Fractional comparative analysis of Camassa-Holm and Degasperis-Procesi equations

Yousef Jawarneh; Humaira Yasmin; M. Mossa Al-Sawalha; Rasool Shah; Asfandyar Khan; Yousef Jawarneh; Humaira Yasmin; M. Mossa Al-Sawalha; Rasool Shah; Asfandyar Khan

doi:10.3934/math.20231318

AIMS Mathematics

2023, Volume 8, Issue 11: 25845-25862. doi: 10.3934/math.20231318

Previous Article Next Article

Research article Special Issues

Fractional comparative analysis of Camassa-Holm and Degasperis-Procesi equations

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Basic Sciences, Preparatory Year Deanship, King Faisal University, Al-Ahsa, 31982, Saudi Arabia
3.
Department of Computer Science and Mathematics, Lebanese American University, Beirut Lebanon
4.
Department of Mathematics, Abdul Wali Khan University Mardan 23200, Pakistan

Received: 07 August 2023 Revised: 24 August 2023 Accepted: 28 August 2023 Published: 07 September 2023
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

This paper focuses on novel approaches to finding solitary wave (SW) solutions for the modified Degasperis-Procesi and fractionally modified Camassa-Holm equations. The study presents two innovative methodologies: the Yang transformation decomposition technique and the homotopy perturbation transformation method. These methods use the Caputo sense fractional order derivative, the Yang transformation, the adomian decomposition technique, and the homotopy perturbation method. The inquiry effectively solves the fractional Camassa-Holm and Degasperis-Procesi equations, which also provides a detailed numerical and graphical comparison of the solutions found. The results, which include accurate solutions, derived solutions, and absolute error displayed in tabular style, demonstrate the effectiveness of the suggested procedures. These procedures are iterative, which results in several answers. The estimated absolute error attests to the correctness and simplicity of these solutions. Especially in plasma physics, these approaches may be expanded to handle various linear and nonlinear physical issues, including the evolution equations controlling nonlinear waves.
- Yang transform,
- Degasperis-Procesi and Camassa-Holm equations,
- series solution,
- adomian decomposition method,
- homotopy perturbation method
Citation: Yousef Jawarneh, Humaira Yasmin, M. Mossa Al-Sawalha, Rasool Shah, Asfandyar Khan. Fractional comparative analysis of Camassa-Holm and Degasperis-Procesi equations[J]. AIMS Mathematics, 2023, 8(11): 25845-25862. doi: 10.3934/math.20231318

Related Papers:

Abstract

This paper focuses on novel approaches to finding solitary wave (SW) solutions for the modified Degasperis-Procesi and fractionally modified Camassa-Holm equations. The study presents two innovative methodologies: the Yang transformation decomposition technique and the homotopy perturbation transformation method. These methods use the Caputo sense fractional order derivative, the Yang transformation, the adomian decomposition technique, and the homotopy perturbation method. The inquiry effectively solves the fractional Camassa-Holm and Degasperis-Procesi equations, which also provides a detailed numerical and graphical comparison of the solutions found. The results, which include accurate solutions, derived solutions, and absolute error displayed in tabular style, demonstrate the effectiveness of the suggested procedures. These procedures are iterative, which results in several answers. The estimated absolute error attests to the correctness and simplicity of these solutions. Especially in plasma physics, these approaches may be expanded to handle various linear and nonlinear physical issues, including the evolution equations controlling nonlinear waves.

