Kink phenomena of the time-space fractional Oskolkov equation

M. Mossa Al-Sawalha; Humaira Yasmin; Ali M. Mahnashi; M. Mossa Al-Sawalha; Humaira Yasmin; Ali M. Mahnashi

doi:10.3934/math.20241502

AIMS Mathematics

2024, Volume 9, Issue 11: 31163-31179. doi: 10.3934/math.20241502

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Kink phenomena of the time-space fractional Oskolkov equation

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

Received: 14 September 2024 Revised: 20 October 2024 Accepted: 28 October 2024 Published: 01 November 2024
MSC : 32W50, 34A25, 83C15

In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.
- fractional Oskolkov equation,
- Bäcklund transformation,
- non-linear differential equations,
- exact solutions
Citation: M. Mossa Al-Sawalha, Humaira Yasmin, Ali M. Mahnashi. Kink phenomena of the time-space fractional Oskolkov equation[J]. AIMS Mathematics, 2024, 9(11): 31163-31179. doi: 10.3934/math.20241502

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Abstract

In this study, we applied the Riccati-Bernoulli sub-ODE method and Bäcklund transformation to analyze the time-space fractional Oskolkov equation for kink solutions by matching the coefficients and optimal series parameters. The time-space fractional Oskolkov equation is used to analyze the behavior of solitons for different applications such as fluid dynamics and viscoelastic flow. The kink solutions derived have important consequences for stability analysis and interaction dynamic in these systems, and these are useful in controlling the physical behaviour of systems described by this equation. Such effects are illustrated by 2D and 3D plots, showing that the proposed model can handle both fractional and integer-order solitons with different but equally efficient outcomes. This research contributes to a valuable analytical method that can determine and manage processes in diversified systems based on fractional differential equations. This work provides a basis for subsequent analysis in other branches of science and technology in which the fractional Oskolkov model is used.

