SCIENTIA SINICA Informationis, Volume 48 , Issue 7 : 871-887(2018) https://doi.org/10.1360/N112017-00295

High-accuracy analysis of finite-element method for two-term mixed time-fractional diffusion-wave equations

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  • ReceivedMar 27, 2018
  • AcceptedApr 8, 2018
  • PublishedJul 12, 2018


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  • Table 1   Temporal numerical results with $\alpha_1~=0.3,0.5,~\alpha~=~1.9,1.3$ at $~t_n=1$
    $\alpha_1,~\alpha$ $\tau$ $\|u^n-U^n\|_0$ Rate $\|u^n-U^n\|_1$ Rate $\|I_hu^n-U^n\|_1$ Rate $\|u^n-I_{2h}U^n\|_1$ Rate
    $t_n/3$ 4.373E$-$1 8.276E$-$1 8.978E$-$1 9.202E$-$1
    $\alpha_1=0.3$ $t_n/6$ 2.140E$-$1 1.031 3.985E$-$1 1.054 4.327E$-$1 1.053 4.387E$-$1 1.069
    $\alpha=1.9$ $t_n/12$ 1.017E$-$1 1.073 1.879E$-$1 1.084 2.037E$-$1 1.087 2.054E$-$1 1.094
    $t_n/24$ 4.768E$-$2 1.093 8.793E$-$2 1.096 9.574E$-$2 1.089 9.639E$-$2 1.092
    $t_n/3$ 1.619E$-$1 3.377E$-$1 3.536E$-$1 3.378E$-$1
    $\alpha_1=0.5$ $t_n/6$ 5.854E$-$2 1.467 1.214E$-$1 1.476 1.287E$-$1 1.458 1.307E$-$1 1.370
    $\alpha=1.3$ $t_n/12$ 2.069E$-$2 1.501 4.297E$-$2 1.499 4.632E$-$2 1.474 4.592E$-$2 1.509
    $t_n/24$ 7.242E$-$3 1.515 1.506E$-$2 1.496 1.622E$-$2 1.514 1.622E$-$2 1.505
  • Table 2   Spatial numerical results with $\alpha~=1.5,~\alpha_1=0.5,\tau=10^{-3}$ at $~t=0.3,0.5,0.9,1$
    $t$ $~m\times~n~$ $\|u^n-U^n\|_0$ Rate $\|u^n-U^n\|_1$ Rate $\|I_hu^n-U^n\|_1$ Rate $\|u^n-I_{2h}U^n\|_1$ Rate
    $4\times4$ 6.434E$-$4 6.034E$-$3 3.063E$-$3 3.954E$-$3
    $t=0.3$ $8\times8$ 1.525E$-$4 2.077 2.964E$-$3 1.026 7.922E$-$4 1.951 9.969E$-$4 1.987
    $16\times16$ 3.755E$-$5 2.021 1.475E$-$3 1.006 2.009E$-$4 1.979 2.507E$-$4 1.992
    $32\times32$ 9.352E$-$6 2.006 7.369E$-$4 1.002 5.168E$-$5 1.959 6.378E$-$5 1.975
    $4\times4$ 2.076E$-$3 1.930E$-$2 9.576E$-$3 1.247E$-$2
    $t=0.5$ $8\times8$ 4.923E$-$4 2.076 9.495E$-$3 1.024 2.478E$-$3 1.950 3.146E$-$3 1.986
    $16\times16$ 1.212E$-$4 2.022 4.728E$-$3 1.006 6.273E$-$4 1.982 7.900E$-$4 1.993
    $32\times32$ 3.004E$-$5 2.012 2.362E$-$3 1.001 1.598E$-$4 1.973 1.997E$-$4 1.984
    $4\times4$ 8.746E$-$3 7.893E$-$2 3.731E$-$2 4.951E$-$2
    $t=0.9$ $8\times8$ 2.079E$-$3 2.073 3.893E$-$2 1.020 9.676E$-$3 1.947 1.252E$-$2 1.984
    $16\times16$ 5.122E$-$4 2.021 1.939E$-$2 1.005 2.446E$-$3 1.984 3.140E$-$3 1.995
    $32\times32$ 1.268E$-$4 2.014 9.692E$-$3 1.001 6.181E$-$4 1.984 7.897E$-$4 1.992
    $4\times4$ 1.146E$-$2 1.024E$-$1 4.787E$-$2 6.384E$-$2
    $t=1$ $8\times8$ 2.726E$-$3 2.072 5.058E$-$2 1.019 1.242E$-$2 1.946 1.614E$-$2 1.983
    $16\times16$ 6.719E$-$4 2.020 2.521E$-$2 1.005 3.139E$-$3 1.984 4.051E$-$3 1.995
    $32\times32$ 1.665E$-$4 2.013 1.259E$-$2 1.001 7.926E$-$4 1.946 1.018E$-$3 1.993
  • Table 3   Spatial numerical results with $\alpha~=1.3,~\alpha_1=0.1,\tau=10^{-3}$ at $t=0.2,0.4,0.6,0.8$
    $t$ $~m\times~n~$ $\|u^n-U^n\|_0$ Rate $\|u^n-U^n\|_1$ Rate $\|I_hu^n-U^n\|_1$ Rate $\|u^n-I_{2h}U^n\|_1$ Rate
    $4\times4$ 1.155E$-$3 1.084E$-$2 5.508E$-$3 7.106E$-$3
    $t=0.2$ $8\times8$ 2.738E$-$4 2.077 5.323E$-$3 1.026 1.424E$-$3 1.951 1.792E$-$3 1.988
    $16\times16$ 6.744E$-$5 2.021 2.649E$-$3 1.006 3.611E$-$4 1.979 4.504E$-$4 1.992
    $32\times32$ 1.679E$-$5 2.006 1.323E$-$3 1.002 9.267E$-$5 1.962 1.144E$-$4 1.977
    $4\times4$ 6.156E$-$3 5.706E$-$2 2.813E$-$2 3.671E$-$2
    $t=0.4$ $8\times8$ 1.460E$-$3 2.076 2.808E$-$2 1.023 7.277E$-$3 1.950 9.261E$-$3 1.987
    $16\times16$ 3.599E$-$4 2.020 1.398E$-$2 1.006 1.837E$-$3 1.986 2.322E$-$3 1.996
    $32\times32$ 8.944E$-$5 2.009 6.984E$-$3 1.001 4.631E$-$4 1.988 5.831E$-$4 1.994
    $4\times4$ 1.656E$-$2 1.506E$-$1 7.195E$-$2 9.508E$-$2
    $t=0.6$ $8\times8$ 3.935E$-$3 2.073 7.425E$-$2 1.021 1.864E$-$2 1.948 2.402E$-$2 1.985
    $16\times16$ 9.707E$-$4 2.019 3.699E$-$2 1.005 4.705E$-$3 1.987 6.021E$-$3 1.996
    $32\times32$ 2.415E$-$4 2.007 1.848E$-$2 1.001 1.182E$-$3 1.993 1.508E$-$3 1.997
    $4\times4$ 3.374E$-$2 2.998E$-$1 1.389E$-$1 1.859E$-$1
    $t=0.8$ $8\times8$ 8.033E$-$3 2.071 1.480E$-$1 1.018 3.606E$-$2 1.945 4.701E$-$2 1.983
    $16\times16$ 1.983E$-$3 2.018 7.378E$-$2 1.004 9.099E$-$3 1.987 1.178E$-$2 1.996
    $32\times32$ 4.936E$-$4 2.006 3.686E$-$2 1.001 2.283E$-$3 1.995 2.950E$-$3 1.998