×

Numerical methods for fractional partial differential equations. (English) Zbl 1513.65291

Summary: In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), which are divided into the time-fractional, space-fractional, and space-time-fractional partial differential equations (PDEs). Besides, fast algorithms for the FPDEs are included in order to stimulate more efficient algorithms for high-dimensional FPDEs.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Software:

FODE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alikhanov, A. A., A new difference scheme for the time fractional diffusion equation, J. Comput. Phys., 280, 424-438 (2015) · Zbl 1349.65261 · doi:10.1016/j.jcp.2014.09.031
[2] Alikhanov, A. A., Numerical methods of solutions of boundary value problems for the multi-term variable distributed order diffusion equation, Appl. Math. Comput., 268, 12-22 (2015) · Zbl 1410.65294
[3] Alpert, B.; Greengard, L.; Hagstrom, T., Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM. J. Numer. Anal., 37, 1138-1164 (2000) · Zbl 0963.65104 · doi:10.1137/S0036142998336916
[4] Baffet, D.; Hesthaven, J. S., A kernel compression scheme for fractional differential equations, SIAM. J. Numer. Anal., 55, 496-520 (2017) · Zbl 1359.65106 · doi:10.1137/15M1043960
[5] Bu, W. P.; Tang, Y. F.; Yang, J. Y., Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276, 26-38 (2014) · Zbl 1349.65441 · doi:10.1016/j.jcp.2014.07.023
[6] Bu, W. P.; Tang, Y. F.; Wu, Y. C.; Yang, J. Y., Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations, J. Comput. Phys., 293, 264-279 (2015) · Zbl 1349.65440 · doi:10.1016/j.jcp.2014.06.031
[7] Bu, W. P.; Tang, Y. F.; Wu, Y. C.; Yang, J. Y., Crank-Nicolson ADI Galerkin finite element method for two-dimensional fractional FitzHugh-Nagumo monodomain model, Appl. Math. Comput., 257, 355-364 (2015) · Zbl 1339.65170
[8] Bu, W. P.; Xiao, A.; Zeng, W., Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput., 1-20 (2017) · Zbl 1375.65110 · doi:10.1007/s10915-017-0360-8
[9] Buzbee, B. L.; Golub, G. H.; Nielson, C. W., On direct methods for solving Poisson’s equations, SIAM. J. Numer. Anal., 7, 627-656 (1970) · Zbl 0217.52902 · doi:10.1137/0707049
[10] Cao, J. X.; Li, C. P.; Chen, Y., High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II), Fract. Calc. Appl. Anal., 18, 735-761 (2015) · Zbl 1325.65121 · doi:10.1515/fca-2015-0045
[11] Cao, W.; Zeng, F. H.; Zhang, Z.; Karniadakis, G. E., Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions, SIAM J. Sci. Comput., 38, A3070-A3093 (2016) · Zbl 1355.65104 · doi:10.1137/16M1070323
[12] Cem, Ç.; Duman, M., Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231, 1743-1750 (2012) · Zbl 1242.65157 · doi:10.1016/j.jcp.2011.11.008
[13] Chen, M. H.; Deng, W. H., A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation, Appl. Math. Model., 38, 3244-3259 (2014) · Zbl 1427.65149 · doi:10.1016/j.apm.2013.11.043
[14] Chen, M. H.; Deng, W. H., Fourth order accurate scheme for the space fractional diffusion equations, SIAM. J. Numer. Anal., 52, 1418-1438 (2014) · Zbl 1318.65048 · doi:10.1137/130933447
[15] Chen, M. H.; Deng, W. H.; Wu, Y. J., Superlinearly convergent algorithms for the two-dimensional space-time Caputo-Riesz fractional diffusion equation, Appl. Numer. Math., 70, 22-41 (2013) · Zbl 1283.65082 · doi:10.1016/j.apnum.2013.03.006
[16] Chen, A.; Du, Q.; Li, C. P.; Zhou, Z., Asymptotically compatible schemes for space-time nonlocal diffusion equations, Chaos, Solitons & Fractals (2017) · Zbl 1422.65279 · doi:10.1016/j.chaos.2017.03.061
[17] Chen, A.; Li, C. P., A novel compact ADI scheme for the time-fractional subdiffusion equation in two space dimensions, Int. J. Comput. Math., 93, 889-914 (2016) · Zbl 1390.65062 · doi:10.1080/00207160.2015.1009905
[18] Chen, A.; Li, C. P., Numerical solution of fractional diffusion-wave equation, Numer. Funct. Anal. Optim., 37, 19-39 (2016) · Zbl 1382.65236 · doi:10.1080/01630563.2015.1078815
[19] Chen, C. M.; Liu, F.; Turner, I.; Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 886-897 (2007) · Zbl 1165.65053 · doi:10.1016/j.jcp.2007.05.012
[20] Chen, C.-M.; Liu, F.; Turner, I.; Anh, V., Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation, Numer. Algorithms, 54, 1-21 (2010) · Zbl 1191.65116 · doi:10.1007/s11075-009-9320-1
[21] Chen, H.; Lü, S.; Chen, W., Finite difference/spectral approximations for the distributed order time fractional reaction-diffusion equation on an unbounded domain, J. Comput. Phys., 315, 84-97 (2016) · Zbl 1349.65507 · doi:10.1016/j.jcp.2016.03.044
[22] Chen, S.; Shen, J.; Wang, L.-L., Generalized Jacobi functions and their applications to fractional differential equations, Math. Comput., 85, 1603-1638 (2016) · Zbl 1335.65066 · doi:10.1090/mcom3035
[23] Chen, X.; Zeng, F. H.; Karniadakis, G. E., A tunable finite difference method for fractional differential equations with non-smooth solutions, Comput. Methods Appl. Mech. Eng., 318, 193-214 (2017) · Zbl 1439.65082 · doi:10.1016/j.cma.2017.01.020
