Traveling-wave and numerical investigations to nonlinear equations via modern computational techniques

Taghread Ghannam Alharbi; Abdulghani Alharbi; Taghread Ghannam Alharbi; Abdulghani Alharbi

doi:10.3934/math.2024595

AIMS Mathematics

2024, Volume 9, Issue 5: 12188-12210. doi: 10.3934/math.2024595

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Research article

Traveling-wave and numerical investigations to nonlinear equations via modern computational techniques

Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia; taghread1@hotmail.com
Correction on: AIMS Mathematics 9: 14310-14311

Received: 30 November 2023 Revised: 28 January 2024 Accepted: 06 February 2024 Published: 28 March 2024
MSC : 35A24, 35Q51, 65N06, 65N40, 65N50

In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G$ ^\prime $)-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh.
- Gilson-Pickering equation,
- exact wave solutions,
- numerical solutions,
- stability,
- accuracy,
- Parabolic-Monge-Ampere
Citation: Taghread Ghannam Alharbi, Abdulghani Alharbi. Traveling-wave and numerical investigations to nonlinear equations via modern computational techniques[J]. AIMS Mathematics, 2024, 9(5): 12188-12210. doi: 10.3934/math.2024595

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Abstract

In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G$ ^\prime $)-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh.

References

[1]	M. B. Almatrafi, A. Alharbi, New soliton wave solutions to a nonlinear equation arising in plasma physics, Comput. Model. Eng. Sci., 137 (2023), 827–841. https://doi.org/10.32604/cmes.2023.027344 doi: 10.32604/cmes.2023.027344
[2]	C. Dai, J. Zhang, Jacobian elliptic function method for nonlinear differential-difference equations, Chaos Solitons Fract., 27 (2006), 1042–1047. https://doi.org/10.1016/j.chaos.2005.04.071 doi: 10.1016/j.chaos.2005.04.071
[3]	C. Wei, B. Tian, D. Yang, S. Liu, Jacobian-elliptic-function and rogue-periodic-wave solutions of a high-order nonlinear Schrödinger equation in an inhomogeneous optical fiber, Chin. J. Phys., 81 (2023), 354–361. https://doi.org/10.1016/j.cjph.2022.11.023 doi: 10.1016/j.cjph.2022.11.023
[4]	A. M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., 187 (2007), 1131–1142. https://doi.org/10.1016/j.amc.2006.09.013 doi: 10.1016/j.amc.2006.09.013
[5]	M. B. Almatrafi, Solitary wave solutions to a fractional model using the improved modified extended tanh-function method, Fractal Fract., 7 (2023), 252. https://doi.org/10.3390/fractalfract7030252 doi: 10.3390/fractalfract7030252
[6]	S. A. Khuri, A complex tanh-function method applied to nonlinear equations of Schrödinger type, Chaos Solitons Fract., 20 (2004), 1037–1040. https://doi.org/10.1016/j.chaos.2003.09.042 doi: 10.1016/j.chaos.2003.09.042
[7]	A. Aasaraai, The application of modified F-expansion method solving the Maccari's system, Br. J. Math. Comput. Sci., 11 (2015), 1–14. https://doi.org/10.9734/BJMCS/2015/19938 doi: 10.9734/BJMCS/2015/19938
[8]	C. L. Bai, C. J. Bai, H. Zhao, A new generalized algebraic method and its application in nonlinear evolution equations with variable coefficients, Z. Naturforsch. A, 60 (2005), 211–220. https://doi.org/10.1515/zna-2005-0401 doi: 10.1515/zna-2005-0401
