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Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves

  • * Corresponding author: H. Jafari

    * Corresponding author: H. Jafari
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  • This paper aimed at obtaining the traveling-wave solution of the nonlinear time fractional regularized long-wave equation. In this approach, firstly, the time fractional derivative is accomplished using a finite difference with convergence order $ \mathcal{O}(\delta t^{2-\alpha}) $ for $ 0 < \alpha< 1 $ and the nonlinear term is linearized by a linearization technique. Then, the spatial terms are approximated with the help of the radial basis function (RBF). Numerical stability of the method is analyzed by applying the Von-Neumann linear stability analysis. Three invariant quantities corresponding to mass, momentum and energy are evaluated for further validation. Numerical results demonstrate the accuracy and validity of the proposed method.

    Mathematics Subject Classification: 34K37, 91G80, 97N50.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  The distributed nodes in the computational domain with a stencil

    Figure 2.  The behavior of approximate solution for distinct values of $ \mathrm{d}\; (\alpha = 0.9) $ (left panel) and $ \alpha\; (\mathrm{d} = 0.03) $ (right panel)

    Figure 3.  The behavior of approximate solution by letting $ \mathrm{d} = 0.03, $ $ \nu = 1, $ $ h = 0.1, $ $ {\rm{ \mathsf{ τ}}} = 0.001, $ and $ c = 1 $ for $ \alpha \in \{ 0.5, 0.75\} $

    Figure 4.  The plots of single solitary wave solution and their computational errors by letting $ \alpha = 1, $ $ {\rm{ \mathsf{ τ}}} = 0.01 $, $ h = 0.125, $ and $ c = 0.8 $ for $ \mathrm{d} = 0.1 $ (up) and $ \mathrm{d} = 0.03 $ (down) at time $ T = 20 $

    Table 1.  Invariants and errors for single soliton by taking $ {\rm{ \mathsf{ τ}}} = 0.01 $, $ N = 1000 $, $ \mathrm{d} = 0.1 $ and $ c = 0.5 $ when $ \alpha = 1 $ in the domain $ [-80,100] $

    Method $ T $ $ N_I $ Cond($ M $) $ L_{\infty} $ $ L_2 $ $ I_1 $ $ I_2 $ $ I_3 $
    RBF-FD $ 5 $ $ 631 $ $ 5.8562E+02 $ $ 1.8585E-09 $ $ 1.5072E-08 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    GRBF $ 5 $ $ - $ $ 7.7014E+06 $ $ 4.4431E-07 $ $ 2.4354E-06 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    RBF-FD $ 10 $ $ 631 $ $ 5.8562E+02 $ $ 3.7224E-09 $ $ 3.0141E-08 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    GRBF $ 10 $ $ - $ $ 7.7014E+06 $ $ 8.7584E-07 $ $ 4.8478E-06 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    RBF-FD $ 15 $ $ 631 $ $ 5.8562E+02 $ $ 5.5814E-09 $ $ 4.5208E-08 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    GRBF $ 15 $ $ - $ $ 7.7014E+06 $ $ 1.3066E-06 $ $ 7.2527E-06 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    RBF-FD $ 20 $ $ 631 $ $ 5.8562E+02 $ $ 7.4388E-09 $ $ 6.0272E-08 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    GRBF $ 20 $ $ - $ $ 7.7014E+06 $ $ 1.7364E-06 $ $ 9.6557E-06 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    RBF-FD $ 25 $ $ 631 $ $ 5.8562E+02 $ $ 9.2911E-09 $ $ 7.5333E-08 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
    GRBF $ 25 $ $ - $ $ 7.7014E+06 $ $ 2.1649E-06 $ $ 1.2058E-05 $ $ 3.9759698 $ $ 0.80962219 $ $ 2.5764285 $
     | Show Table
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    Table 2.  Invariants and errors for single soliton by letting $ {\rm{ \mathsf{ τ}}} = 0.01 $, $ N = 1000 $, $ \mathrm{d} = 0.03 $ and $ c = 0.75 $ when $ \alpha = 1 $ in the domain $ [-80,100] $