[24] Cui, M. R., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228, 7792-7804 (2009) · Zbl 1179.65107 · doi:10.1016/j.jcp.2009.07.021
[25] Dehghan, M.; Abbaszadeh, M.; Mohebbi, A., Analysis of a meshless method for the time fractional diffusion-wave equation, Numer. Algorithms, 73, 445-476 (2016) · Zbl 1352.65298 · doi:10.1007/s11075-016-0103-1
[26] D’Elia, M.; Gunzburger, M., The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66, 1245-1260 (2013) · Zbl 1345.35128 · doi:10.1016/j.camwa.2013.07.022
[27] Deng, W. H., Finite element method for the space and time fractional Fokker-Planck equation, SIAM. J. Numer. Anal., 47, 204-226 (2008) · Zbl 1416.65344 · doi:10.1137/080714130
[28] Deng, W. H.; Hesthaven, J. S., Local discontinuous Galerkin methods for fractional diffusion equations, Math. Model. Numer. Anal., 47, 1845-1864 (2013) · Zbl 1282.35400 · doi:10.1051/m2an/2013091
[29] Ding, H. F.; Li, C. P., Mixed spline function method for reaction-subdiffusion equations, J. Comput. Phys., 242, 103-123 (2013) · Zbl 1297.65091 · doi:10.1016/j.jcp.2013.02.014
[30] Ding, H. F.; Li, C. P., High-order algorithms for Riesz derivative and their applications (III), Fract. Calc. Appl. Anal., 19, 19-55 (2016) · Zbl 1332.65122 · doi:10.1515/fca-2016-0003
[31] Ding, H.F. and Li, C.P., Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations, Fract. Calc. Appl. Anal. 20(3) (2017), pp. 722-764. · Zbl 1365.65194
[32] Ding, H.F. and Li, C.P., High-order algorithms for Riesz derivative and their applications (V), Numer. Methods Partial Differ. Equ. (2017), doi: . · Zbl 1376.65024
[33] Ding, H.F., Li, C.P., and Yi, Q.. A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application, IMA J. Appl. Math. (2017), doi.org/doi: . · Zbl 1471.65019
[34] Ding, H. F.; Li, C. P., High-order compact difference schemes for the modified anomalous subdiffusion equation, Numer. Methods Partial Differ. Equ., 32, 1, 213-242 (2016) · Zbl 1339.65109 · doi:10.1002/num.21992
[35] Ding, H. F.; Li, C. P., High-order numerical algorithms for Riesz derivatives via constructing new generating functions, J. Sci. Comput., 71, 2, 759-784 (2017) · Zbl 1398.65030 · doi:10.1007/s10915-016-0317-3
[36] Ding, H. F.; Li, C. P.; Chen, Y. Q., High-order algorithms for Riesz derivative and their applications (I), Abstract and Applied Analysis, 2014 (2014) · Zbl 1434.65113
[37] Ding, H. F.; Li, C. P.; Chen, Y. Q., High-order algorithms for Riesz derivative and their applications (II), J. Comput. Phys., 293, 218-237 (2015) · Zbl 1349.65284 · doi:10.1016/j.jcp.2014.06.007
[38] Du, R.; Cao, W. R.; Sun, Z. Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34, 2998-3007 (2010) · Zbl 1201.65154 · doi:10.1016/j.apm.2010.01.008
[39] Du, Q.; Gunzburger, M.; Lehoucq, R. B.; Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54, 667-696 (2012) · Zbl 1422.76168 · doi:10.1137/110833294
[40] Ervin, V. J.; Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differ. Equ., 22, 558-576 (2006) · Zbl 1095.65118 · doi:10.1002/num.20112
[41] Fatone, L.; Funaro, D., Optimal collocation nodes for fractional derivative operators, SIAM J. Scient. Comput., 37, A1504-A1524 (2015) · Zbl 1328.65252 · doi:10.1137/140993697
[42] Feng, L. B.; Zhuang, P.; Liu, F.; Turner, I.; Gu, Y. T., Finite element method for space-time fractional diffusion equation, Numer. Algorithms, 72, 749-767 (2016) · Zbl 1343.65122 · doi:10.1007/s11075-015-0065-8
[43] Fix, G. J.; Roop, J. P., Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48, 1017-1033 (2004) · Zbl 1069.65094 · doi:10.1016/j.camwa.2004.10.003
[44] Gao, G. H.; Sun, Z. Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230, 586-595 (2011) · Zbl 1211.65112 · doi:10.1016/j.jcp.2010.10.007
[45] Gao, G. H.; Sun, Z. Z., Two alternating direction implicit difference schemes for two-dimensional distributed-order fractional diffusion equations, J. Sci. Comput., 66, 1281-1312 (2016) · Zbl 1373.65055 · doi:10.1007/s10915-015-0064-x
[46] Gao, G. H.; Sun, H. W.; Sun, Z. Z., Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence, J. Comput. Phys., 280, 510-528 (2015) · Zbl 1349.65295 · doi:10.1016/j.jcp.2014.09.033
[47] Gao, G. H.; Sun, H. W.; Sun, Z. Z., Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 298, 337-359 (2015) · Zbl 1349.65296 · doi:10.1016/j.jcp.2015.05.047
[48] Gao, G. H.; Sun, Z. Z.; Zhang, H. W., A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259, 33-50 (2014) · Zbl 1349.65088 · doi:10.1016/j.jcp.2013.11.017
[49] Guo, B.-Y.; Shen, J.; Wang, L.-L., Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math., 59, 1011-1028 (2009) · Zbl 1171.33006 · doi:10.1016/j.apnum.2008.04.003
[50] Hao, Z. P.; Cao, W. R., An improved algorithm based on finite difference schemes for fractional boundary value problems with nonsmooth solution, Journal of Scientific Computation (2017) · Zbl 1377.26009 · doi:10.1007/s10915-017-0417-8
[51] Hao, Z. P.; Sun, Z. Z.; Cao, W. R., A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys., 281, 787-805 (2015) · Zbl 1352.65238 · doi:10.1016/j.jcp.2014.10.053
