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Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case
October  2021, 14(10): 3685-3701. doi: 10.3934/dcdss.2020466

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

 1 School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran 2 Department of Mathematics, University of Mazandaran, Babolsar, Iran 3 Department of Mathematical Sciences, University of South Africa, UNISA 0003, South Africa 4 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan

* Corresponding author: H. Jafari

Received  February 2020 Revised  June 2020 Published  October 2021 Early access  November 2020

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.

Citation: Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3685-3701. doi: 10.3934/dcdss.2020466
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##### References:
The distributed nodes in the computational domain with a stencil
The behavior of approximate solution for distinct values of $\mathrm{d}\; (\alpha = 0.9)$ (left panel) and $\alpha\; (\mathrm{d} = 0.03)$ (right panel)
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 $\alpha \in \{ 0.5, 0.75\}$
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 $T = 20$
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 $25$ $-$ $7.7014E+06$ $2.1649E-06$ $1.2058E-05$ $3.9759698$ $0.80962219$ $2.5764285$
 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 $25$ $-$ $7.7014E+06$ $2.1649E-06$ $1.2058E-05$ $3.9759698$ $0.80962219$ $2.5764285$
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 $25$ $-$ $1.3531E+07$ $6.4814E-05$ $3.4796E-04$ $2.1072996$ $0.12717442$ $0.38841718$
 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 $25$ $-$ $1.3531E+07$ $6.4814E-05$ $3.4796E-04$ $2.1072996$ $0.12717442$ $0.38841718$
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] $1.15E-04$ $3.02E-04$ $3.97988$ $0.81046$ $2.57900$
 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] $1.15E-04$ $3.02E-04$ $3.97988$ $0.81046$ $2.57900$
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 $0.01$ $0.08519576412$ $0.07279537400$ $0.08537651205$ $0.02$ $0.08357213920$ $0.07279537400$ $0.08264519947$ $0.03$ $0.08214278398$ $0.07128811205$ $0.08181623410$ $0.04$ $0.08083623113$ $0.07066631588$ $0.08090025798$ $0.05$ $0.07962032856$ $0.07198119193$ $0.07973865076$ $0.06$ $0.07847675139$ $0.06956555612$ $0.07781678793$ $0.07$ $0.07739366105$ $0.06906896549$ $0.07695183111$ $0.08$ $0.07636277983$ $0.06860131956$ $0.07614922701$ $0.09$ $0.07590726375$ $0.06815922842$ $0.07520979105$ $0.10$ $0.07443460092$ $0.06774010956$ $0.07487357774$
 $T$ $\alpha = 0.8$ Method of [12] Method of [13] Present Method $0.01$ $0.08519576412$ $0.07279537400$ $0.08537651205$ $0.02$ $0.08357213920$ $0.07279537400$ $0.08264519947$ $0.03$ $0.08214278398$ $0.07128811205$ $0.08181623410$ $0.04$ $0.08083623113$ $0.07066631588$ $0.08090025798$ $0.05$ $0.07962032856$ $0.07198119193$ $0.07973865076$ $0.06$ $0.07847675139$ $0.06956555612$ $0.07781678793$ $0.07$ $0.07739366105$ $0.06906896549$ $0.07695183111$ $0.08$ $0.07636277983$ $0.06860131956$ $0.07614922701$ $0.09$ $0.07590726375$ $0.06815922842$ $0.07520979105$ $0.10$ $0.07443460092$ $0.06774010956$ $0.07487357774$
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 $0.01$ $0.08603543519$ $0.07872750691$ $0.08640643159$ $0.02$ $0.08484592054$ $0.07786464651$ $0.08489204894$ $0.03$ $0.08373410921$ $0.07706362124$ $0.08360036158$ $0.04$ $0.08267550004$ $0.07630395167$ $0.08286912350$ $0.05$ $0.08165864444$ $0.07557650967$ $0.08227833554$ $0.06$ $0.08067684029$ $0.07487606663$ $0.08184132412$ $0.07$ $0.07972566456$ $0.07419921448$ $0.07986678865$ $0.08$ $0.07880196853$ $0.07354354582$ $0.07819674713$ $0.09$ $0.07790338981$ $0.07290726375$ $0.07660742243$ $0.10$ $0.07702808602$ $0.07228897179$ $0.07661724331$
 $T$ $\alpha = 0.9$ Method of [12] Method of [13] Present Method $0.01$ $0.08603543519$ $0.07872750691$ $0.08640643159$ $0.02$ $0.08484592054$ $0.07786464651$ $0.08489204894$ $0.03$ $0.08373410921$ $0.07706362124$ $0.08360036158$ $0.04$ $0.08267550004$ $0.07630395167$ $0.08286912350$ $0.05$ $0.08165864444$ $0.07557650967$ $0.08227833554$ $0.06$ $0.08067684029$ $0.07487606663$ $0.08184132412$ $0.07$ $0.07972566456$ $0.07419921448$ $0.07986678865$ $0.08$ $0.07880196853$ $0.07354354582$ $0.07819674713$ $0.09$ $0.07790338981$ $0.07290726375$ $0.07660742243$ $0.10$ $0.07702808602$ $0.07228897179$ $0.07661724331$
Invariants for single solitary wave by taking $\mathrm{d} = 0.03,$ $h = 0.1,$ and $c = 0.95$ when $\alpha = 0.5$
 $T$ $I_1$ $I_2$ $I_3$ $0.00$ $0.197708647779586$ $0.128293299043990$ $0.387537096937904$ $0.01$ $0.197709389335031$ $0.126849748687847$ $0.387166785333068$ $0.02$ $0.197709389335031$ $0.126832805997773$ $0.387113999130940$ $0.03$ $0.197705310408835$ $0.126802946718958$ $0.387058367254051$ $0.04$ $0.197698761277219$ $0.126780218019371$ $0.387001260827619$ $0.05$ $0.197690652584066$ $0.126757804752181$ $0.386943215828569$ $0.06$ $0.197681450241377$ $0.126735635338625$ $0.386884508751169$ $0.07$ $0.197671429969532$ $0.126713659452037$ $0.386825304512479$ $0.08$ $0.197660770590404$ $0.126691840813316$ $0.386765710882986$ $0.09$ $0.197649595685292$ $0.126670152526776$ $0.386705802887836$ $0.10$ $0.197637994751921$ $0.126648574139578$ $0.386645635247663$
 $T$ $I_1$ $I_2$ $I_3$ $0.00$ $0.197708647779586$ $0.128293299043990$ $0.387537096937904$ $0.01$ $0.197709389335031$ $0.126849748687847$ $0.387166785333068$ $0.02$ $0.197709389335031$ $0.126832805997773$ $0.387113999130940$ $0.03$ $0.197705310408835$ $0.126802946718958$ $0.387058367254051$ $0.04$ $0.197698761277219$ $0.126780218019371$ $0.387001260827619$ $0.05$ $0.197690652584066$ $0.126757804752181$ $0.386943215828569$ $0.06$ $0.197681450241377$ $0.126735635338625$ $0.386884508751169$ $0.07$ $0.197671429969532$ $0.126713659452037$ $0.386825304512479$ $0.08$ $0.197660770590404$ $0.126691840813316$ $0.386765710882986$ $0.09$ $0.197649595685292$ $0.126670152526776$ $0.386705802887836$ $0.10$ $0.197637994751921$ $0.126648574139578$ $0.386645635247663$
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