Semi-analytical and numerical study of fractal fractional nonlinear system under Caputo fractional derivative

Obaid Algahtani; Sayed Saifullah; Amir Ali; Obaid Algahtani; Sayed Saifullah; Amir Ali

doi:10.3934/math.2022920

AIMS Mathematics

2022, Volume 7, Issue 9: 16760-16774. doi: 10.3934/math.2022920

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Research article

Semi-analytical and numerical study of fractal fractional nonlinear system under Caputo fractional derivative

1.
Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan

Received: 12 May 2022 Revised: 29 June 2022 Accepted: 04 July 2022 Published: 13 July 2022
MSC : 35Bxx, 35Qxx, 37Mxx, 65Mxx, 41Axx

The article aims to investigate the fractional Drinfeld-Sokolov-Wilson system with fractal dimensions under the power-law kernel. The integral transform with the Adomian decomposition technique is applied to investigate the general series solution as well as study the applications of the considered model with fractal-fractional dimensions. For validity, a numerical case with appropriate subsidiary conditions is considered with a detailed numerical/physical interpretation. The absolute error in the considered exact and obtained series solutions is also presented. From the obtained results, it is revealed that minimizing the fractal dimension reinforces the amplitude of the solitary wave solution. Moreover, one can see that reducing the fractional order $ \alpha $ marginally reduces the amplitude as well as alters the nature of the solitonic waves. It is also revealed that for insignificant values of time, solutions of the coupled system in the form of solitary waves are in good agreement. However, when one of the parameters (fractal/fractional) is one and time increases, the amplitude of the system also increases. From the error analysis, it is noted that the absolute error in the solutions reduces rapidly when $ x $ enlarges at small-time $ t $, whereas, increment in iterations decreases error in the system. Finally, the results show that the considered method is a significant mathematical approach for studying linear/nonlinear FPDE's and therefore can be extensively applied to other physical models.
- integral transform,
- Adomian decomposition method,
- Drinfeld-Sokolov-Wilson equation,
- fractal-fractional operator,
- Caputo fractional operator
Citation: Obaid Algahtani, Sayed Saifullah, Amir Ali. Semi-analytical and numerical study of fractal fractional nonlinear system under Caputo fractional derivative[J]. AIMS Mathematics, 2022, 7(9): 16760-16774. doi: 10.3934/math.2022920

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Abstract

The article aims to investigate the fractional Drinfeld-Sokolov-Wilson system with fractal dimensions under the power-law kernel. The integral transform with the Adomian decomposition technique is applied to investigate the general series solution as well as study the applications of the considered model with fractal-fractional dimensions. For validity, a numerical case with appropriate subsidiary conditions is considered with a detailed numerical/physical interpretation. The absolute error in the considered exact and obtained series solutions is also presented. From the obtained results, it is revealed that minimizing the fractal dimension reinforces the amplitude of the solitary wave solution. Moreover, one can see that reducing the fractional order $ \alpha $ marginally reduces the amplitude as well as alters the nature of the solitonic waves. It is also revealed that for insignificant values of time, solutions of the coupled system in the form of solitary waves are in good agreement. However, when one of the parameters (fractal/fractional) is one and time increases, the amplitude of the system also increases. From the error analysis, it is noted that the absolute error in the solutions reduces rapidly when $ x $ enlarges at small-time $ t $, whereas, increment in iterations decreases error in the system. Finally, the results show that the considered method is a significant mathematical approach for studying linear/nonlinear FPDE's and therefore can be extensively applied to other physical models.

