Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel

M. Mossa Al-Sawalha; Rasool Shah; Kamsing Nonlaopon; Imran Khan; Osama Y. Ababneh; M. Mossa Al-Sawalha; Rasool Shah; Kamsing Nonlaopon; Imran Khan; Osama Y. Ababneh

doi:10.3934/math.2023186

AIMS Mathematics

2023, Volume 8, Issue 2: 3730-3746. doi: 10.3934/math.2023186

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Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Abdul Wali khan university, Mardan 23200, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
4.
Department of mathematics and statistics, Bacha Khan University Charsadda, Pakistan
5.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan

Received: 06 September 2022 Revised: 01 October 2022 Accepted: 07 October 2022 Published: 25 November 2022
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.
- Kaup-Kupershmidt equation,
- Yang transform,
- new iterative transform method,
- Caputo-Fabrizio operator
Citation: M. Mossa Al-Sawalha, Rasool Shah, Kamsing Nonlaopon, Imran Khan, Osama Y. Ababneh. Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel[J]. AIMS Mathematics, 2023, 8(2): 3730-3746. doi: 10.3934/math.2023186

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Abstract

In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.

References

[1]	A. A. Kilbas, H M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[2]	D. Baleanu, Z. B. Guvenc, J. A. Tenreiro Machado, New trends in nanotechnology and fractional calculus applications, Springer, 2010.
[3]	M. Turkyilmazoglu, An efficient computational method for differential equations of fractional type, CMES-Comput. Model. Eng. Sci., 133 (2022), 1–9. http://dx.doi.org/10.32604/cmes.2022.020781 doi: 10.32604/cmes.2022.020781
[4]	A. Akgul, M. Inc, E. Karatas, D. Baleanu, Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Adv. Difference Equ., 2015 (2015), 220. http://doi.org/10.1186/s13662-015-0558-8 doi: 10.1186/s13662-015-0558-8
[5]	B. Riemann, Versuch einer allgemeinen Auffassung der Integration und Differentiation, Cambridge: Cambridge University Press, 1847.
[6]	M. Caputo, Elasticita e Dissipazione, Bologna: Zanichelli, 1969.
[7]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1998.
[8]	K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, Wiley, 1993.
[9]	J. Liouville, Memoire sur quelques questions de geometrie et de mecanique, et sur un nouveaugenre de calcul pour resoudre ces questions, J. Ecole Polytech, 1832, 1–69.
[10]	W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709–726. https://doi.org/10.1016/j.jmaa.2006.10.040 doi: 10.1016/j.jmaa.2006.10.040
[11]	N. A. Shah, H. A. Alyousef, S. A. El-Tantawy, J. D. Chung, Analytical investigation of fractional-order Korteweg-de-Vries-type equations under Atangana-Baleanu-Caputo operator: modeling nonlinear waves in a plasma and fluid, Symmetry, 14 (2022), 739. https://doi.org/10.3390/sym14040739 doi: 10.3390/sym14040739
[12]	M. M. Al-Sawalha, N. Amir, R. Shah, M. Yar, Novel analysis of fuzzy fractional Emden-Fowler equations within new iterative transform method, J. Funct. Spaces, 2022 (2022), 7731135. https://doi.org/10.1155/2022/7731135 doi: 10.1155/2022/7731135
[13]	P. Veeresha, D. G. Prakasha, H. M. Baskonus, Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method, Math. Sci., 13 (2019), 115–128. https://doi.org/10.1007/S40096-019-0284-6 doi: 10.1007/S40096-019-0284-6
[14]	D. G. Prakasha, P. Veeresha, H. M. Baskonus, Analysis of the dynamics of hepatitis E virus using the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus, 134 (2019), 241. https://doi.org/10.1140/epjp/i2019-12590-5 doi: 10.1140/epjp/i2019-12590-5