    Method $ T $ $ N_I $ Cond($ M $) $ L_{\infty} $ $ L_2 $ $ I_1 $ $ I_2 $ $ I_3 $
    RBF-FD $ 5 $ $ 631 $ $ 5.7100E+02 $ $ 2.4958E-07 $ $ 8.2317E-07 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    GRBF $ 5 $ $ - $ $ 1.3531E+07 $ $ 1.6526E-05 $ $ 5.2640E-05 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    RBF-FD $ 10 $ $ 631 $ $ 5.7100E+02 $ $ 3.6639E-07 $ $ 1.8503E-06 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    GRBF $ 10 $ $ - $ $ 1.3531E+07 $ $ 2.6723E-05 $ $ 1.2832E-04 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    RBF-FD $ 15 $ $ 631 $ $ 5.7100E+02 $ $ 4.1549E-07 $ $ 2.7871E-06 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    GRBF $ 15 $ $ - $ $ 1.3531E+07 $ $ 3.2940E-05 $ $ 2.0597E-04 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    RBF-FD $ 20 $ $ 631 $ $ 5.7100E+02 $ $ 4.5519E-07 $ $ 3.6033E-06 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    GRBF $ 20 $ $ - $ $ 1.3531E+07 $ $ 3.6973E-05 $ $ 2.7849E-04 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    RBF-FD $ 25 $ $ 631 $ $ 5.7100E+02 $ $ 1.1159E-06 $ $ 4.2789E-06 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
    GRBF $ 25 $ $ - $ $ 1.3531E+07 $ $ 6.4814E-05 $ $ 3.4796E-04 $ $ 2.1072996 $ $ 0.12717442 $ $ 0.38841718 $
     | Show Table
    DownLoad: CSV

    Table 3.  Invariants and errors norms by letting $ T = 20, {\rm{ \mathsf{ τ}}} = 0.1 $, $ N = 1000 $, $ \mathrm{d} = 0.1 $ and $ c = 0.75 $ when $ \alpha = 1 $ in the domain $ [-80,100] $

    Methods$ L_{\infty} $$ L_2 $ $ I_1 $ $ I_2 $ $ I_3 $
    RBF-FD $ 7.44E-09 $$ 6.03E-08 $ $ 3.97600 $$ 0.809656 $ $ 2.57640 $
    [6] $ 8.60E-05 $$ 2.20E-04 $ $ 3.97989 $$ 0.81046 $ $ 2.57902 $
    [22] $ 1.40E-05 $$ 1.50E-04 $ $ 3.96466 $$ 0.80963 $ $ 2.56971 $
    [21] $ 2.27E-04 $$ 5.32E-04 $ $ 3.97803 $$ 0.80972 $ $ 2.57657 $
    [5] $ 2.10E-03 $$ 5.50E-03 $ $ 3.97997 $$ 0.81045 $ $ 2.57901 $
    [18] $ 1.15E-04 $$ 3.02E-04 $ $ 3.97988 $$ 0.81046 $ $ 2.57900 $
     | Show Table
    DownLoad: CSV

    Table 4.  The approximate solution $ u(x, t) $ for $ \mathrm{d} = 0.03 $ and $ x = 2 $ by taking $ {\rm{ \mathsf{ τ}}} = 0.0001, $ $ h = 0.1 $ and $ c = 1 $