References

[1]	F. Rahman, A. Ali, S. Saifullah, Analysis of time-fractional $\phi ^{4}$-equation with singular and non-singular kernels, Int. J. Appl. Comput. Math, 7 (2021), 192. https://doi.org/10.1007/s40819-021-01128-w doi: 10.1007/s40819-021-01128-w
[2]	K. D. Kucche, S. T. Sutar, Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative, Chaos Soliton. Fract., 143 (2021), 110556. https://doi.org/10.1016/j.chaos.2020.110556 doi: 10.1016/j.chaos.2020.110556
[3]	S. Ahmad, A Ullah, A. Akgül, M. De la Sen, A study of fractional order Ambartsumian equation involving exponential decay kernel, AIMS Mathematics, 6 (2021), 9981–9997. https://doi.org/10.3934/math.2021580 doi: 10.3934/math.2021580
[4]	M. A. Khan, M. Farhan, S. Islam, E. Bonyah, Modeling the transmission dynamics of avian influenza with saturation and psychological effect, DCDS-S, 12 (2019), 455–474. https://doi.org/10.3934/dcdss.2019030 doi: 10.3934/dcdss.2019030
[5]	S. Saifullah, A. Ali, M. Irfan, K. Shah, Time-fractional Klein–Gordon equation with solitary/shock waves solutions, Math. Probl. Eng., 2021 (2021), 6858592. https://doi.org/10.1155/2021/6858592 doi: 10.1155/2021/6858592
[6]	A. Atangana, J. F. Gómez-Aguilar, Numerical approximation of Riemann–Liouville definition of fractional derivative: From Riemann–Liouville to Atangana–Baleanu, Numer. Meth. Part. Differ. Equ., 34 (2018), 1502–1523. https://doi.org/10.1002/num.22195 doi: 10.1002/num.22195
[7]	M. M. Khader, K. M. Saad, Z. Hammouch, D. Baleanu, A spectral collocation method for solving fractional KdV and KdV-Burgers equations with non-singular kernel derivatives, Appl. Numer. Math., 161 (2021), 137–146. https://doi.org/10.1016/j.apnum.2020.10.024 doi: 10.1016/j.apnum.2020.10.024
[8]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel; Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[9]	D. Baleanu, A. Fernandez, A. Akgül, On a fractional operator combining proportional and classical sifferintegrals, Mathematics, 8 (2020), 360. https://doi.org/10.3390/math8030360 doi: 10.3390/math8030360
[10]	W. Chen, H. G. Sun, X. D. Zhang, D. Koro$\breve{s}$ak, Anomalous diffusion modeling by fractal and fractional derivatives, Comput. Math. Appl., 59 (2010), 1754–1758. https://doi.org/10.1016/j.camwa.2009.08.020 doi: 10.1016/j.camwa.2009.08.020
[11]	H. G. Sun, M. M. Meerschaert, Y. Zhang, J. T. Zhu, W. Chen, A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resour., 52 (2013), 292–295. https://doi.org/10.1016/j.advwatres.2012.11.005 doi: 10.1016/j.advwatres.2012.11.005
[12]	A. Ali, A. U. Khan, O. Algahtani, S. Saifullah, Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels, AIMS Mathematics, 7 (2022), 14975–14990. https://doi.org/10.3934/math.2022820 doi: 10.3934/math.2022820
[13]	S. Saifullah, A. Ali, K. Shah, C. Promsakon, Investigation of fractal fractional nonlinear Drinfeld–Sokolov–Wilson system with non-singular operators, Res. Phys., 33 (2022), 105145. https://doi.org/10.1016/j.rinp.2021.105145 doi: 10.1016/j.rinp.2021.105145
[14]	A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. https://doi.org/10.1016/j.chaos.2017.04.027 doi: 10.1016/j.chaos.2017.04.027
[15]	S. Saifullah, A. Ali, E. F. D. Goufo, Investigation of complex behaviour of fractal fractional chaotic attractor with mittag-leffler Kernel, Chaos Soliton. Fract., 152 (2021), 111332. https://doi.org/10.1016/j.chaos.2021.111332 doi: 10.1016/j.chaos.2021.111332
[16]	A. Akgül, I. Siddique, Analysis of MHD Couette flow by fractal-fractional differential operators, Chaos Soliton. Fract., 146, (2021), 110893. https://doi.org/10.1016/j.chaos.2021.110893 doi: 10.1016/j.chaos.2021.110893
[17]	A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination? Chaos Soliton. Fract., 136 (2020), 109860. https://doi.org/10.1016/j.chaos.2020.109860 doi: 10.1016/j.chaos.2020.109860
[18]	H. Jafari, C. M. Khalique, M. Nazari, Application of the Laplace decomposition method for solving linear and nonlinear fractional diffusion-wave equations, Appl. Math. Lett., 24 (2011), 1799–1805. https://doi.org/10.1016/j.aml.2011.04.037 doi: 10.1016/j.aml.2011.04.037
[19]	J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Jpn., 51 (1982), 3390–3397. https://doi.org/10.1143/JPSJ.51.3390 doi: 10.1143/JPSJ.51.3390
[20]	K. khan, Z. khan, A Ali, M. Irfan, Investigation of Hirota equation: Modified double Laplace decomposition method, Phys. Scr., 96 (2021), 104006.
[21]	V. G. Drinfeld, V. V. Sokolov, Equations of Korteweg-de Vries type and simple lie algebras, Dokl. Akad. Nauk SSSR, 258 (1981), 11–16.