[15]	M. M. Al-Sawalha, O. Y. Ababneh, R. Shah, K. Nonlaopon, Numerical analysis of fractional-order Whitham-Broer-Kaup equations with non-singular kernel operators, AIMS Mathematics, 8 (2023), 2308–2336. https://doi.org/10.3934/math.2023120 doi: 10.3934/math.2023120
[16]	M. M. Al-Sawalha, R. P. Agarwal, O. Y. Ababneh, W. Weera, A reliable way to deal with fractional-order equations that describe the unsteady flow of a polytropic gas, Mathematics, 10 (2022), 2293. http://doi.org/10.3390/math10132293 doi: 10.3390/math10132293
[17]	D. Baleanu, G. C. Wu, S. D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals, 102 (2017), 99–105. https://doi.org/10.1016/j.chaos.2017.02.007 doi: 10.1016/j.chaos.2017.02.007
[18]	C. S. Drapaka, S. Sivaloganathan, A fractional model of continuum mechanics, J. Elasticity, 107 (2012), 105–123.
[19]	M. M. Al-Sawalha, A. S. Alshehry, K. Nonlaopon, R. Shah, O. Y. Ababneh, Fractional view analysis of delay differential equations via numerical method, AIMS Mathematics, 7 (2022), 20510–20523. https://doi.org/10.3934/math.20221123 doi: 10.3934/math.20221123
[20]	J. M. Cruz-Duarte, J. Rosales-Garcia, C. RodrigoCorrea-Cely, A. Garcia-Perez, J. GabrielAvina-Cervantes, A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications, Commun. Nonlinear Sci. Numer. Simul., 61 (2018), 138–148. https://doi.org/10.1016/j.cnsns.2018.01.020 doi: 10.1016/j.cnsns.2018.01.020
[21]	B. K. Singh, A novel approach for numeric study of 2D biological population model, Cogent Math., 3 (2016), 1261527. https://doi.org/10.1080/23311835.2016.1261527 doi: 10.1080/23311835.2016.1261527
[22]	E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Phys. A, 284 (2000), 376–384. https://doi.org/10.1016/S0378-4371(00)00255-7 doi: 10.1016/S0378-4371(00)00255-7
[23]	X. Gong, L. Wang, Y. Mou, H. Wang, X. Wei, W. Zheng, et al., Improved four-channel PBTDPA control strategy using force feedback bilateral teleoperation system, Int. J. Control, 20 (2022), 1002–1017. https://doi.org/10.1007/s12555-021-0096-y doi: 10.1007/s12555-021-0096-y
[24]	F. W. Meng, A. P. Pang, X. F. Dong, C. Han, X. P. Sha, H-infinity optimal performance design of an unstable plant under Bode integral constraint, Complexity, 2018 (2018), 4942906. https://doi.org/ 10.1155/2018/4942906 doi: 10.1155/2018/4942906
[25]	H. Zheng, S. F. Jin, A multi-source fluid queue based stochastic model of the probabilistic offloading strategy in a MEC system with multiple mobile devices and a single MEC server, Int. J. Appl. Math. Comput. Sci., 32 (2022), 125–138. https://doi.org/10.34768/amcs-2022-0010 doi: 10.34768/amcs-2022-0010
[26]	S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput., 177 (2006), 488–494. https://doi.org/10.1016/j.amc.2005.11.025 doi: 10.1016/j.amc.2005.11.025
[27]	M. Alqhtani, K. M. Saad, R. Shah, W. Weera, W. M. Hamanah, Analysis of the fractional-order local poisson equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
[28]	M. M. Al-Sawalha, A. S. Alshehry, K. Nonlaopon, R. Shah, O. Y. Ababneh, Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm, AIMS Mathematics, 7 (2022), 19739–19757. https://doi.org/10.3934/math.20221082 doi: 10.3934/math.20221082
[29]	G. C. Wu, A fractional variational iteration method for solving fractional nonlinear differential equations, Comput. Math. Appl., 61 (2011), 2186–2190. https://doi.org/10.1016/j.camwa.2010.09.010 doi: 10.1016/j.camwa.2010.09.010
