A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher’s equation. (English) Zbl 1304.35713
Summary: In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher’s equation, which is obtained from the standard one-dimensional Fisher’s equation by replacing the first-order time derivative with a fractional derivative (of order \(\alpha\), with \(0<\alpha <1\)). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order \(O(h^{k+1}+\tau^{2-\alpha})\), where \(h\), \(\tau\) and \(k\) are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
time-fractional Fisher’s equation; local discontinuous Galerkin finite element method; Caputo fractional derivative; fractional differential equationReferences:
| [1] | DOI: 10.1016/S0377-0427(01)00356-9 · Zbl 0992.65108 · doi:10.1016/S0377-0427(01)00356-9 |
| [2] | DOI: 10.1007/BF02462380 · Zbl 0423.35079 · doi:10.1007/BF02462380 |
| [3] | Brazhnik P., SIAM J. Appl. Math 60 (2) pp 371– (1999) |
| [4] | Brazhnik P.K., SIAM J. Appl. Math 60 (2) pp 371– (1999) |
| [5] | DOI: 10.1002/num.1690110206 · Zbl 0819.65124 · doi:10.1002/num.1690110206 |
| [6] | Castillo P.E., Internat. J. Numer. Methods Engrg 81 pp 1475– (2010) |
| [7] | Cockburn B., M2AN Math. Model. Numer. Anal 25 pp 337– (1991) |
| [8] | DOI: 10.1137/S0036142997316712 · Zbl 0927.65118 · doi:10.1137/S0036142997316712 |
| [9] | DOI: 10.1016/0021-9991(89)90183-6 · Zbl 0677.65093 · doi:10.1016/0021-9991(89)90183-6 |
| [10] | Cockburn B., Math. Comp 54 pp 545– (1990) |
| [11] | DOI: 10.1137/S0036142900371544 · Zbl 1041.65080 · doi:10.1137/S0036142900371544 |
| [12] | DOI: 10.1016/j.jcp.2009.07.021 · Zbl 1179.65107 · doi:10.1016/j.jcp.2009.07.021 |
| [13] | DOI: 10.1016/j.chaos.2005.08.180 · Zbl 1142.65418 · doi:10.1016/j.chaos.2005.08.180 |
| [14] | DOI: 10.1006/jmaa.2000.7194 · doi:10.1006/jmaa.2000.7194 |
| [15] | DOI: 10.1023/B:NUMA.0000027736.85078.be · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be |
| [16] | DOI: 10.1016/j.physa.2005.10.058 · doi:10.1016/j.physa.2005.10.058 |
| [17] | DOI: 10.1111/j.1469-1809.1937.tb02153.x · doi:10.1111/j.1469-1809.1937.tb02153.x |
| [18] | DOI: 10.1007/s002850000047 · Zbl 0982.92028 · doi:10.1007/s002850000047 |
| [19] | DOI: 10.1016/j.na.2005.01.094 · Zbl 1073.35123 · doi:10.1016/j.na.2005.01.094 |
| [20] | DOI: 10.1016/j.amc.2008.12.089 · Zbl 1162.65394 · doi:10.1016/j.amc.2008.12.089 |
| [21] | DOI: 10.1016/S0045-7825(98)00108-X · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X |
| [22] | DOI: 10.1016/j.cam.2007.04.005 · Zbl 1132.26313 · doi:10.1016/j.cam.2007.04.005 |
| [23] | DOI: 10.1515/IJNSNS.2009.10.9.1127 · doi:10.1515/IJNSNS.2009.10.9.1127 |
| [24] | Kolmogorov A.N., Moscow Univ. Bull. Math 1 pp 1– (1937) |
| [25] | DOI: 10.1016/j.jcp.2003.11.013 · Zbl 1055.65109 · doi:10.1016/j.jcp.2003.11.013 |
| [26] | DOI: 10.1016/j.jcp.2007.02.001 · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [27] | DOI: 10.1002/num.1690100505 · Zbl 0810.65131 · doi:10.1002/num.1690100505 |
| [28] | DOI: 10.1002/(SICI)1098-2426(199701)13:1<51::AID-NUM4>3.0.CO;2-L · Zbl 0872.65080 · doi:10.1002/(SICI)1098-2426(199701)13:1<51::AID-NUM4>3.0.CO;2-L |
| [29] | DOI: 10.1002/(SICI)1098-2426(199811)14:6<815::AID-NUM5>3.0.CO;2-T · Zbl 0929.76093 · doi:10.1002/(SICI)1098-2426(199811)14:6<815::AID-NUM5>3.0.CO;2-T |
| [30] | DOI: 10.1016/S0898-1221(03)80028-7 · Zbl 1036.65071 · doi:10.1016/S0898-1221(03)80028-7 |
| [31] | Miller K.S., An introduction to the fractional calculus and fractional differential equations (1993) · Zbl 0789.26002 |
| [32] | DOI: 10.1016/j.amc.2004.03.014 · Zbl 1063.65055 · doi:10.1016/j.amc.2004.03.014 |
| [33] | DOI: 10.1016/j.amc.2005.11.025 · Zbl 1096.65131 · doi:10.1016/j.amc.2005.11.025 |
| [34] | DOI: 10.1080/00207160903023581 · Zbl 1192.65137 · doi:10.1080/00207160903023581 |
| [35] | DOI: 10.1016/j.cam.2005.06.028 · Zbl 1092.65088 · doi:10.1016/j.cam.2005.06.028 |
| [36] | DOI: 10.1088/0305-4470/23/21/003 · Zbl 0729.35065 · doi:10.1088/0305-4470/23/21/003 |
| [37] | Podlubny I., The Laplace transform method for linear differential equations of fractional order (1994) |
| [38] | Podlubny I., Fractional differential equations (1999) · Zbl 0924.34008 |
| [39] | Reed W.H., Tech. Rep. LA-UR-73-479 (1973) |
| [40] | DOI: 10.1016/j.camwa.2009.03.043 · Zbl 1189.65255 · doi:10.1016/j.camwa.2009.03.043 |
| [41] | DOI: 10.1016/j.mcm.2011.07.016 · Zbl 1235.65115 · doi:10.1016/j.mcm.2011.07.016 |
| [42] | DOI: 10.1080/02681119808806258 · Zbl 0912.35084 · doi:10.1080/02681119808806258 |
| [43] | DOI: 10.1017/S0334270000008602 · Zbl 0728.65110 · doi:10.1017/S0334270000008602 |
| [44] | DOI: 10.1016/0375-9601(88)90027-8 · doi:10.1016/0375-9601(88)90027-8 |
| [45] | DOI: 10.1016/S0096-3003(03)00738-0 · Zbl 1054.65107 · doi:10.1016/S0096-3003(03)00738-0 |
| [46] | DOI: 10.1108/09615531311323782 · Zbl 1357.65181 · doi:10.1108/09615531311323782 |
| [47] | DOI: 10.1016/j.jcp.2004.11.001 · Zbl 1072.65130 · doi:10.1016/j.jcp.2004.11.001 |
| [48] | DOI: 10.1137/070679764 · Zbl 1173.65063 · doi:10.1137/070679764 |
| [49] | DOI: 10.1137/S0036142901390378 · Zbl 1021.65050 · doi:10.1137/S0036142901390378 |
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