References

[1]	S. Noor, W. Albalawi, R. Shah, M. M. Al-Sawalha, S. M. E. Ismaeel, Mathematical frameworks for investigating fractional nonlinear coupled Korteweg-de Vries and Burger's equations, Front. Phys., 12 (2024), 1374452. https://doi.org/10.3389/fphy.2024.1374452 doi: 10.3389/fphy.2024.1374452
[2]	S. Noor, W. Albalawi, R. Shah, M. M. Al-Sawalha, S. M. E. Ismaeel, S. A. El-Tantawy, On the approximations to fractional nonlinear damped Burger's-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods, Front. Phys., 12 (2024), 1374481. https://doi.org/10.3389/fphy.2024.1374481 doi: 10.3389/fphy.2024.1374481
[3]	H. Yasmin, A. S. Alshehry, A. H. Ganie, A. M. Mahnashi, R. Shah, Perturbed Gerdjikov-Ivanov equation: soliton solutions via Backlund transformation, Optik, 298 (2024), 171576. https://doi.org/10.1016/j.ijleo.2023.171576 doi: 10.1016/j.ijleo.2023.171576
[4]	S. A. El-Tantawy, H. A. Alyousef, R. T. Matoog, R. Shah, On the optical soliton solutions to the fractional complex structured $(1+1)$-dimensional perturbed Gerdjikov-Ivanov equation, Phys. Scr., 99 (2024), 035249. https://doi.org/10.1088/1402-4896/ad241b doi: 10.1088/1402-4896/ad241b
[5]	H. Jafari, N. Kadkhoda, D. Baleanu, Fractional Lie group method of the time-fractional Boussinesq equation, Nonlinear Dyn., 81 (2015), 1569–1574. https://doi.org/10.1007/s11071-015-2091-4 doi: 10.1007/s11071-015-2091-4
[6]	N. Raza, M. R. Aslam, H. Rezazadeh, Analytical study of resonant optical solitons with variable coefficients in Kerr and non-Kerr law mediaa, Opt. Quant. Electron., 51 (2019), 59. https://doi.org/10.1007/s11082-019-1773-4 doi: 10.1007/s11082-019-1773-4
[7]	W. Gao, H. Rezazadeh, Z. Pinar, H. M. Baskonus, S. Sarwar, G. Yel, Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Quant. Electron., 52 (2020), 52. https://doi.org/10.1007/s11082-019-2162-8 doi: 10.1007/s11082-019-2162-8
[8]	H. I. Abdel-Gawad, M. S. Osman, On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math. J., 53 (2013), 661–680. https://doi.org/10.5666/KMJ.2013.53.4.680 doi: 10.5666/KMJ.2013.53.4.680
[9]	H. I. Abdel-Gawad, M. Osman, On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients, J. Adv. Res., 6 (2014), 593–599. https://doi.org/10.1016/j.jare.2014.02.004 doi: 10.1016/j.jare.2014.02.004
[10]	M. N. Ali, M. S. Osman, S. M. Husnine, On the analytical solutions of conformable time-fractional extended ZakharovCKuznetsov equation through $(G'/G^2)$-expansion method and the modified Kudryashov method, SeMA J., 76 (2019), 15–25. https://doi.org/10.1007/s40324-018-0152-6 doi: 10.1007/s40324-018-0152-6
[11]	K. K. Ali, M. S. Osman, M. Abdel-Aty, New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via Sine-Gordon expansion method, Alex. Eng. J., 59 (2020), 1191–1196. https://doi.org/10.1016/j.aej.2020.01.037 doi: 10.1016/j.aej.2020.01.037
[12]	A. Kumar, R. D. Pankaj, Tanh-coth scheme for traveling wave solutions for nonlinear wave interaction model, J. Egypt. Math. Soc., 23 (2015), 282–285. https://doi.org/10.1016/j.joems.2014.05.002 doi: 10.1016/j.joems.2014.05.002
[13]	D. R. G. Domairry, A. Mohsenzadeh, M. Famouri, The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 85–95. https://doi.org/10.1016/j.cnsns.2007.07.009 doi: 10.1016/j.cnsns.2007.07.009
[14]	Z. Y. Zhang, Exact traveling wave solutions of the perturbed Klein-Gordon equation with quadratic nonlinearity in $(1+1)$-dimension, part I: without local inductance and dissipation effect, Turk. J. Phys., 37 (2013), 259–267. https://doi.org/10.3906/fiz-1205-13 doi: 10.3906/fiz-1205-13
[15]	K. A. Abro, A. Atangana, Dual fractional modeling of rate type fluid through non-local differentiation, Numer. Methods Partial Differ. Equ., 38 (2020), 390–405. https://doi.org/10.1002/num.22633 doi: 10.1002/num.22633
[16]	S. Alshammari, K. Moaddy, R. Shah, M. Alshammari, Z. Alsheekhhussain, M. M. Al-sawalha, et al., Analysis of solitary wave solutions in the fractional-order Kundu-Eckhaus system, Sci. Rep., 14 (2024), 3688. https://doi.org/10.1038/s41598-024-53330-7 doi: 10.1038/s41598-024-53330-7