[22]	G. Wilson, The affine Lie algebra $C^{1}_2$ and an equation of Hirota and Satsuma, Phys. Lett. A, 89 (1982), 332–334. https://doi.org/10.1016/0375-9601(82)90186-4 doi: 10.1016/0375-9601(82)90186-4
[23]	M. Inc, On numerical doubly periodic wave solutions of the coupled Drinfeld–Sokolov–Wilson equation by the decomposition method, Appl. Math. Comput., 172 (2006), 421–430. https://doi.org/10.1016/j.amc.2005.02.012 doi: 10.1016/j.amc.2005.02.012
[24]	K. Khan, M. A. Akbar, M. N. Alam, Traveling wave solutions of the nonlinear Drinfel'd–Sokolov–Wilson equation and modified Benjamin–Bona–Mahony equations, J. Egypt. Math. Soc., 21 (2013), 233–240. https://doi.org/10.1016/j.joems.2013.04.010 doi: 10.1016/j.joems.2013.04.010
[25]	O. Tasbozan, M. Senol, A. Kurt, O. Özkanc New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves, Ocean Eng., 161 (2018), 62–68. https://doi.org/10.1016/j.oceaneng.2018.04.075 doi: 10.1016/j.oceaneng.2018.04.075
[26]	H. M. Srivastava, K. M. Saad, Some new and modified fractional analysis of the time-fractional Drinfeld–Sokolov–Wilson system, Chaos, 30 (2020), 113104. https://doi.org/10.1063/5.0009646 doi: 10.1063/5.0009646
[27]	P. J. Olver, Applications of lie group to fifferential equations, Springer Verlag, 1986. https://doi.org/10.1007/978-1-4684-0274-2
[28]	R. Hirota, Direct methods in soliton theory, In: Solitons, Berlin, Heidelberg: Springer, 1980. https://doi.org/10.1007/978-3-642-81448-8_5
[29]	J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700–708. https://doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
[30]	A. M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Appl. Math. Comput., 188 (2007), 1467–1475. https://doi.org/10.1016/j.amc.2006.11.013 doi: 10.1016/j.amc.2006.11.013
[31]	C. A. Gómez S, A. H. Salas, The Cole-Hopf transformation and improved tanh-coth method applied to new integrable system (KdV6), Appl. Math. Comput., 204 (2008), 957–962. https://doi.org/10.1016/j.amc.2008.08.006 doi: 10.1016/j.amc.2008.08.006
[32]	H. Fatoorehchi, M. Alidadi, The extended Laplace transform method for mathematical analysis of the Thomas–Fermi equation, Chinese J. Phys., 55 (2017), 2548–2558. https://doi.org/10.1016/j.cjph.2017.10.001 doi: 10.1016/j.cjph.2017.10.001
[33]	H. Fatoorehchi, H. Abolghasemi, Series solution of nonlinear differential equations by a novel extension of the Laplace transform method, Int. J. Comput. Math., 93 (2016), 1299–1319. https://doi.org/10.1080/00207160.2015.1045421 doi: 10.1080/00207160.2015.1045421
[34]	H. Fatoorehchi, R. Rach, A method for inverting the Laplace transforms of two classes of rational transfer functions in control engineering, Alex. Eng. J., 59 (2020), 4879–4887. https://doi.org/10.1016/j.aej.2020.08.052 doi: 10.1016/j.aej.2020.08.052
[35]	J. Saelao, N. Yokchoo, The solution of Klein–Gordon equation by using modified Adomian decomposition method, Math. Comput. Simulat., 171 (2020), 94–102. https://doi.org/10.1016/j.matcom.2019.10.010 doi: 10.1016/j.matcom.2019.10.010
[36]	L. Bougoffa, R. C. Rach, A. Mennouni, A convenient technique for solving linear and nonlinear Abel integral equations by the Adomian decomposition method, Appl. Math. Comput., 218 (2011), 1785–1793. https://doi.org/10.1016/j.amc.2011.06.062 doi: 10.1016/j.amc.2011.06.062
[37]	O. González-Gaxiola, A. Biswas, Optical solitons with Radhakrishnan–Kundu–Lakshmanan equation by Laplace–Adomian decomposition method, Optik, 179 (2019), 434–442. https://doi.org/10.1016/j.ijleo.2018.10.173 doi: 10.1016/j.ijleo.2018.10.173
[38]	A. Ali, Z. Gul, W. A. Khan, S. Ahmad, S. Zeb, Investigation of fractional order sine-Gordon equation using Laplace Adomian decomposition method, Fractals, 29 (2021), 1–10. https://doi.org/10.1142/S0218348X21501218 doi: 10.1142/S0218348X21501218
[39]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, 2006.
[40]	H. Khan, R. Shah, P. Kumam, D. Baleanu, M. Arif, Laplace decomposition for solving nonlinear system of fractional order partial differential equations, Adv. Differ. Equ., 2020 (2020), 375. https://doi.org/10.1186/s13662-020-02839-y doi: 10.1186/s13662-020-02839-y
[41]	G. Adomian, Modification of the decomposition approach to heat equation, J. Math. Anal. Appl., 124 (1987), 290–291.
[42]	W. M. Zhang, Solitary solutions and singular periodic solutions of the Drinfeld-Sokolov-Wilson equation by variational approach, Appl. Math. Sci., 5 (2011), 1887–1894.
[43]	D. Kumar, G. C. Paul, A. R. Seadawy, M. T. Darvishi, A variety of novel closed‐form soliton solutions to the family of Boussinesq‐like equations with different types, J. Ocean Eng. Sci., 2021. (In Press). https://doi.org/10.1016/j.joes.2021.10.007

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