References

[1]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998. https://doi.org/10.1016/s0076-5392(99)x8001-5
[2]	B. L. Guo, X. K. Pu, F. H. Huang, Fractional partial differential equations and their numerical solutions, World Scientific, 2015. https://doi.org/10.1142/9543
[3]	J. G. Liu, X. J. Yang, L. L. Geng, X. J. Yu, On fractional symmetry group scheme to the higher-dimensional space and time fractional dissipative Burgers equation, Int. J. Geom. Methods M., 19 (2022), 2250173. https://doi.org/10.1142/S0219887822501730 doi: 10.1142/S0219887822501730
[4]	M. I. Abbas, Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative, Miskolc Math. Notes, 22 (2021), 491–502. https://doi.org/10.18514/MMN.2021.3470 doi: 10.18514/MMN.2021.3470
[5]	J. G. Liu, Y. F. Zhang, J. J. Wang, Investigation of the time fractional generalized (2+1)-dimensional Zakharov-Kuznetsov equation with single-power law nonlinearity, Fractals, 31 (2023), 2350033. https://doi.org/10.1142/S0218348X23500330 doi: 10.1142/S0218348X23500330
[6]	S. Y. Lu, M. Z. Zhe, L. R. Yin, Z. T. Yin, X. Liu, W. F. Zheng, The multi-modal fusion in visual question answering: A review of attention mechanisms, PeerJ Comput. Sci., 9 (2023), e1440. https://doi.org/10.7717/peerj-cs.1400 doi: 10.7717/peerj-cs.1400
[7]	J. G. Liu, X. J. Yang, Symmetry group analysis of several coupled fractional partial differential equations, Chaos Soliton. Fract., 173 (2023), 113603. https://doi.org/10.1016/j.chaos.2023.113603 doi: 10.1016/j.chaos.2023.113603
[8]	A. Atangana, D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017), D4016005. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001091 doi: 10.1061/(ASCE)EM.1943-7889.0001091
[9]	H. Fu, G. C. Wu, G. Yang, L. L. Huang, Continuous time random walk to a general fractional Fokker-Planck equation on fractal media, Eur. Phys. J-Spec. Top., 230 (2021), 3927–3933. https://doi.org/10.1140/epjs/s11734-021-00323-6 doi: 10.1140/epjs/s11734-021-00323-6
[10]	A. Shafee, Y. Alkhezi, R. Shah, Efficient solution of fractional system partial differential equations using laplace residual power series method, Fractal Fract., 7 (2023), 429. https://doi.org/10.3390/fractalfract7060429 doi: 10.3390/fractalfract7060429
[11]	H. Yasmin, A. S. Alshehry, A. M. Saeed, R. Shah, K. Nonlaopon, Application of the q-homotopy analysis transform method to fractional-order kolmogorov and Rosenau-Hyman models within the Atangana-Baleanu operator, Symmetry, 15 (2023), 671. https://doi.org/10.3390/sym15030671 doi: 10.3390/sym15030671
[12]	C. Yang, J. S. Zhang, Z. W. Huang, Numerical study on cavitation-vortex-noise correlation mechanism and dynamic mode decomposition of a hydrofoil, Phys. Fluids, 34 (2022), 125105. https://doi.org/10.1063/5.0128169 doi: 10.1063/5.0128169
[13]	A. Akgül, S. A. Khoshnaw, Application of fractional derivative on nonlinear biochemical reaction models, Int. J. Intell. Netw., 1 (2020), 52–58. https://doi.org/10.1016/j.ijin.2020.05.0019 doi: 10.1016/j.ijin.2020.05.0019
[14]	J. Song, A. Mingotti, J. H. Zhang, L. Peretto, H. Wen, Accurate damping factor and frequency estimation for damped real-valued sinusoidal signals, IEEE T. Tnstrum. Meas., 71 (2022), 6503504. https://doi.org/10.1109/TIM.2022.3220300 doi: 10.1109/TIM.2022.3220300
[15]	T. A. A. Ali, Z. Xiao, H. B. Jiang, B. Li, A class of digital integrators based on trigonometric quadrature rules, IEEE T. Ind. Electron., 2023, 1–11. https://doi.org/10.1109/TIE.2023.3290247
[16]	C. Q. Guo, J. P. Hu, J. S. Hao, S. Celikovsky, X. M. Hu, Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions, Kybernetika, 59 (2023), 342–364. https://doi.org/10.14736/kyb-2023-3-0342 doi: 10.14736/kyb-2023-3-0342
[17]	C. Q. Guo, J. P. Hu, Y. Z. Wu, S. Celikovsky, Non-singular fixed-time tracking control of uncertain nonlinear Pure-Feedback systems with practical state constraints, IEEE T. Circuits-I, 70 (2023), 3746–3758. https://doi.org/10.1109/TCSI.2023.3291700 doi: 10.1109/TCSI.2023.3291700