[9]	S. K. Mohanty, O. V. Kravchenko, M. K. Deka, A. N. Dev, D. V. Churikov, The exact solutions of the 2+1-dimensional Kadomtsev-Petviashvili equation with variable coefficients by extended generalized $G^\prime / G$-expansion method, J. King Saud Univ. Sci., 35 (2023), 102358. https://doi.org/10.1016/j.jksus.2022.102358 doi: 10.1016/j.jksus.2022.102358
[10]	S. K. Mohanty, O. V. Kravchenko, A. N. Dev, Exact traveling wave solutions of the Schamel Burgers' equation by using generalized-improved and generalized $G^\prime / G$-expansion methods, Results Phys., 33 (2022), 105124. https://doi.org/10.1016/j.rinp.2021.105124 doi: 10.1016/j.rinp.2021.105124
[11]	Y. Qiu, B. Tian, D. Xian, L. Xian, New exact solutions of nontraveling wave and local excitation of dynamic behavior for GGKdV equation, Results Phys., 49 (2023), 106463. https://doi.org/10.1016/j.rinp.2023.106463 doi: 10.1016/j.rinp.2023.106463
[12]	A. R. Alharbi, M. B. Almatrafi, Exact solitary wave and numerical solutions for geophysical KdV equation, J. King Saud Univ. Sci., 34 (2022), 102087. https://doi.org/10.1016/j.jksus.2022.102087 doi: 10.1016/j.jksus.2022.102087
[13]	A. R. Alharbi, Traveling-wave and numerical solutions to a Novikov-Veselov system via the modified mathematical methods, AIMS Math., 8 (2023), 1230–1250. https://doi.org/10.3934/math.2023062 doi: 10.3934/math.2023062
[14]	A. Alharbi, M. B. Almatrafi, M. A. E. Abdelrahman, Constructions of the travelling wave solutions to the MRLW equation and their stability and accuracy arising in plasma physics, Int. J. Appl. Comput. Math., 9 (2023), 46. https://doi.org/10.1007/s40819-023-01520-8 doi: 10.1007/s40819-023-01520-8
[15]	A. R. Alharbi, A study of traveling wave structures and numerical investigation of two-dimensional Riemann problems with their stability and accuracy, Comput. Model. Eng. Sci., 134 (2023), 2193–2209. https://doi.org/10.32604/cmes.2022.018445 doi: 10.32604/cmes.2022.018445
[16]	A. Alharbi, M. B. Almatrafi, Exact and numerical solitary wave structures to the variant Boussinesq system, Symmetry, 12 (2020), 1473. https://doi.org/10.3390/sym12091473 doi: 10.3390/sym12091473
[17]	T. Han, Z. Li, K. Zhang, Exact solutions of the stochastic fractional long-short wave interaction system with multiplicative noise in generalized elastic medium, Results Phys., 44 (2023), 106174. https://doi.org/10.1016/j.rinp.2022.106174 doi: 10.1016/j.rinp.2022.106174
[18]	T. Han, Z. Li, C. Li, Bifurcation analysis, stationary optical solitons and exact solutions for generalized nonlinear Schrödinger equation with nonlinear chromatic dispersion and quintuple power-law of refractive index in optical fibers, Phys. A, 615 (2023), 128599. https://doi.org/10.1016/j.physa.2023.128599 doi: 10.1016/j.physa.2023.128599
[19]	T. Han, L. Zhao, Bifurcation, sensitivity analysis and exact traveling wave solutions for the stochastic fractional Hirota-Maccari system, Results Phys., 47 (2023), 106349. https://doi.org/10.1016/j.rinp.2023.106349 doi: 10.1016/j.rinp.2023.106349
[20]	T. Han, Z. Zhao, K. Zhang, C. Tang, Chaotic behavior and solitary wave solutions of stochastic-fractional Drinfel'd-Sokolov-Wilson equations with Brownian motion, Results Phys., 51 (2023), 106657. https://doi.org/10.1016/j.rinp.2023.106657 doi: 10.1016/j.rinp.2023.106657