[52] Hejazi, H.; Moroney, T.; Liu, F., Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, J. Comput. Appl. Math., 255, 684-697 (2014) · Zbl 1291.65280 · doi:10.1016/j.cam.2013.06.039
[53] Hosseini, V. R.; Shivanian, E.; Chen, W., Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, J. Comput. Phys., 312, 307-332 (2016) · Zbl 1352.65348 · doi:10.1016/j.jcp.2016.02.030
[54] Hu, Y.; Li, C. P.; Li, H. F., The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case, Chaos, Solit. Fract (2017) · Zbl 1422.65157 · doi:10.1016/j.chaos.2017.03.038
[55] Hu, X.; Liu, F.; Turner, I.; Anh, V., An implicit numerical method of a new time distributed-order and two-sided space-fractional advection-dispersion equation, Numer. Algorithms, 72, 393-407 (2016) · Zbl 1343.65110 · doi:10.1007/s11075-015-0051-1
[56] Hu, X. L.; Zhang, L. M., An analysis of a second order difference scheme for the fractional subdiffusion system, Appl. Math. Model., 40, 1634-1649 (2016) · Zbl 1446.65067 · doi:10.1016/j.apm.2015.08.010
[57] Huang, J. F.; Nie, N. M.; Tang, Y. F., A second order finite difference-spectral method for space fractional diffusion equations, Science China Math., 57, 1303-1317 (2014) · Zbl 1305.65185 · doi:10.1007/s11425-013-4716-8
[58] Huang, J. F.; Tang, Y. F.; Vázquez, L.; Yang, J. Y., Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algorithms, 64, 707-720 (2013) · Zbl 1284.65103 · doi:10.1007/s11075-012-9689-0
[59] Ilic, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation, I, Fract. Calc. Appl. Anal., 8, 323-341 (2005) · Zbl 1126.26009
[60] Ilic, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 333-349 (2006) · Zbl 1132.35507
[61] Jia, J. H.; Wang, H., A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299, 842-862 (2015) · Zbl 1352.65430 · doi:10.1016/j.jcp.2015.06.028
[62] Jiang, S.; Greengard, L.; Wang, S., Efficient sum-of-exponentials approximations for the heat kernel and their applications, Adv. Comput. Math., 41, 529-551 (2015) · Zbl 1318.31010 · doi:10.1007/s10444-014-9372-1
[63] Jiang, Y. G.; Ma, J. T., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235, 3285-3290 (2011) · Zbl 1216.65130 · doi:10.1016/j.cam.2011.01.011
[64] Jiang, S.; Zhang, J.; Zhang, Q.; Zhang, Z., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21, 650-678 (2017) · Zbl 1488.65247 · doi:10.4208/cicp.OA-2016-0136
[65] Jin, B.; Zhou, Z., A finite element method with singularity reconstruction for fractional boundary value problems, ESAIM Math. Model. Numer. Anal., 49, 1261-1283 (2015) · Zbl 1332.65115 · doi:10.1051/m2an/2015010
[66] Jin, B.; Lazarov, R.; Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM. J. Numer. Anal., 51, 445-466 (2013) · Zbl 1268.65126 · doi:10.1137/120873984
[67] Jin, B.; Lazarov, R.; Liu, Y.; Zhou, Z., The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281, 825-843 (2015) · Zbl 1352.65350 · doi:10.1016/j.jcp.2014.10.051
[68] Jin, B.; Lazarov, R.; Lu, X.; Zhou, Z., A simple finite element method for boundary value problems with a Riemann-Liouville derivative, J. Comput. Appl. Math., 293, 94-111 (2016) · Zbl 1328.65172 · doi:10.1016/j.cam.2015.02.058
[69] Jin, B.; Lazarov, R.; Pasciak, J.; Rundell, W., Variational formulation of problems involving fractional order differential operators, Math. Comput., 84, 2665-2700 (2015) · Zbl 1321.65127 · doi:10.1090/mcom/2960
[70] Jin, B.; Lazarov, R.; Pasciak, J.; Zhou, Z., Error analysis of a finite element method for the space-fractional parabolic equation, SIAM. J. Numer. Anal., 52, 2272-2294 (2014) · Zbl 1310.65126 · doi:10.1137/13093933X
[71] Jin, B.; Lazarov, R.; Sheen, D.; Zhou, Z., Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data, Fract. Calc. Appl. Anal., 19, 69-93 (2016) · Zbl 1333.65111 · doi:10.1515/fca-2016-0005
[72] Jin, B.; Lazarov, R.; Zhou, Z., A Petrov-Galerkin finite element method for fractional convection-diffusion equation, SIAM. J. Numer. Anal., 54, 481-503 (2016) · Zbl 1335.65092 · doi:10.1137/140992278
[73] Jin, B.; Lazarov, R.; Zhou, Z., Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Scient. Comput., 38, A146-A170 (2016) · Zbl 1381.65082 · doi:10.1137/140979563
[74] Ke, R. H.; Ng, M. K.; Sun, H. W., A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations, J. Comput. Phys., 303, 203-211 (2015) · Zbl 1349.65404 · doi:10.1016/j.jcp.2015.09.042
[75] Khaliq, A. Q.M.; Liang, X.; Furati, K. M., A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75, 147-172 (2017) · Zbl 1365.65195 · doi:10.1007/s11075-016-0200-1
[76] Langlands, T.; Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205, 719-736 (2005) · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025
[77] Le, K. N.; McLean, W.; Mustapha, K., Numerical solution of the time-fractional Fokker-Planck equation with general forcing, SIAM. J. Numer. Anal., 54, 1763-1784 (2016) · Zbl 1404.65122 · doi:10.1137/15M1031734
[78] Lei, S. L.; Sun, H. W., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, 715-725 (2013) · Zbl 1297.65095 · doi:10.1016/j.jcp.2013.02.025
[79] Li, C. P.; Cai, M., High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations: Revisited, Numer. Funct. Anal. Optim · Zbl 1373.26005 · doi:10.1080/01630563.2017.1291521
