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

Omid Nikan; Seyedeh Mahboubeh Molavi-Arabshai; Hossein Jafari

doi:10.3934/dcdss.2020466

Article Contents

2021, Volume 14, Issue 10: 3685-3701. Doi: 10.3934/dcdss.2020466

This issue Previous Article Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case Next Article Some new bounds analogous to generalized proportional fractional integral operator with respect to another function

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

Omid Nikan^1,,
Seyedeh Mahboubeh Molavi-Arabshai^1, and
Hossein Jafari^2,3,4, ,

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
^* Corresponding author: H. Jafari

Received: February 2020

Revised: June 2020

Early access: November 2020

Published: October 2021

Abstract / Introduction Full Text(HTML) Figure(4) / Table(6) Related Papers Cited by

Abstract

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.

Keywords:

Fractional regularized long-wave equation,
Caputo fractional derivative,
RBF-FD,
Stability.

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

Citation:

Full Text(HTML)

Figure 1. The distributed nodes in the computational domain with a stencil

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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)

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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\} $

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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 $

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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 $

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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 $

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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 $

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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 $
$ T $	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 $

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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 $
$ T $	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 $

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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 $

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