Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform

Meshari Alesemi; Meshari Alesemi

doi:10.3934/math.20241419

AIMS Mathematics

2024, Volume 9, Issue 10: 29269-29295. doi: 10.3934/math.20241419

Previous Article Next Article

Research article Special Issues

Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform

Meshari Alesemi ^,

Department of Mathematics, College of Science, University of Bisha, P.O. Box 511, Bisha 61922, Saudi Arabia

Received: 25 August 2024 Revised: 21 September 2024 Accepted: 08 October 2024 Published: 16 October 2024
MSC : 34G20, 35A20, 35A22, 35R11

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics.
- Boussinesq equation,
- Mohand transform iterative method,
- Mohand residual power series method,
- fractional-order differential equation,
- Caputo operator
Citation: Meshari Alesemi. Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform[J]. AIMS Mathematics, 2024, 9(10): 29269-29295. doi: 10.3934/math.20241419

Related Papers:

Abstract

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics.

References

[1]	X. Zhang, Y. Feng, Z. Luo, J. Liu, A spatial sixth-order numerical scheme for solving fractional partial differential equation, Appl. Math. Lett., 159 (2025), 109265. https://doi.org/10.1016/j.aml.2024.109265 doi: 10.1016/j.aml.2024.109265
[2]	O. O. Omotola, A. J. Temiatyo, A. A. Sufiat, S. A. Ezekiel, Application of Mohand transform coupled with homotopy perturbation method to solve Newel-White-Segel equation, Ann. Math. Comput. Sci., 21 (2024), 162–180. https://doi.org/10.56947/amcs.v21.267 doi: 10.56947/amcs.v21.267
[3]	M. Nadeem, S. W. Yao, Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method, Int. J. Numer. Methods Heat Fluid Flow, 31 (2021), 273–292. https://doi.org/10.1108/HFF-02-2020-0111 doi: 10.1108/HFF-02-2020-0111
[4]	A. A. Alderremy, R. Shah, N. Iqbal, S. Aly, K. Nonlaopon, Fractional series solution construction for nonlinear fractional reaction-diffusion Brusselator model utilizing Laplace residual power series, Symmetry, 14 (2022), 1944. https://doi.org/10.3390/sym14091944 doi: 10.3390/sym14091944
[5]	J. Fang, M. Nadeem, M. Habib, S. Karim, H. A. Wahash, A new iterative method for the approximate solution of Klein-Gordon and Sine-Gordon equations, J. Funct. Spaces, 2022 (2022), 5365810. https://doi.org/10.1155/2022/5365810 doi: 10.1155/2022/5365810
[6]	S. Alshammari, M. M. Al-Sawalha, R. Shah, Approximate analytical methods for a fractional-order nonlinear system of Jaulent-Miodek equation with energy-dependent Schrödinger potential, Fractal Fract., 7 (2023), 140. https://doi.org/10.3390/fractalfract7020140 doi: 10.3390/fractalfract7020140
[7]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, Series on Complexity, Nonlinearity and Chaos: Volume 3, World Scientific, 2012. https://doi.org/10.1142/8180
[8]	M. M. Al-Sawalha, A. Khan, O. Y. Ababneh, T. Botmart, Fractional view analysis of Kersten-Krasil'shchik coupled KdV-mKdV systems with non-singular kernel derivatives, AIMS Math., 7 (2022), 18334–18359. https://doi.org/10.3934/math.20221010 doi: 10.3934/math.20221010
[9]	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
[10]	S. Guo, S. Wang, Twisted relative Rota-Baxter operators on Leibniz conformal algebras, Commun. Algebra, 52 (2024), 3946–3959. https://doi.org/10.1080/00927872.2024.2337276 doi: 10.1080/00927872.2024.2337276
[11]	X. Xi, J. Li, Z. Wang, H. Tian, R. Yang, The effect of high-order interactions on the functional brain networks of boys with ADHD, Eur. Phys. J. Spec. Top., 233 (2024), 817–829. https://doi.org/10.1140/epjs/s11734-024-01161-y doi: 10.1140/epjs/s11734-024-01161-y
[12]	Z. Wang, M. Chen, X. Xi, H. Tian, R. Yang, Multi-chimera states in a higher order network of FitzHugh-Nagumo oscillators, Eur. Phys. J. Spec. Top., 233 (2024), 779–786. https://doi.org/10.1140/epjs/s11734-024-01143-0 doi: 10.1140/epjs/s11734-024-01143-0
[13]	G. Gallavotti, Breakdown and regeneration of time reversal symmetry in nonequilibrium statistical mechanics, Phys. D, 112 (1998), 250–257. https://doi.org/10.1016/S0167-2789(97)00214-5 doi: 10.1016/S0167-2789(97)00214-5
[14]	B. H. Lavenda, Concepts of stability and symmetry in irreversible thermodynamics. I, Found. Phys., 2 (1972), 161–179. https://doi.org/10.1007/BF00708499 doi: 10.1007/BF00708499
[15]	G. Russo, J. J. E. Slotine, Symmetries, stability, and control in nonlinear systems and networks, Phys. Rev. E, 84 (2011), 041929. https://doi.org/10.1103/PhysRevE.84.041929 doi: 10.1103/PhysRevE.84.041929