[80] Li, C. P.; Deng, W. H., Remarks on fractional derivatives, Appl. Math. Comput., 187, 777-784 (2007) · Zbl 1125.26009
[81] Li, C. P.; Ding, H. F., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model., 38, 3802-3821 (2014) · Zbl 1429.65188 · doi:10.1016/j.apm.2013.12.002
[82] Li, C. K.; Li, C. P., On defining the distributions \(####\) and \(####\) by fractional derivatives, Appl. Math. Comput., 246, 502-513 (2014) · Zbl 1338.46053
[83] Li, C. K.; Li, C. P., Remarks on fractional derivatives of distributions, Tbilisi Math. J., 10, 1, 1-18 (2017) · Zbl 1358.46038 · doi:10.1515/tmj-2017-0001
[84] Li, X. J.; Xu, C. J., A space-time spectral method for the time fractional diffusion equation, SIAM. J. Numer. Anal., 47, 2108-2131 (2009) · Zbl 1193.35243 · doi:10.1137/080718942
[85] Li, X. J.; Xu, C. J., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8, 1016-1051 (2010) · Zbl 1364.35424
[86] Li, C. P.; Zeng, F. H., Finite difference methods for fractional differential equations, Int. J. Bifur. Chaos, 22, 427-432 (2012) · Zbl 1258.34018
[87] Li, C. P.; Zeng, F. H.; Li, C. P.; Wu, Y. J.; Ye, R. S., Finite element methods for fractional differential equation, in Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis, 49-68 (2013), World Scientific: World Scientific, Singapore · Zbl 1319.65070
[88] Li, C. P.; Zeng, F. H., Numerical Methods for Fractional Calculus (2015), Chapman and Hall/CRC Press: Chapman and Hall/CRC Press, Boca Raton, FL · Zbl 1326.65033
[89] Li, C. P.; Zeng, F. H.; Liu, F., Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15, 383-406 (2012) · Zbl 1276.26016
[90] Li, C. P.; Zhao, Z. G.; Chen, Y. Q., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62, 855-875 (2011) · Zbl 1228.65190 · doi:10.1016/j.camwa.2011.02.045
[91] Li, H. F.; Cao, J. X.; Li, C. P., High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III), J. Comput. Appl. Math., 299, 159-175 (2016) · Zbl 1382.65251 · doi:10.1016/j.cam.2015.11.037
[92] Li, C. K.; Li, C. P.; Kacsmar, B.; Lacroix, R.; Tilbury, K., The Abel’s integral equations in distribution, Advances in Analysis, 2, 2, 88-104 (2017)
[93] Li, C. P.; Wu, R. F.; Ding, H. F., High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations, Commun. Appl. Ind. Math., 6, 1-32 (2014) · Zbl 1329.65182
[94] Li, L. M.; Xu, D.; Luo, M., Alternating direction implicit Galerkin finite element method for the two-dimensional fractional diffusion-wave equation, J. Comput. Phys., 255, 471-485 (2013) · Zbl 1349.65456 · doi:10.1016/j.jcp.2013.08.031
[95] Li, C. P.; Yi, Q.; Chen, A., Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys., 316, 614-631 (2016) · Zbl 1349.65246 · doi:10.1016/j.jcp.2016.04.039
[96] Liang, X.; Khaliq, A. Q.M.; Bhatt, H.; Furati, K. M., The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations, Numer. Algorithms (2017) · Zbl 1380.65162 · doi:10.1007/s11075-017-0291-3
[97] Lin, Y. M.; Li, X. J.; Xu, C. J., Finite difference/spectral approximations for the fractional cable equation, Math. Comput., 80, 1369-1396 (2011) · Zbl 1220.78107 · doi:10.1090/S0025-5718-2010-02438-X
[98] Lin, Y. M.; Xu, C. J., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001
[99] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Finite element approximation for a modified anomalous subdiffusion equation, Appl. Math. Model., 35, 4103-4116 (2011) · Zbl 1221.65257 · doi:10.1016/j.apm.2011.02.036
[100] Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160-176 (2009) · Zbl 1170.65107 · doi:10.1016/j.cam.2009.02.013
[101] Liu, F.; Zhuang, P.; Turner, I.; Burrage, K.; Anh, V., A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38, 3871-3878 (2014) · Zbl 1429.65213 · doi:10.1016/j.apm.2013.10.007
[102] Liu, Q.; Zeng, F. H.; Li, C. P., Finite difference method for time-space-fractional Schrödinger equation, Int. J. Comput. Math., 92, 1439-1451 (2015) · Zbl 1325.65124 · doi:10.1080/00207160.2014.945440
[103] Liu, Y.; Du, Y. W.; Li, H.; Wang, J. F., An H1-Galerkin mixed finite element method for time fractional reaction-diffusion equation, Journal of Applied Mathematics and Computating, 47, 103-117 (2015) · Zbl 1319.65097 · doi:10.1007/s12190-014-0764-7
[104] Liu, F.; Zhuang, P.; Liu, Q., Numerical Methods for Fractional Partial Differential Equations and Their Applications (2015), Science Press: Science Press, Beijing
[105] Liu, F.; Zhuang, P.; Turner, I.; Anh, V.; Burrage, K., A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys., 293, 252-263 (2015) · Zbl 1349.65316 · doi:10.1016/j.jcp.2014.06.001
[106] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear. Dyn., 29, 57-98 (2002) · Zbl 1018.93007 · doi:10.1023/A:1016586905654
[107] Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17, 704-719 (1986) · Zbl 0624.65015 · doi:10.1137/0517050
[108] Lynch, V.; Carreras, B.; del Castillo-Negrete, D.; Ferreira-Mejias, K.; Hicks, H., Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192, 406-421 (2003) · Zbl 1047.76075 · doi:10.1016/j.jcp.2003.07.008