[30]	D. G. Prakasha, P. Veeresha, M. S. Rawashdeh, Numerical solution for (2+1)-dimensional time-fractional coupled Burger equations using fractional natural decomposition method, Math. Methods Appl. Sci., 42 (2019), 3409–3427. https://doi.org/10.1002/mma.5533 doi: 10.1002/mma.5533
[31]	S. Mukhtar, R. Shah, S. Noor, The numerical investigation of a fractional-order multi-dimensional model of Navier-Stokes equation via novel techniques, Symmetry, 14 (2022), 1102. https://doi.org/10.3390/sym14061102 doi: 10.3390/sym14061102
[32]	M. Areshi, A. Khan, R. Shah, K. Nonlaopon, Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform. AIMS Mathematics, 7 (2022), 6936–6958. https://doi.org/10.3934/math.2022385
[33]	E. M. Elsayed, R. Shah, K. Nonlaopon, The analysis of the fractional-order Navier-Stokes equations by a novel approach, J. Funct. Spaces, 2022 (2022), 8979447. https://doi.org/10.1155/2022/8979447 doi: 10.1155/2022/8979447
[34]	Z. H. Xie, X. A. Feng, X. J. Chen, Partial least trimmed squares regression, Chemometr. Intell. Lab. Syst., 221 (2022), 104486. https://doi.org/10.1016/j.chemolab.2021.104486 doi: 10.1016/j.chemolab.2021.104486
[35]	V. N. Kovalnogov, M. I. Kornilova, Y. A. Khakhalev, D. A. Generalov, T. E. Simos, C. Tsitouras, Fitted modifications of Runge-Kutta-Nystrom pairs of orders 7(5) for addressing oscillatory problems, Math. Methods Appl. Sci., 2022. https://doi.org/10.1002/mma.8510
[36]	K. Liu, Z. X. Yang, W. F. Wei, B. Gao, D. L. Xin, C. M. Sun, et al., Novel detection approach for thermal defects: Study on its feasibility and application to vehicle cables, High Volt., 2022. https://doi.org/10.1049/hve2.12258
[37]	B. A. Kupershmidt, A super Korteweg-de-Vries equations: an integrable system, Phys. Lett. A, 102 (1984), 213–215. https://doi.org/10.1016/0375-9601(84)90693-5 doi: 10.1016/0375-9601(84)90693-5
[38]	E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Solitons Fractals, 16 (2003), 819–839. https://doi.org/10.1016/S0960-0779(02)00472-1 doi: 10.1016/S0960-0779(02)00472-1
[39]	M. Inc, On numerical soliton solution of the Kaup-Kupershmidt equation and convergence analysis of the decomposition method, Appl. Math. Comput., 172 (2006), 72–85. https://doi.org/10.1016/j.amc.2005.01.120 doi: 10.1016/j.amc.2005.01.120
[40]	M. A. Ablowitz, P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge: Cambridge University Press, 1991.
[41]	H. W. Tam, X. B. Hu, Two integrable differential-difference equations exhibiting soliton solutions of the Kaup-Kupershimdt equation type, Phys. Lett. A, 272 (2000), 174–183. https://doi.org/10.1016/S0375-9601(00)00422-9 doi: 10.1016/S0375-9601(00)00422-9
[42]	X. J. Yang, A new integral transform method for solving steady heat-transfer problem, Thermal Sci., 20 (2016), s639–s642. https://doi.org/10.2298/TSCI16S3639Y doi: 10.2298/TSCI16S3639Y
[43]	M. Caputo, M. Fabrizio, On the singular kernels for fractional derivatives. Some applications to partial differential equations, Prog. Fract. Differ. Appl., 7 (2021), 79–82. http://doi.org/10.18576/pfda/070201 doi: 10.18576/pfda/070201
[44]	S. Ahmad, A. Ullah, A. Akgul, M. De la Sen, A novel homotopy perturbation method with applications to nonlinear fractional order KdV and Burger equation with exponential-decay kernel, J. Funct. Spaces, 2021 (2021), 8770488. https://doi.org/10.1155/2021/8770488 doi: 10.1155/2021/8770488
[45]	D. G. Prakasha, N. S. Malagi, P. Veeresha, B. C. Prasannakumara, An efficient computational technique for time-fractional Kaup-Kupershmidt equation, Numer. Methods Partial Differential Equations, 37 (2021), 1299–1316. https://doi.org/10.1002/num.22580 doi: 10.1002/num.22580

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