[18]	Q. T. Meng, Q. Ma, Y. Shi, Adaptive fixed-time stabilization for a class of uncertain nonlinear systems, IEEE T. Automat. Contr., 2023, 1–8. https://doi.org/10.1109/TAC.2023.3244151
[19]	K. Diethelm, A. D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, In: Scientific computing in chemical engineering II, Springer, 1999,217–224. https://doi.org/10.1007/978-3-642-60185-9_24
[20]	M. D. Aloko, O. J. Fenuga, S. A. Okunuga, Solutions of some non-linear Volterra integro-differential equations of the second kind using modified variational iteration method, FUW Trends Sci. Technol. J., 4 (2019), 298–303.
[21]	F. Mainardi, Fractional calculus, In: Fractals and fractional calculus in continuum mechanics, Springer, 1997,291–348.
[22]	J. Bai, X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492–2502. https://doi.org/10.1109/TIP.2007.904971 doi: 10.1109/TIP.2007.904971
[23]	P. L. Butzer, U. Westphal, An introduction to fractional calculus, In: Applications of fractional calculus in physics, World Scientific, 2010, 1–86. https://doi.org/10.1142/9789812817747-0001
[24]	D. Kumar, F. Tchier, J. Singh, D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. https://doi.org/10.3390/e20040259 doi: 10.3390/e20040259
[25]	J. R. Loh, A. Isah, C. Phang, Y. T. Toh, On the new properties of Caputo-Fabrizio operator and its application in deriving shifted Legendre operational matrix, Appl. Numer. Math., 132 (2018), 138–153. https://doi.org/10.1016/j.apnum.2018.05.016 doi: 10.1016/j.apnum.2018.05.016
[26]	D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2018), 1413–1423. https://doi.org/10.1029/2000WR900032 doi: 10.1029/2000WR900032
[27]	B. A. Carreras, V. E. Lynch, G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096–5103. https://doi.org/10.1063/1.1416180 doi: 10.1063/1.1416180
[28]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Probing families of optical soliton solutions in fractional perturbed Radhakrishnan-Kundu-Lakshmanan model with improved versions of extended direct algebraic method, Fractal Fract., 7 (2023), 512. https://doi.org/10.3390/fractalfract7070512 doi: 10.3390/fractalfract7070512
[29]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Investigating families of soliton solutions for the complex structured coupled fractional Biswas-Arshed model in birefringent fibers using a novel analytical technique, Fractal Fract., 7 (2023), 491. https://doi.org/10.3390/fractalfract7070491 doi: 10.3390/fractalfract7070491
[30]	H. Yasmin, A. S. Alshehry, A. Khan, R. Shah, K. Nonlaopon, Numerical analysis of the fractional-order Belousov-Zhabotinsky system, Symmetry, 15 (2023), 834. https://doi.org/10.3390/sym15040834 doi: 10.3390/sym15040834
[31]	H. C. Li, R. Peng, Z. A. Wang, On a diffusive susceptible-infected-susceptible epidemic model with mass action mechanism and birth-death effect: Analysis, simulations, and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129–2153. https://doi.org/10.1137/18M1167863 doi: 10.1137/18M1167863
[32]	H. Y. Jin, Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differ. Equ., 260 (2016), 162–196. https://doi.org/10.1016/j.jde.2015.08.040 doi: 10.1016/j.jde.2015.08.040
[33]	W. B. Lyu, Z. A. Wang, Logistic damping effect in chemotaxis models with density-suppressed motility, Adv. Nonlinear Anal., 12 (2023), 336–355. https://doi.org/10.1515/anona-2022-0263 doi: 10.1515/anona-2022-0263
[34]	Q. K. Li, H. Lin, X. Tan, S. L. Du, H $\infty$ Consensus for multiagent-based supply chain systems under switching topology and uncertain demands, IEEE T. Syst. Man Cy-S., 50 (2020), 4905–4918. https://doi:10.1109/TSMC.2018.2884510. doi: 10.1109/TSMC.2018.2884510
[35]	B. Wang, Y. M. Zhang, W. Zhang, A composite adaptive fault-tolerant attitude control for a ouadrotor UAV with multiple uncertainties, J. Syst. Sci. Complex., 35 (2022), 81–104. https://doi.org/10.1007/s11424-022-1030-y doi: 10.1007/s11424-022-1030-y