[109] Ma, J. T.; Liu, J. Q.; Zhou, Z. Q., Convergence analysis of moving finite element methods for space fractional differential equations, J. Comput. Appl. Math., 255, 661-670 (2014) · Zbl 1291.65303 · doi:10.1016/j.cam.2013.06.021
[110] Machado, J. A.T.; Mainardi, F.; Kiryakova, V.; Atanacković, T., Fractional calculus: D’où venons-nous? Que sommes-nous? Où allons-nous?, Fract. Calc. Appl. Anal., 19, 1074-1104 (2016) · Zbl 1351.26017 · doi:10.1515/fca-2016-0059
[111] McLean, W.; Mustapha, K., A second-order accurate numerical method for a fractional wave equation, Numer. Math., 105, 481-510 (2007) · Zbl 1111.65113 · doi:10.1007/s00211-006-0045-y
[112] McLean, W.; Mustapha, K., Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation, Numer. Algorithms, 52, 69-88 (2009) · Zbl 1177.65194 · doi:10.1007/s11075-008-9258-8
[113] McLean, W.; Thomée, V., Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal., 24, 439-463 (2004) · Zbl 1068.65146 · doi:10.1093/imanum/24.3.439
[114] McLean, W.; Thomée, V., Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equ. Appl., 22, 57-94 (2010) · Zbl 1195.65122 · doi:10.1216/JIE-2010-22-1-57
[115] Meerschaert, M. M.; Scheffler, H. P.; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211, 249-261 (2006) · Zbl 1085.65080 · doi:10.1016/j.jcp.2005.05.017
[116] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346 · doi:10.1016/j.cam.2004.01.033
[117] Meerschaert, M. M.; Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math., 56, 80-90 (2006) · Zbl 1086.65087 · doi:10.1016/j.apnum.2005.02.008
[118] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[119] Metzler, R.; Nonnenmacher, T. F., Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys., 284, 67-90 (2002) · doi:10.1016/S0301-0104(02)00537-2
[120] Morgado, M. L.; Rebelo, M., Numerical approximation of distributed order reaction diffusion equations, J. Comput. Appl. Math., 275, 216-227 (2015) · Zbl 1298.35242 · doi:10.1016/j.cam.2014.07.029
[121] Mustapha, K., An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements, IMA J. Numer. Anal., 31, 719-739 (2011) · Zbl 1219.65091 · doi:10.1093/imanum/drp057
[122] Mustapha, K., Time-stepping discontinuous Galerkin methods for fractional diffusion problems, Numer. Math., 130, 497-516 (2015) · Zbl 1320.65144 · doi:10.1007/s00211-014-0669-2
[123] Mustapha, K.; Abdallah, B.; Furati, K. M., A discontinuous Petrov-Galerkin method for time-fractional diffusion equations, SIAM. J. Numer. Anal., 52, 2512-2529 (2014) · Zbl 1323.65109 · doi:10.1137/140952107
[124] Mustapha, K.; Abdallah, B.; Furati, K. M.; Nour, M., A discontinuous Galerkin method for tiem fractional diffusion equations with variable coefficients, Numer. Algorithms, 73, 517-534 (2016) · Zbl 1352.65363 · doi:10.1007/s11075-016-0106-y
[125] Mustapha, K.; AlMustawa, J., A finite difference method for an anomalous sub-diffusion equation, theory and applications, Numer. Algorithms, 61, 525-543 (2012) · Zbl 1263.65082 · doi:10.1007/s11075-012-9547-0
[126] Mustapha, K.; Brunner, H.; Mustapha, H.; Schötzau, D., An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type, SIAM. J. Numer. Anal., 49, 1369-1396 (2011) · Zbl 1230.65143 · doi:10.1137/100797114
[127] Mustapha, K.; McLean, W., Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer. Algorithms, 56, 159-184 (2011) · Zbl 1211.65127 · doi:10.1007/s11075-010-9379-8
[128] Mustapha, K.; McLean, W., Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM. J. Numer. Anal., 51, 491-515 (2013) · Zbl 1267.26005 · doi:10.1137/120880719
[129] Mustapha, K.; Schötzau, D., Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA J. Numer. Anal., 34, 1426-1446 (2014) · Zbl 1310.65128 · doi:10.1093/imanum/drt048
[130] Nasir, H. M.; Gunawardana, B. L.K.; Abeyrathna, H. M.N. P., A second order finite difference approximation for the fractional diffusion equation, Int. J. Appl. Phys. Math., 3, 237-243 (2013) · doi:10.7763/IJAPM.2013.V3.212
[131] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press, New York · Zbl 0428.26004
[132] Ortigueira, M. D., Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci., 2006 (2006) · Zbl 1122.26007 · doi:10.1155/IJMMS/2006/48391
[133] Pan, J. Y.; Ke, R. H.; Ng, M. K.; Sun, H.-W., Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Scient. Comput., 36, A2698-A2719 (2014) · Zbl 1314.65112 · doi:10.1137/130931795
[134] Pang, H. K.; Sun, H. W., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231, 693-703 (2012) · Zbl 1243.65117 · doi:10.1016/j.jcp.2011.10.005
[135] Podlubny, I., Fractional Differential Equations (1998), Academic Press: Academic Press, New York · Zbl 0922.45001
[136] Qiu, L. L.; Deng, W. H.; Hesthaven, J. S., Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes, J. Comput. Phys., 298, 678-694 (2015) · Zbl 1349.65476 · doi:10.1016/j.jcp.2015.06.022
[137] Ren, J. C.; Sun, Z. Z.; Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232, 456-467 (2013) · Zbl 1291.35428 · doi:10.1016/j.jcp.2012.08.026
[138] Roop, J.P., Variational solution of the fractional advection dispersion equation, PhD thesis, Clemson University, 2004.