[16]	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. https://doi.org/10.1080/17455030.2022.2044541
[17]	S. Lin, J. Zhang, C. Qiu, Asymptotic analysis for one-stage stochastic linear complementarity problems and applications, Mathematics, 11 (2023), 482. https://doi.org/10.3390/math11020482 doi: 10.3390/math11020482
[18]	L. Liu, S. Zhang, L. Zhang, G. Pan, J. Yu, Multi-UUV maneuvering counter-game for dynamic target scenario based on fractional-order recurrent neural network, IEEE Trans. Cybernetics, 53 (2023), 4015–4028. https://doi.org/10.1109/TCYB.2022.3225106 doi: 10.1109/TCYB.2022.3225106
[19]	C. Yang, A. M. Wazwaz, Q. Zhou, W. Liu, Transformation of soliton states for a (2+1) dimensional fourth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain, Laser Phys., 29 (2019), 035401. https://doi.org/10.1088/1555-6611/aaffc9 doi: 10.1088/1555-6611/aaffc9
[20]	X. Liu, H. Triki, Q. Zhou, M. Mirzazadeh, W. Liu, A. Biswas, et al., Generation and control of multiple solitons under the influence of parameters, Nonlinear Dyn., 95 (2019), 143–150. https://doi.org/10.1007/s11071-018-4556-8 doi: 10.1007/s11071-018-4556-8
[21]	C. Yang, W. Liu, Q. Zhou, D. Mihalache, B. A. Malomed, One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation, Nonlinear Dyn., 95 (2019), 369–380. https://doi.org/10.1007/s11071-018-4569-3 doi: 10.1007/s11071-018-4569-3
[22]	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
[23]	A. U. K. Niazi, N. Iqbal, R. Shah, F. Wannalookkhee, K. Nonlaopon, Controllability for fuzzy fractional evolution equations in credibility space, Fractal Fract., 5 (2021), 112. https://doi.org/10.3390/fractalfract5030112 doi: 10.3390/fractalfract5030112
[24]	J. L. Bona, M. Chen, J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: Ⅱ. The nonlinear theory, Nonlinearity, 17 (2004), 925. https://doi.org/10.1088/0951-7715/17/3/010 doi: 10.1088/0951-7715/17/3/010
[25]	B. Mehdinejadiani, H. Jafari, D. Baleanu, Derivation of a fractional Boussinesq equation for modelling unconfined groundwater, Eur. Phys. J. Spec. Top., 222 (2013), 1805–1812. https://doi.org/10.1140/epjst/e2013-01965-1 doi: 10.1140/epjst/e2013-01965-1
[26]	S. E. Serrano, Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity, Water Resour. Res., 31 (1995), 2733–2742. https://doi.org/10.1029/95WR02038 doi: 10.1029/95WR02038
[27]	T. Botmart, R. P. Agarwal, M. Naeem, A. Khan, R. Shah, On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators, AIMS Math., 7 (2022), 12483–12513. https://doi.org/10.3934/math.2022693 doi: 10.3934/math.2022693
[28]	P. A. Clarkson, T. J. Priestley, Symmetries of a Generalised Boussinesq equation, IMS Technical Report, 1996.
[29]	D. Kaya, Explicit solutions of generalized nonlinear Boussinesq equations, J. Appl. Math., 1 (2001), 29–37.
[30]	G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. https://doi.org/10.1016/0022-247X(88)90170-9 doi: 10.1016/0022-247X(88)90170-9
[31]	G. Adomian, Solving frontier problems of physics: the decomposition method, Springer Science and Business Media, Vol. 60, 2013.
[32]	A. M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput., 81 (1997), 265–274. https://doi.org/10.1016/S0096-3003(95)00327-4 doi: 10.1016/S0096-3003(95)00327-4
[33]	O. A. Arqub, Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5 (2013), 31–52.
[34]	O. A. Arqub, A. El-Ajou, Z. A. Zhour, S. Momani, Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique, Entropy, 16 (2014), 471–493. https://doi.org/10.3390/e16010471 doi: 10.3390/e16010471
[35]	T. Eriqat, A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, S. Momani, A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations, Chaos Soliton. Fract., 138 (2020), 109957. https://doi.org/10.1016/j.chaos.2020.109957 doi: 10.1016/j.chaos.2020.109957
[36]	M. Alquran, M. Alsukhour, M. Ali, I. Jaradat, Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems, Nonlinear Eng., 10 (2021), 282–292. https://doi.org/10.1515/nleng-2021-0022 doi: 10.1515/nleng-2021-0022
[37]	M. Mohand, A. Mahgoub, The new integral transform "Mohand Transform", Adv. Theor. Appl. Math., 12 (2017), 113–120.
[38]	M. Nadeem, J. H. He, A. Islam, The homotopy perturbation method for fractional differential equations: part 1 Mohand transform, Int. J. Numer. Methods Heat Fluid Flow, 31 (2021), 3490–3504. https://doi.org/10.1108/HFF-11-2020-0703 doi: 10.1108/HFF-11-2020-0703

Reader Comments

Your name:*

Email:*
© 2024 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)