[17]	S. A. El-Tantawy, R. T. Matoog, R. Shah, A. W. Alrowaily, S. M. E. Ismaeel, On the shock wave approximation to fractional generalized Burger-Fisher equations using the residual power series transform method, Phys. Fluids, 36 (2024), 023105. https://doi.org/10.1063/5.0187127 doi: 10.1063/5.0187127
[18]	Z. Alsheekhhussain, K. Moaddy, R. Shah, S. Alshammari, M. Alshammari, M. M. Al-Sawalha, et al., Extension of the optimal auxiliary function method to solve the system of a fractional-order Whitham-Broer-Kaup equation, Fractal Fract., 8 (2023), 1. https://doi.org/10.3390/fractalfract8010001 doi: 10.3390/fractalfract8010001
[19]	M. Alqhtani, K. M. Saad, R. Shah, W. M. Hamanah, Discovering novel soliton solutions for $(3+ 1)$-modified fractional Zakharov-Kuznetsov equation in electrical engineering through an analytical approach, Opt. Quant. Electron., 55 (2023), 1149. https://doi.org/10.1007/s11082-023-05407-2 doi: 10.1007/s11082-023-05407-2
[20]	S. Noor, B. M. Alotaibi, R. Shah, S. M. E. Ismaeel, S. A. El-Tantawy, On the solitary waves and nonlinear oscillations to the fractional Schrödinger-KdV equation in the framework of the Caputo operator, Symmetry, 15 (2023), 1616. https://doi.org/10.3390/sym15081616 doi: 10.3390/sym15081616
[21]	H. Yasmin, A. A. Alderremy, R. Shah, A. Hamid Ganie, S. Aly, Iterative solution of the fractional Wu-Zhang equation under Caputo derivative operator, Front. Phys., 12 (2024), 1333990. https://doi.org/10.3389/fphy.2024.1333990 doi: 10.3389/fphy.2024.1333990
[22]	Y. Kai, S. Chen, K. Zhang, Z. Yin, Exact solutions and dynamic properties of a nonlinear fourth-order time-fractional partial differential equation, Waves Random Complex Media, 2022, 1–12. https://doi.org/10.1080/17455030.2022.2044541
[23]	Y. Kai, Z. Yin, Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation, Phys. Lett. A, 452 (2022), 128430. https://doi.org/10.1016/j.physleta.2022.128430 doi: 10.1016/j.physleta.2022.128430
[24]	C. Zhu, M. Al-Dossari, S. Rezapour, B. Gunay, On the exact soliton solutions and different wave structures to the $(2+ 1)$ dimensional Chaffee-Infante equation, Results Phys., 57 (2024), 107431. https://doi.org/10.1016/j.rinp.2024.107431 doi: 10.1016/j.rinp.2024.107431
[25]	M. D. Ortigueira, J. A. T. Machado, Fractional calculus applications in signals and systems, Signal Process., 86 (2006), 2503–2504. https://doi.org/10.1016/j.sigpro.2006.02.001 doi: 10.1016/j.sigpro.2006.02.001
[26]	M. A. Abdou, An analytical method for space-time fractional nonlinear differential equations arising in plasma physics, J. Ocean Eng. Sci., 2 (2017), 288–292. https://doi.org/10.1016/j.joes.2017.09.002 doi: 10.1016/j.joes.2017.09.002
[27]	C. Baishya, S. J. Achar, P. Veeresha, D. G. Prakasha, Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos, 31 (2021), 043130. https://doi.org/10.1063/5.0028905 doi: 10.1063/5.0028905
[28]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the modified Korteweg-de Vries equation in deterministic case and random case, J. Phys. Math., 8 (2017), 1–6. https://doi.org/10.4172/2090-0902.1000214 doi: 10.4172/2090-0902.1000214
[29]	S. Meng, F. Meng, H. Chi, H. Chen, A. Pang, A robust observer based on the nonlinear descriptor systems application to estimate the state of charge of lithium-ion batteries, J. Franklin Inst., 360 (2023), 11397–11413. https://doi.org/10.1016/j.jfranklin.2023.08.037 doi: 10.1016/j.jfranklin.2023.08.037
[30]	M. A. E. Abdelrahman, M. A. Sohaly, Solitary waves for the nonlinear Schrödinger problem with the probability distribution function in the stochastic input case, Eur. Phys. J. Plus, 132 (2017), 339. https://doi.org/10.1140/epjp/i2017-11607-5 doi: 10.1140/epjp/i2017-11607-5
[31]	S. Kumar, B. Mohan, R. Kumar, Lump, soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics, Nonlinear Dyn., 110 (2022), 693–704. https://doi.org/10.1007/s11071-022-07647-5 doi: 10.1007/s11071-022-07647-5