[139] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach, Amsterdam · Zbl 0818.26003
[140] Sheen, D.; Sloan, I. H.; Thomée, V., A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comput., 69, 177-196 (2000) · Zbl 0936.65109 · doi:10.1090/S0025-5718-99-01098-4
[141] Sheen, D.; Sloan, I. H.; Thomée, V., A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer. Anal., 23, 269-299 (2003) · Zbl 1022.65108 · doi:10.1093/imanum/23.2.269
[142] Shen, S. J.; Liu, F.; Anh, V., Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation, Numer. Algorithms, 56, 383-403 (2011) · Zbl 1214.65046 · doi:10.1007/s11075-010-9393-x
[143] Song, F. Y.; Xu, C. J., Spectral direction splitting methods for two-dimensional space fractional diffusion equations, J. Comput. Phys., 299, 196-214 (2015) · Zbl 1352.65400 · doi:10.1016/j.jcp.2015.07.011
[144] Sousa, E., Finite difference approximations for a fractional advection diffusion problem, J. Comput. Phys., 228, 4038-4054 (2009) · Zbl 1169.65126 · doi:10.1016/j.jcp.2009.02.011
[145] Sousa, E.; Li, C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, Appl. Numer. Math., 90, 22-37 (2015) · Zbl 1326.65111 · doi:10.1016/j.apnum.2014.11.007
[146] Sun, Z. Z.; Gao, G. H., The Finite Difference Methods for Fractional Differential Equations (2015), Chinese Scientific Press: Chinese Scientific Press, Beijing
[147] Sun, H.; Sun, Z.-Z.; Gao, G.-H., Some temporal second order difference schemes for fractional wave equations, Numer. Methods Partial Differ. Equ., 32, 970-1001 (2016) · Zbl 1352.65269 · doi:10.1002/num.22038
[148] Sun, H.; Sun, Z. Z.; Gao, G. H., Some high order difference schemes for the space and time fractional Bloch-Torrey equations, Appl. Math. Comput., 281, 356-380 (2016) · Zbl 1410.65329
[149] Sun, Z.-Z.; Wu, X. N., A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56, 193-209 (2006) · Zbl 1094.65083 · doi:10.1016/j.apnum.2005.03.003
[150] Tadjeran, C.; Meerschaert, M. M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220, 813-823 (2007) · Zbl 1113.65124 · doi:10.1016/j.jcp.2006.05.030
[151] Tadjeran, C.; Meerschaert, M. M.; Scheffler, H.-P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213, 205-213 (2006) · Zbl 1089.65089 · doi:10.1016/j.jcp.2005.08.008
[152] Tian, W. Y.; Deng, W. H.; Wu, Y. J., Polynomial spectral collocation method for space fractional advection-diffusion equation, Numer. Methods Partial Differ. Equ., 30, 514-535 (2014) · Zbl 1287.65093 · doi:10.1002/num.21822
[153] Tian, X. C.; Du, Q., Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM. J. Numer. Anal., 51, 3458-3482 (2013) · Zbl 1295.82021 · doi:10.1137/13091631X
[154] Tian, W. Y.; Zhou, H.; Deng, W. H., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput., 84, 1703-1727 (2015) · Zbl 1318.65058 · doi:10.1090/S0025-5718-2015-02917-2
[155] Tuan, V. K.; Gorenflo, R., Extrapolation to the limit for numerical fractional differentiation, ZAMM J. Appl. Math. Mech., 75, 646-648 (1995) · Zbl 0860.65011 · doi:10.1002/zamm.19950750826
[156] Uchaikin, U. U., Fractional Derivative for Physicists and Engineers (2013), Higher Education Press: Higher Education Press, Beijing · Zbl 1312.26002
[157] Vong, S.; Wang, Z. B., A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, J. Comput. Phys., 274, 268-282 (2014) · Zbl 1352.65273 · doi:10.1016/j.jcp.2014.06.022
[158] Wang, Y.-M., A compact finite difference method for solving a class of time fractional convection-subdiffusion equations, BIT Numer. Math., 55, 1187-1217 (2015) · Zbl 1348.65120 · doi:10.1007/s10543-014-0532-y
[159] Wang, H.; Basu, T. S., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Scient. Comput., 34, A2444-A2458 (2012) · Zbl 1256.35194 · doi:10.1137/12086491X
[160] Wang, H.; Du, N., A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation, J. Comput. Phys., 253, 50-63 (2013) · Zbl 1349.65341 · doi:10.1016/j.jcp.2013.06.040
[161] Wang, H.; Du, N., Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations, J. Comput. Phys., 258, 305-318 (2014) · Zbl 1349.65342 · doi:10.1016/j.jcp.2013.10.040
[162] Wang, Z. B.; Vong, S., Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277, 1-15 (2014) · Zbl 1349.65348 · doi:10.1016/j.jcp.2014.08.012
[163] Wang, H.; Wang, K. X., An \(####\) alternating-direction finite difference method for two-dimensional fractional diffusion equations, J. Comput. Phys., 230, 7830-7839 (2011) · Zbl 1229.65165 · doi:10.1016/j.jcp.2011.07.003