[36]	G. Yel, H. M. Baskonus, H. Bulut, Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method, Opt. Quant. Electron., 49 (2017), 1–10. https://doi.org/10.1007/s11082-017-1127-z doi: 10.1007/s11082-017-1127-z
[37]	A. M. Zidan, A. Khan, R. Shah, M. K. Alaoui, W. Weera, Evaluation of time-fractional Fisher's equations with the help of analytical methods, AIMS Mathematics, 7 (2022), 18746–18766. https://doi:10.3934/math.20221031 doi: 10.3934/math.20221031
[38]	S. Alyobi, R. Shah, A. Khan, N. A. Shah, K. Nonlaopon, Fractional analysis of nonlinear boussinesq equation under Atangana-Baleanu-Caputo operator, Symmetry, 14 (2022), 2417. https://doi.org/10.3390/sym14112417 doi: 10.3390/sym14112417
[39]	A. A. Alderremy, S. Aly, R. Fayyaz, A. Khan, R. Shah, N. Wyal, The analysis of fractional-order nonlinear systems of third order KdV and Burgers equations via a novel transform, Complexity, 2022 (2022), 4935809. https://doi.org/10.1155/2022/4935809 doi: 10.1155/2022/4935809
[40]	N. J. Ford, J. Y. Xiao, Y. B. Yan, A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal., 14 (2011), 454–474. https://doi.org/10.2478/s13540-011-0028-2 doi: 10.2478/s13540-011-0028-2
[41]	M. Eslami, B. F. Vajargah, M. Mirzazadeh, A. Biswas, Application of first integral method to fractional partial differential equations, Indian J. Phys., 88 (2014), 177–184. https://doi.org/10.1007/s12648-013-0401-6 doi: 10.1007/s12648-013-0401-6
[42]	N. A. Shah, Y. S. Hamed, K. M. Abualnaja, J. D. Chung, R. Shah, A. Khan, A comparative analysis of fractional-order kaup-kupershmidt equation within different operators, Symmetry, 14 (2022), 986. https://doi.org/10.3390/sym14050986 doi: 10.3390/sym14050986
[43]	M. K. Alaoui, K. Nonlaopon, A. M. Zidan, A. Khan, R. Shah, Analytical investigation of fractional-order cahn-hilliard and gardner equations using two novel techniques, Mathematics, 10 (2022), 1643. https://doi.org/10.3390/math10101643 doi: 10.3390/math10101643
[44]	S. Y. Lu, Y. M. Ding, M. Z. Liu, Z. T. Yin, L. R. Yin, W. F. Zheng, Multiscale feature extraction and fusion of image and text in VQA, Int. J. Comput. Int. Sys., 16 (2023), 54. https://doi.org/10.1007/s44196-023-00233-6 doi: 10.1007/s44196-023-00233-6
[45]	F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871–3878. https://doi.org/10.1016/j.apm.2013.10.007 doi: 10.1016/j.apm.2013.10.007
[46]	A. M. Wazwaz, Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352 (2006), 500–504. https://doi.org/10.1016/j.physleta.2005.12.036 doi: 10.1016/j.physleta.2005.12.036
[47]	J. S. Kamdem, Z. J. Qiao, Decomposition method for the Camassa-Holm equation, Chaos Soliton. Fract., 31 (2007), 437–447. https://doi.org/10.1016/j.chaos.2005.09.071 doi: 10.1016/j.chaos.2005.09.071
[48]	R. Liu, Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation, Commun. Pur. Appl. Anal., 9 (2010), 77–90. https://doi.org/10.3934/cpaa.2010.9.77 doi: 10.3934/cpaa.2010.9.77
[49]	G. Omel'yanov, J. N. Rodriguez, Solitary wave solutions to a generalization of the mKdV equation, Russ. J. Math. Phys., 30 (2023), 246–256. https://doi.org/10.1134/S1061920823020103 doi: 10.1134/S1061920823020103
[50]	P. Sunthrayuth, R. Ullah, A. Khan, R. Shah, J. Kafle, I. Mahariq, et al., Numerical analysis of the fractional-order nonlinear system of Volterra integro-differential equations, J. Funct. Space., 2021 (2021), 1537958. https://doi.org/10.1155/2021/1537958 doi: 10.1155/2021/1537958
[51]	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 doi: 10.3934/math.2022385
[52]	M. K. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. S. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 3248376. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376
[53]	X. J. Yang, A new integral transform method for solving steady heat-transfer problem, Therm. Sci., 20 (2016), 639–642. https://doi.org/10.2298/TSCI16S3639Y doi: 10.2298/TSCI16S3639Y

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)