[21]	K. K. Ali, R. Yilmazer, A. Yokus, H. Bulut, Analytical solutions for the (3+1)-dimensional nonlinear extended quantum Zakharov-Kuznetsov equation in plasma physics, Phys. A, 548 (2020), 124327. https://doi.org/10.1016/j.physa.2020.124327 doi: 10.1016/j.physa.2020.124327
[22]	C. Gilson, A. Pickering, Factorization and Painlevé analysis of a class of nonlinear third-order partial differential equations, J. Phys. A, 28 (1995), 2871. https://doi.org/10.1088/0305-4470/28/10/017 doi: 10.1088/0305-4470/28/10/017
[23]	N. A. Mohamed, A. S. Rashed, A. Melaibari, H. M. Sedighi, M. A. Eltaher, Effective numerical technique applied for Burgers' equation of (1+1)-, (2+1)-dimensional, and coupled forms, Math. Meth. Appl. Sci., 44 (2021), 10135–10153. https://doi.org/10.1002/mma.7395 doi: 10.1002/mma.7395
[24]	N. A. Mohamed, Solving one and two-dimensional unsteady Burgers' equation using fully implicit finite difference schemes, Arab J. Basic Appl. Sci., 26 (2019), 254–268. https://doi.org/10.1080/25765299.2019.1613746 doi: 10.1080/25765299.2019.1613746
[25]	N. Mohamed, Fully implicit scheme for solving Burgers' equation based on finite difference method, Egypt. Int. J. Eng. Sci. Technol., 26 (2018), 1687–8493. https://doi.org/10.21608/eijest.2018.97263 doi: 10.21608/eijest.2018.97263
[26]	M. Mohamed, S. M. Mabrouk, A. S. Rashed, Mathematical investigation of the infection dynamics of COVID-19 using the fractional differential quadrature method, Computation, 11 (2023), 198. https://doi.org/10.3390/computation11100198 doi: 10.3390/computation11100198
[27]	T. G. Alharbi, A. Alharbi, A study of traveling wave structures and numerical investigations into the coupled nonlinear Schrödinger equation using advanced mathematical techniques, Mathematics, 11 (2023), 4597. https://doi.org/10.3390/math11224597 doi: 10.3390/math11224597
[28]	A. R. Alharbi, M. B. Almatrafi, Analytical and numerical solutions for the variant Boussinseq equations, J. Taibah Univ. Sci., 14 (2020), 454–462. https://doi.org/10.1080/16583655.2020.1746575 doi: 10.1080/16583655.2020.1746575
[29]	A. R. Alharbi, Numerical solution of thin-film flow equations using adaptive moving mesh methods, Keele University Press, 2016.
[30]	A. Alharbi, S. Naire, An adaptive moving mesh method for thin film flow equations with surface tension, J. Comput. Appl. Math., 319 (2017), 365–384. https://doi.org/10.1016/j.cam.2017.01.019 doi: 10.1016/j.cam.2017.01.019
[31]	C. J. Budd, W. Huang, R. D. Russell, Adaptivity with moving grids, Acta Numer., 18 (2009), 111–241. https://doi.org/10.1017/S0962492906400015
[32]	W. Huang, R. D. Russell, Adaptive moving mesh methods, Springer, 2010. https://doi.org/10.1007/978-1-4419-7916-2
[33]	S. H. Alhejaili, A. Alharbi, Structure of analytical and numerical wave solutions for the nonlinear (1+1)-coupled Drinfel'd-Sokolov-Wilson system arising in shallow water waves, Mathematics, 11 (2023), 4598. https://doi.org/10.3390/math11224598 doi: 10.3390/math11224598
[34]	A. R. Alharbi, Numerical investigation for the GRLW equation using parabolic Monge Ampere equation, Int. J. Math. Comput. Sci., 15 (2020), 443–462.
[35]	C. J. Budd, J. F. Williams, Moving mesh generation using the parabolic Monge-Ampère equation, SIAM J. Sci. Comput., 31 (2009), 3438–3465. https://doi.org/10.1137/080716773 doi: 10.1137/080716773