[164] Wang, H.; Wang, K. X.; Sircar, T., A direct \(####\) finite difference method for fractional diffusion equations, J. Comput. Phys., 229, 8095-8104 (2010) · Zbl 1198.65176 · doi:10.1016/j.jcp.2010.07.011
[165] Wang, H.; Yang, D. P., Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM. J. Numer. Anal., 51, 1088-1107 (2013) · Zbl 1277.65059 · doi:10.1137/120892295
[166] Wang, H.; Yang, D. P.; Zhu, S. F., Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations, SIAM. J. Numer. Anal., 52, 1292-1310 (2014) · Zbl 1320.65182 · doi:10.1137/130932776
[167] Wang, Y.; Liu, Y.; Li, H.; Wang, J., Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation, Eur. Phys. J. Plus, 131, 1-16 (2016) · doi:10.1140/epjp/i2016-16001-3
[168] Wang, H.; Yang, D. P.; Zhu, S. F., Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, J. Sci. Comput., 70, 429-449 (2017) · Zbl 1359.65271 · doi:10.1007/s10915-016-0196-7
[169] Wu, C. H.; Lu, L. Z., Implicit numerical approximation scheme for the fractional Fokker-Planck equation, Appl. Math. Comput., 216, 1945-1955 (2010) · Zbl 1196.82101
[170] Xu, Q. W.; Hesthaven, J. S., Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM. J. Numer. Anal., 52, 405-423 (2014) · Zbl 1297.26018 · doi:10.1137/130918174
[171] Xu, Q. W.; Hesthaven, J. S., Stable multi-domain spectral penalty methods for fractional partial differential equations, J. Comput. Phys., 257, 241-258 (2014) · Zbl 1349.35414 · doi:10.1016/j.jcp.2013.09.041
[172] Yang, J. Y.; Huang, J. F.; Liang, D. M.; Tang, Y. F., Numerical solution of fractional diffusion-wave equation based on fractional multistep method, Appl. Math. Model., 38, 3652-3661 (2014) · Zbl 1427.65196 · doi:10.1016/j.apm.2013.11.069
[173] Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34, 200-218 (2010) · Zbl 1185.65200 · doi:10.1016/j.apm.2009.04.006
[174] Ye, H.; Liu, F.; Anh, V., Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains, J. Comput. Phys., 298, 652-660 (2015) · Zbl 1349.65353 · doi:10.1016/j.jcp.2015.06.025
[175] Yu, B.; Jiang, X. Y.; Xu, H. Y., A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation, Numer. Algorithms, 68, 923-950 (2015) · Zbl 1314.65114 · doi:10.1007/s11075-014-9877-1
[176] Yuste, S. B., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216, 264-274 (2006) · Zbl 1094.65085 · doi:10.1016/j.jcp.2005.12.006
[177] Yuste, S. B.; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM. J. Numer. Anal., 42, 1862-1874 (2005) · Zbl 1119.65379 · doi:10.1137/030602666
[178] Zayernouri, M.; Ainsworth, M.; Karniadakis, G. E., A unified Petrov-Galerkin spectral method for fractional PDEs, Comput. Methods Appl. Mech. Eng., 283, 1545-1569 (2015) · Zbl 1425.65127 · doi:10.1016/j.cma.2014.10.051
[179] Zayernouri, M.; Cao, W.; Zhang, Z.; Karniadakis, G. E., Spectral and discontinuous spectral element methods for fractional delay equations, SIAM Journal on Scientific Computing, 36, B904-B929 (2014) · Zbl 1314.34159 · doi:10.1137/130935884
[180] Zayernouri, M.; Karniadakis, G. E., Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252, 495-517 (2013) · Zbl 1349.34095 · doi:10.1016/j.jcp.2013.06.031
[181] Zayernouri, M.; Karniadakis, G. E., Exponentially accurate spectral and spectral element methods for fractional ODEs, J. Comput. Phys., 257, 460-480 (2014) · Zbl 1349.65257 · doi:10.1016/j.jcp.2013.09.039
[182] Zayernouri, M.; Karniadakis, G. E., Discontinuous spectral element methods for time-and space-fractional advection equations, SIAM J. Scient. Comput., 36, B684-B707 (2014) · Zbl 1304.35757 · doi:10.1137/130940967
[183] Zayernouri, M.; Karniadakis, G. E., Fractional spectral collocation method, SIAM J. Sci. Comput., 36, A40-A62 (2014) · Zbl 1294.65097 · doi:10.1137/130933216
[184] Zayernouri, M.; Karniadakis, G. E., Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comput. Phys., 293, 312-338 (2015) · Zbl 1349.65531 · doi:10.1016/j.jcp.2014.12.001
[185] Zeng, F. H., Second-order stable finite difference schemes for the time-fractional diffusion-wave equation, J. Sci. Comput., 65, 411-430 (2015) · Zbl 1408.65058 · doi:10.1007/s10915-014-9966-2
[186] Zeng, F.H. and Li, C.P., Fractional differentiation matrices with applications, preprint (2014), Available at arXiv:1404.4429.
[187] Zeng, F.H. and Li, C.P., A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation, submitted, 2014; revised 2017.