    $ T $$ \alpha = 0.8 $
    Method of [12]Method of [13]Present Method
    $ 0.01 $ $ 0.08519576412 $ $ 0.07279537400 $ $ 0.08537651205 $
    $ 0.02 $$ 0.08357213920 $ $ 0.07279537400 $ $ 0.08264519947 $
    $ 0.03 $$ 0.08214278398 $ $ 0.07128811205 $$ 0.08181623410 $
    $ 0.04 $$ 0.08083623113 $ $ 0.07066631588 $ $ 0.08090025798 $
    $ 0.05 $$ 0.07962032856 $ $ 0.07198119193 $ $ 0.07973865076 $
    $ 0.06 $$ 0.07847675139 $ $ 0.06956555612 $$ 0.07781678793 $
    $ 0.07 $$ 0.07739366105 $ $ 0.06906896549 $$ 0.07695183111 $
    $ 0.08 $$ 0.07636277983 $ $ 0.06860131956 $$ 0.07614922701 $
    $ 0.09 $$ 0.07590726375 $ $ 0.06815922842 $$ 0.07520979105 $
    $ 0.10 $$ 0.07443460092 $ $ 0.06774010956 $$ 0.07487357774 $
     | Show Table
    DownLoad: CSV

    Table 5.  The approximate solution $ u(x, t) $ for $ \mathrm{d} = 0.03 $ and $ x = 2 $ by letting $ {\rm{ \mathsf{ τ}}} = 0.0001, $ $ h = 0.1 $ and $ c = 1 $

    $ T $$ \alpha = 0.9 $
    Method of [12]Method of [13]Present Method
    $ 0.01 $$ 0.08603543519 $ $ 0.07872750691 $$ 0.08640643159 $
    $ 0.02 $$ 0.08484592054 $ $ 0.07786464651 $$ 0.08489204894 $
    $ 0.03 $$ 0.08373410921 $ $ 0.07706362124 $$ 0.08360036158 $
    $ 0.04 $$ 0.08267550004 $ $ 0.07630395167 $$ 0.08286912350 $
    $ 0.05 $$ 0.08165864444 $ $ 0.07557650967 $$ 0.08227833554 $
    $ 0.06 $$ 0.08067684029 $ $ 0.07487606663 $$ 0.08184132412 $
    $ 0.07 $$ 0.07972566456 $ $ 0.07419921448 $$ 0.07986678865 $
    $ 0.08 $$ 0.07880196853 $ $ 0.07354354582 $$ 0.07819674713 $
    $ 0.09 $$ 0.07790338981 $ $ 0.07290726375 $$ 0.07660742243 $
    $ 0.10 $$ 0.07702808602 $ $ 0.07228897179 $$ 0.07661724331 $
     | Show Table
    DownLoad: CSV

    Table 6.  Invariants for single solitary wave by taking $ \mathrm{d} = 0.03, $ $ h = 0.1, $ and $ c = 0.95 $ when $ \alpha = 0.5 $

    $ T $ $ I_1 $$ I_2 $$ I_3 $
    $ 0.00 $ $ 0.197708647779586 $ $ 0.128293299043990 $ $ 0.387537096937904 $
    $ 0.01 $ $ 0.197709389335031 $$ 0.126849748687847 $$ 0.387166785333068 $
    $ 0.02 $ $ 0.197709389335031 $$ 0.126832805997773 $$ 0.387113999130940 $
    $ 0.03 $ $ 0.197705310408835 $$ 0.126802946718958 $$ 0.387058367254051 $
    $ 0.04 $ $ 0.197698761277219 $$ 0.126780218019371 $$ 0.387001260827619 $
    $ 0.05 $ $ 0.197690652584066 $$ 0.126757804752181 $$ 0.386943215828569 $
    $ 0.06 $ $ 0.197681450241377 $$ 0.126735635338625 $$ 0.386884508751169 $
    $ 0.07 $ $ 0.197671429969532 $$ 0.126713659452037 $$ 0.386825304512479 $
    $ 0.08 $ $ 0.197660770590404 $$ 0.126691840813316 $$ 0.386765710882986 $
    $ 0.09 $ $ 0.197649595685292 $$ 0.126670152526776 $$ 0.386705802887836 $
    $ 0.10 $ $ 0.197637994751921 $$ 0.126648574139578 $$ 0.386645635247663 $
     | Show Table
    DownLoad: CSV
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