[36]	A. R. Alharbi, Numerical solutions to two-dimensional fourth order parabolic thin film equations using the Parabolic Monge-Ampere method, AIMS Math., 8 (2023), 16463–16478. https://doi.org/10.3934/math.2023841 doi: 10.3934/math.2023841
[37]	K. L. di Pietro, A. E. Lindsay, Monge-Ampére simulation of fourth order PDEs in two dimensions with application to elastic-electrostatic contact problems, J. Comput. Phys., 349 (2017), 328–350. https://doi.org/10.1016/j.jcp.2017.08.032 doi: 10.1016/j.jcp.2017.08.032
[38]	A. Chen, W. Huang, S. Tang, Bifurcations of travelling wave solutions for the Gilson-Pickering equation, Nonlinear Anal., 10 (2009), 1468–1218. https://doi.org/10.1016/j.nonrwa.2008.07.005 doi: 10.1016/j.nonrwa.2008.07.005
[39]	P. A. Clarkson, E. L. Mansfield, T. J. Priestley, Symmetries of a class of nonlinear third-order partial differential equations, Math. Comput. Modell., 25 (1997), 195–212. https://doi.org/10.1016/S0895-7177(97)00069-1 doi: 10.1016/S0895-7177(97)00069-1
[40]	K. K. Ali, H. Dutta, R. Yilmazer, S. Noeiaghdam, Wave solutions of Gilson-Pickering equation, arXiv, 2019. https://doi.org/10.48550/arXiv.1907.06254
[41]	M. Bilal, A. R. Seadawy, M. Younis, S. T. R. Rizvi, K. El-Rashidy, S. F. Mahmoud, Analytical wave structures in plasma physics modelled by Gilson-Pickering equation by two integration norms, Results Phys., 23 (2021), 103959. https://doi.org/10.1016/j.rinp.2021.103959 doi: 10.1016/j.rinp.2021.103959
[42]	H. M. Baskonus, Complex soliton solutions to the Gilson-Pickering model, Axioms, 8 (2019), 18. https://doi.org/10.3390/axioms8010018 doi: 10.3390/axioms8010018
[43]	Y. Kai, Y. Li, L. Huang, Topological properties and wave structures of Gilson-Pickering equation, Chaos Solitons Fract., 157 (2022), 111899. https://doi.org/10.1016/j.chaos.2022.111899 doi: 10.1016/j.chaos.2022.111899
[44]	A. Yokuş, H. Durur, K. A. Abro, D. Kaya, Role of Gilson-Pickering equation for the different types of soliton solutions: a nonlinear analysis, Eur. Phys. J. Plus, 135 (2020), 657. https://doi.org/10.1140/epjp/s13360-020-00646-8 doi: 10.1140/epjp/s13360-020-00646-8
[45]	K. K. Ali, H. Dutta, R. Yilmazer, S. Noeiaghdam, On the new wave behaviors of the Gilson-Pickering equation, Front. Phys., 8 (2020), 54. https://doi.org/10.3389/fphy.2020.00054 doi: 10.3389/fphy.2020.00054
[46]	K. K. Ali, M. S. Mehanna, Traveling wave solutions and numerical solutions of Gilson-Pickering equation, Results Phys., 28 (2021), 104596. https://doi.org/10.1016/j.rinp.2021.104596 doi: 10.1016/j.rinp.2021.104596
[47]	M. M. A. Khater, Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson-Pickering equation, Results Phys., 44 (2023), 106193. https://doi.org/10.1016/j.rinp.2022.106193 doi: 10.1016/j.rinp.2022.106193
[48]	K. W. Morton, D. F. Mayers, Numerical solution of partial differential equations: an introduction, Cambridge University Press, 2005. https://doi.org/10.1017/CBO9780511812248
[49]	L. R. Petzold, Description of DASSL: a differential/algebraic system solver, Sandia Natl. Labs., 1982.
[50]	P. N. Brown, A. C. Hindmarsh, L. R. Petzold, Using Krylov methods in the solution of large-scale differential-algebraic systems, SIAM J. Sci. Comput., 15 (1994), 1467–1488. https://doi.org/10.1137/0915088 doi: 10.1137/0915088
[51]	E. J. Walsh, C. Budd, Moving mesh methods for problems in meteorology, University of Bath Press, 2011.

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