[188] Zeng, F. H.; Li, C. P.; Liu, F., High-order explicit-implicit numerical methods for nonlinear anomalous diffusion equations, Eur. Phys. J. Spec. Top., 222, 1885-1900 (2013) · doi:10.1140/epjst/e2013-01971-3
[189] Zeng, F. H.; Li, C. P.; Liu, F.; Turner, I., The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35, A2976-A3000 (2013) · Zbl 1292.65096 · doi:10.1137/130910865
[190] Zeng, F. H.; Liu, F.; Li, C. P.; Burrage, K.; Turner, I.; Anh, V., A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM. J. Numer. Anal., 52, 2599-2622 (2014) · Zbl 1382.65349 · doi:10.1137/130934192
[191] Zeng, F. H.; Li, C. P.; Liu, F.; Turner, I., Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy, SIAM J. Scient. Comput., 37, A55-A78 (2015) · Zbl 1334.65162 · doi:10.1137/14096390X
[192] Zeng, F. H.; Zhang, Z. Q.; Karniadakis, G. E., A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comput., 37, A2710-A2732 (2015) · Zbl 1339.65197 · doi:10.1137/141001299
[193] Zeng, F. H.; Zhang, Z. Q.; Karniadakis, G. E., Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations, J. Comput. Phys., 307, 15-33 (2016) · Zbl 1352.65278 · doi:10.1016/j.jcp.2015.11.058
[194] Zeng, F. H.; Mao, Z.; Karniadakis, G. E., A generalized spectral collocation method with tunable accuracy for fractional differential equations with end-point singularities, SIAM J. Sci. Comput., 39, A360-A383 (2017) · Zbl 1431.65193 · doi:10.1137/16M1076083
[195] Zeng, F.H., Turner, I., and Burrage, K., A stable fast time-stepping method for fractional integral and derivative operators, preprint (2017), Available at arXiv:1703.05480. · Zbl 1406.65047
[196] Zhai, S. Y.; Feng, X. L.; He, Y. N., An unconditionally stable compact ADI method for three-dimensional time-fractional convection-diffusion equation, J. Comput. Phys., 269, 138-155 (2014) · Zbl 1349.65356 · doi:10.1016/j.jcp.2014.03.020
[197] Zhang, Y. N.; Sun, Z. Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230, 8713-8728 (2011) · Zbl 1242.65174 · doi:10.1016/j.jcp.2011.08.020
[198] Zhang, Y. N.; Sun, Z. Z., Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comput., 59, 104-128 (2014) · Zbl 1304.65208 · doi:10.1007/s10915-013-9756-2
[199] Zhang, H.; Liu, F.; Anh, V., Galerkin finite element approximation of symmetric space-fractional partial differential equations, Appl. Math. Comput., 217, 2534-2545 (2010) · Zbl 1206.65234
[200] Zhang, Y.-N.; Sun, Z.-Z.; Wu, H.-W., Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation, SIAM. J. Numer. Anal., 49, 2302-2322 (2011) · Zbl 1251.65132 · doi:10.1137/100812707
[201] Zhang, Y. N.; Sun, Z. Z.; Zhao, X., Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM. J. Numer. Anal., 50, 1535-1555 (2012) · Zbl 1251.65126 · doi:10.1137/110840959
[202] Zhang, Y. N.; Sun, Z. Z.; Liao, H. L., Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys., 265, 195-210 (2014) · Zbl 1349.65359 · doi:10.1016/j.jcp.2014.02.008
[203] Zhang, L.; Sun, H. W.; Pang, H. K., Fast numerical solution for fractional diffusion equations by exponential quadrature rule, J. Comput. Phys., 299, 130-143 (2015) · Zbl 1352.65304 · doi:10.1016/j.jcp.2015.07.001
[204] Zhao, L. J.; Deng, W. H., A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives, Numer. Methods Partial Differ. Equ., 31, 1345-1381 (2015) · Zbl 1332.65131 · doi:10.1002/num.21947
[205] Zhao, Z. G.; Li, C. P., Fractional difference/finite element approximations for the time-space fractional telegraph equation, Appl. Math. Comput., 219, 2975-2988 (2012) · Zbl 1309.65101
[206] Zhao, X.; Sun, Z. Z.; Hao, Z. P., A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput., 36, A2865-A2886 (2014) · Zbl 1328.65187 · doi:10.1137/140961560
[207] Zhao, X.; Sun, Z. Z.; Karniadakis, G. E., Second-order approximations for variable order fractional derivatives: Algorithms and applications, J. Comput. Phys., 293, 184-200 (2015) · Zbl 1349.65092 · doi:10.1016/j.jcp.2014.08.015
[208] Zheng, C., Approximation, stability and fast evaluation of exact artificial boundary condition for the one-dimensional heat equation, J. Comput. Math., 25, 730-745 (2007) · Zbl 1150.65022
[209] Zheng, Y. Y.; Li, C. P.; Zhao, Z. G., A note on the finite element method for the space-fractional advection diffusion equation, Comput. Math. Appl., 59, 1718-1726 (2010) · Zbl 1189.65288 · doi:10.1016/j.camwa.2009.08.071
[210] Zheng, M.; Liu, F.; Turner, I.; Anh, V., A novel high order space-time spectral method for the time fractional Fokker-Planck equation, SIAM J. Scient. Comput., 37, A701-A724 (2015) · Zbl 1320.82052 · doi:10.1137/140980545
[211] Zheng, M.; Liu, F.; Anh, V.; Turner, I., A high-order spectral method for the multi-term time-fractional diffusion equations, Appl. Math. Model., 40, 4970-4985 (2016) · Zbl 1459.65205 · doi:10.1016/j.apm.2015.12.011
[212] Zhou, H.; Tian, W. Y.; Deng, W. H., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 56, 45-66 (2013) · Zbl 1278.65130 · doi:10.1007/s10915-012-9661-0
[213] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM. J. Numer. Anal., 46, 1079-1095 (2008) · Zbl 1173.26006 · doi:10.1137/060673114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.