Comput. Phys. 307 (2016), 15–33.10.1016/j.jcp.2015.11.058Search in Google Scholar

[38] F.H. Zeng, Z.Q. Zhang and G.E. Karniadakis, Second-order numerical methods for multi-term fractional differential equations: smooth and non-smooth solutions. Comput. Methods Appl. Mech. Engrg. 327 (2017), 478–502.10.1016/j.cma.2017.08.029Search in Google Scholar

[39] Y.N. Zhang, Z.Z. Sun and H.L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J. Comput. Phys. 265 (2014), 195–210.10.1016/j.jcp.2014.02.008Search in Google Scholar