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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
References:
[1]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[2]

D. Bhardwaj and R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl., 40 (2000), 1397-1404.  doi: 10.1016/S0898-1221(00)00248-0.

[3]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems, Proc. Camb. Phil. Soc., 73 (1973), 391-405.  doi: 10.1017/S0305004100076945.

[4]

J. L. BonaW. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Phil. Trans. R. Soc. Lond. A., 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.

[5]

A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput., 174 (2006), 833-845.  doi: 10.1016/j.amc.2005.05.032.

[6]

L. R. T. GardnerG. A. Gardner and A. Dogan, A least-squares finite element scheme for the RLW equation, Comm. Numer. Meth. Eng., 12 (1996), 795-804.  doi: 10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-O.

[7]

A. Golbabai and O. Nikan, A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black–Scholes model, Comput. Econ., 55 (2020), 119-141.  doi: 10.1007/s10614-019-09880-4.

[8]

A. Golbabai, O. Nikan and T. Nikazad, Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, Int. J. Appl. Comput. Math., 5 (2019), 50, 22 pp. doi: 10.1007/s40819-019-0635-x.

[9]

B. Y. Guo and W. M Cao, The Fourier pseudospectral method with a restrain operator for the RLW equation, J. Comput. Phys., 74 (1988), 110-126.  doi: 10.1016/0021-9991(88)90072-1.

[10]

A. Houwe, J. Sabi'u, Z. Hammouch and S. Y Doka, Solitary pulses of the conformable derivative nonlinear differential equation governing wave propagation in low-pass electrical transmission line, Phys. Scr., 2019. doi: 10.1088/1402-4896/ab5055.

[11]

D. Kaya, S. Gülbahar, A. Yokuş and M. Gülbahar, Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Difference Equ., 2018 (2018), 77, 16 pp. doi: 10.1186/s13662-018-1531-0.

[12]

D. KumarJ. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.

[13]

D. KumarJ. SinghD. Baleanu and Su shila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Phys. A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002.

[14]

R. Mokhtari and M. Mohammadi, Numerical solution of GRLW equation using Sinc-collocation method, Comput. Phys. Commun., 181 (2010), 1266-1274.  doi: 10.1016/j.cpc.2010.03.015.

[15]

O. Nikan, A. Golbabai and T. Nikazad, Solitary wave solution of the nonlinear KdV-Benjamin-Bona-Mahony-Burgers model via two meshless methods, Eur. Phys. J. Plus., 134 (2019), 367. doi: 10.1140/epjp/i2019-12748-1.

[16]

O. NikanH. Jafari and A. Golbabai, Numerical analysis of the fractional evolution model for heat flow in materials with memory, Alex. Eng. J., 59 (2020), 2627-2637.  doi: 10.1016/j.aej.2020.04.026.

[17]

O. Nikan, J. A. Machado, A. Golbabai and T. Nikazad, Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media, Int. Commun. Heat Mass Transf., 111 (2020), 104443. doi: 10.1016/j.icheatmasstransfer.2019.104443.

[18]

Ö. OruçF. Bulut and A. Esen, Numerical solutions of regularized long wave equation by Haar wavelet method, Mediterr. J. Math., 13 (2016), 3235-3253.  doi: 10.1007/s00009-016-0682-z.

[19]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid. Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.

[20]

I. Podlubny, Fractional Differential Equations, Acdemic Press, San Diego, 1999.

[21]

K. R. Raslan, A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput., 167 (2005), 1101-1118.  doi: 10.1016/j.amc.2004.06.130.

[22]

B. Sakaİ. Dağ and A. Doğan, Galerkin method for the numerical solution of the RLW equation using quadratic B-splines, Int. J. Comput. Math., 81 (2004), 727-739.  doi: 10.1080/00207160310001650043.

[23]

M. Shahriari, B. N. Saray, M. Lakestani and J. Manafian, Numerical treatment of the Benjamin-Bona-Mahony equation using alpert multiwavelets, Eur. Phys. J. Plus, 133 (2018), 201. doi: 10.1140/epjp/i2018-12030-2.

[24]

A. I. Tolstykh and D. A. Shirobokov, On using radial basis functions in a "finite difference mode" with applications to elasticity problems, Comput. Mech., 33 (2003), 68-79.  doi: 10.1007/s00466-003-0501-9.

[25]

N. ValliammalC. RavichandranZ. Hammouch and H. M. Baskonus, A new investigation on fractional-ordered neutral differential systems with state-dependent delay, Int. J. Nonlin. Sci. Num., 20 (2019), 803-809.  doi: 10.1515/ijnsns-2018-0362.

show all references

References:
[1]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. R. Soc. Lond. A., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[2]

D. Bhardwaj and R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl., 40 (2000), 1397-1404.  doi: 10.1016/S0898-1221(00)00248-0.

[3]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems, Proc. Camb. Phil. Soc., 73 (1973), 391-405.  doi: 10.1017/S0305004100076945.