[32]	G. P. Agrawal, Nonlinear fiber optics, In: P. L. Christiansen, M. P. S\o rensen, A. C. Scott, Nonlinear science at the dawn of the 21st century, Lecture Notes in Physics, Springer, 542 (2000), 195–211. https://doi.org/10.1007/3-540-46629-0\_9
[33]	B. Ghanbari, New analytical solutions for the Oskolkov-type equations in fluid dynamics via a modified methodology, Results Phys., 28 (2021), 104610. https://doi.org/10.1016/j.rinp.2021.104610 doi: 10.1016/j.rinp.2021.104610
[34]	E. S. Baranovskii, The Navier-Stokes-Voigt equations with position-dependent slip boundary conditions, Z. Angew. Math. Phys., 74 (2023), 6. https://doi.org/10.1007/s00033-022-01881-y doi: 10.1007/s00033-022-01881-y
[35]	M. M. Roshid, H. O. Roshid, Exact and explicit traveling wave solutions to two nonlinear evolution equations which describe incompressible viscoelastic Kelvin-Voigt fluid, Heliyon, 4 (2018), e00756. https://doi.org/10.1016/j.heliyon.2018.e00756 doi: 10.1016/j.heliyon.2018.e00756
[36]	L. Yu, Y. Lei, Y. Ma, M. Liu, J. Zheng, D. Dan, et al., A comprehensive review of fluorescence correlation spectroscopy, Front. Phys., 9 (2021), 644450. https://doi.org/10.3389/fphy.2021.644450 doi: 10.3389/fphy.2021.644450
[37]	O. A. Ilhan, H. Bulut, T. A. Sulaiman, H. M. Baskonus, Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation, Indian J. Phys., 92 (2018), 999–1007. https://doi.org/10.1007/s12648-018-1187-3 doi: 10.1007/s12648-018-1187-3
[38]	M. N. Alam, M. S. Uddin, C. Tunç, Soliton wave solutions of the Oskolkov equation arising in incompressible visco-elastic Kelvin-Voigt fluid, Abstr. Appl. Anal., 5 (2021), 334–342.
[39]	N. Kheaomaingam, S. Phibanchon, S. Chimchinda, Sine-Gordon expansion method for the kink soliton to Oskolkov equation, J. Phys.: Conf. Ser., 2431 (2023), 012097. https://doi.org/10.1088/1742-6596/2431/1/012097 doi: 10.1088/1742-6596/2431/1/012097
[40]	R. Li, Z. A. B. Sinnah, Z. M. Shatouri, J. Manafian, M. F. Aghdaei, A. Kadi, Different forms of optical soliton solutions to the Kudryashov's quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity, Results Phys., 46 (2023), 106293. https://doi.org/10.1016/j.rinp.2023.106293 doi: 10.1016/j.rinp.2023.106293
[41]	X. Liu, B. Abd Alreda, J. Manafian, B. Eslami, M. F. Aghdaei, M. Abotaleb, et al., Computational modeling of wave propagation in plasma physics over the Gilson-Pickering equation, Results Phys., 50 (2023), 106579. https://doi.org/10.1016/j.rinp.2023.106579 doi: 10.1016/j.rinp.2023.106579
[42]	Y. Qian, J. Manafian, M. Asiri, K. H. Mahmoud, A. I. Alanssari, A. S. Alsubaie, Nonparaxial solitons and the dynamics of solitary waves for the coupled nonlinear Helmholtz systems, Opt. Quant. Electron., 55 (2023), 1022. https://doi.org/10.1007/s11082-023-05232-7 doi: 10.1007/s11082-023-05232-7
[43]	R. Li, J. Manafian, H. A. Lafta, H. A. Kareem, K. F. Uktamov, M. Abotaleb, The nonlinear vibration and dispersive wave systems with cross-kink and solitary wave solutions, Int. J. Geom. Methods Mod. Phys., 19 (2022), 2250151. https://doi.org/10.1142/S0219887822501511 doi: 10.1142/S0219887822501511
[44]	X. F. Yang, Z. C. Deng, Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ., 2015 (2015), 117. https://doi.org/10.1186/s13662-015-0452-4 doi: 10.1186/s13662-015-0452-4
[45]	D. Lu, Q. Shi, New Jacobi elliptic functions solutions for the combined KdV-mKdV equation, Int. J. Nonlinear Sci., 10 (2010), 320–325.
[46]	H. Yao, D. Pugliese, M. Lancry, Y. Dai, Ultrafast laser direct writing nanogratings and their engineering in transparent materials, Laser Photonics Rev., 18 (2024), 2300891. https://doi.org/10.1002/lpor.202300891 doi: 10.1002/lpor.202300891
[47]	H. O. Roshid, M. Roshid, A. Abdeljabbar, M. Begum, H. Basher, Abundant dynamical solitary waves through Kelvin-Voigt fluid via the truncated $M$-fractional Oskolkov model, Results Phys., 55 (2023), 107128. https://doi.org/10.1016/j.rinp.2023.107128 doi: 10.1016/j.rinp.2023.107128

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