[4]

J. L. BonaW. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Phil. Trans. R. Soc. Lond. A., 302 (1981), 457-510.  doi: 10.1098/rsta.1981.0178.

[5]

A. Esen and S. Kutluay, Application of a lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput., 174 (2006), 833-845.  doi: 10.1016/j.amc.2005.05.032.

[6]

L. R. T. GardnerG. A. Gardner and A. Dogan, A least-squares finite element scheme for the RLW equation, Comm. Numer. Meth. Eng., 12 (1996), 795-804.  doi: 10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-O.

[7]

A. Golbabai and O. Nikan, A computational method based on the moving least-squares approach for pricing double barrier options in a time-fractional Black–Scholes model, Comput. Econ., 55 (2020), 119-141.  doi: 10.1007/s10614-019-09880-4.

[8]

A. Golbabai, O. Nikan and T. Nikazad, Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, Int. J. Appl. Comput. Math., 5 (2019), 50, 22 pp. doi: 10.1007/s40819-019-0635-x.

[9]

B. Y. Guo and W. M Cao, The Fourier pseudospectral method with a restrain operator for the RLW equation, J. Comput. Phys., 74 (1988), 110-126.  doi: 10.1016/0021-9991(88)90072-1.

[10]

A. Houwe, J. Sabi'u, Z. Hammouch and S. Y Doka, Solitary pulses of the conformable derivative nonlinear differential equation governing wave propagation in low-pass electrical transmission line, Phys. Scr., 2019. doi: 10.1088/1402-4896/ab5055.

[11]

D. Kaya, S. Gülbahar, A. Yokuş and M. Gülbahar, Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions, Adv. Difference Equ., 2018 (2018), 77, 16 pp. doi: 10.1186/s13662-018-1531-0.

[12]

D. KumarJ. Singh and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves, Math. Methods Appl. Sci., 40 (2017), 5642-5653.  doi: 10.1002/mma.4414.

[13]

D. KumarJ. SinghD. Baleanu and Su shila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Phys. A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002.

[14]

R. Mokhtari and M. Mohammadi, Numerical solution of GRLW equation using Sinc-collocation method, Comput. Phys. Commun., 181 (2010), 1266-1274.  doi: 10.1016/j.cpc.2010.03.015.

[15]

O. Nikan, A. Golbabai and T. Nikazad, Solitary wave solution of the nonlinear KdV-Benjamin-Bona-Mahony-Burgers model via two meshless methods, Eur. Phys. J. Plus., 134 (2019), 367. doi: 10.1140/epjp/i2019-12748-1.

[16]

O. NikanH. Jafari and A. Golbabai, Numerical analysis of the fractional evolution model for heat flow in materials with memory, Alex. Eng. J., 59 (2020), 2627-2637.  doi: 10.1016/j.aej.2020.04.026.

[17]

O. Nikan, J. A. Machado, A. Golbabai and T. Nikazad, Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media, Int. Commun. Heat Mass Transf., 111 (2020), 104443. doi: 10.1016/j.icheatmasstransfer.2019.104443.

[18]

Ö. OruçF. Bulut and A. Esen, Numerical solutions of regularized long wave equation by Haar wavelet method, Mediterr. J. Math., 13 (2016), 3235-3253.  doi: 10.1007/s00009-016-0682-z.

[19]

D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid. Mech., 25 (1966), 321-330.  doi: 10.1017/S0022112066001678.

[20]

I. Podlubny, Fractional Differential Equations, Acdemic Press, San Diego, 1999.

[21]

K. R. Raslan, A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput., 167 (2005), 1101-1118.  doi: 10.1016/j.amc.2004.06.130.

[22]

B. Sakaİ. Dağ and A. Doğan, Galerkin method for the numerical solution of the RLW equation using quadratic B-splines, Int. J. Comput. Math., 81 (2004), 727-739.  doi: 10.1080/00207160310001650043.

[23]

M. Shahriari, B. N. Saray, M. Lakestani and J. Manafian, Numerical treatment of the Benjamin-Bona-Mahony equation using alpert multiwavelets, Eur. Phys. J. Plus, 133 (2018), 201. doi: 10.1140/epjp/i2018-12030-2.

[24]

A. I. Tolstykh and D. A. Shirobokov, On using radial basis functions in a "finite difference mode" with applications to elasticity problems, Comput. Mech., 33 (2003), 68-79.  doi: 10.1007/s00466-003-0501-9.

[25]

N. ValliammalC. RavichandranZ. Hammouch and H. M. Baskonus, A new investigation on fractional-ordered neutral differential systems with state-dependent delay, Int. J. Nonlin. Sci. Num., 20 (2019), 803-809.  doi: 10.1515/ijnsns-2018-0362.

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 $
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 $
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 $
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 $
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 $$ 0.81046 $ $ 2.57900 $
Methods$ L_{\infty} $$ 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 $
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 $
$ 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 $
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 $$ 0.07661724331 $
$ T $$ \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 $
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 $
$ 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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