Triangular Fuzzy Solutions of Coupled Boussinesq-Burgers Equations with Uncertain Parameters

Document Type : Research Paper

Authors

1 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati, India

2 Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati, India.

Abstract

This article outlines the significance of the coupled Boussinesq-Burgers Equations (cBBEs) for the prediction of fluid flow in complicated systems, with special emphasis on shallow water waves (SWW) close to the coasts and lakes. The manuscript also defines the Fuzzy Elzaki Adomian Decomposition Method (FEADM) as a new technique suitable for solving the cBBE’s in crisp and fuzzy manners. Due to uncertainties in initial conditions caused by the fluid velocity or wave height, one parameter is taken to be a triangular fuzzy number (TFNs). Numerical implementations illustrate the performance of the FEADM against Homotopy Perturbation Method (HPM), Optimal Homotopy Asymptotic Method (OHAM) demonstrate that in terms of accuracy and efficiency, the method is superior. The proof of theorems also shows how the FEADM approaches the solution of nonlinear systems under uncertainties. These results of the research work show how useful the FEADM can be for the modeling of fluid dynamics in deterministic and fuzzy environments.

Keywords

Main Subjects

References

[1] H. Abdel-Gawad, M. Tantawy, On controlled propagation of long waves in nonautonomous boussinesq-burgers equations, Nonlinear Dynamics, 87 (2017), 2511-2518. https://doi.org/10.1007/s11071-016-3207-1
[2] Z. Ayati, J. Biazar, On the convergence of homotopy perturbation method, Journal of the Egyptian Mathematical
Society, 23(2) (2015), 424-428. https://doi.org/10.1016/j.joems.2014.06.015
[3] M. Benosman, J. Borggaard, O. San, B. Kramer, Learning-based robust stabilization for reduced-order models of 2d
and 3d boussinesq equations, Applied Mathematical Modelling, 49 (2017), 162-181. https://doi.org/10.1016/j.
apm.2017.04.032
[4] M. J. Dong, S. F. Tian, X. W. Yan, T. T. Zhang, Nonlocal symmetries, conservation laws and interaction solutions
for the classical boussinesq-burgers equation, Nonlinear Dynamics, 95 (2019), 273-291. https://doi.org/10.1007/
s11071-018-4563-9
[5] Z. Eidinejad, R. Saadati, C. Li, M. Inc, J. Vahidi, The multiple exp-function method to obtain soliton solutions of
the conformable date-jimbo-kashiwara-miwa equations, International Journal of Modern Physics B, 38(03) (2024),
2450043. https://doi.org/10.1142/S0217979224500437
[6] Z. Eidinejad, R. Saadati, J. Vahidi, C. Li, Numerical solutions of 2d stochastic time-fractional sine-gordon equation
in the caputo sense, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 36(6)
(2023), e3121. https://doi.org/10.1002/jnm.3121
[7] T. M. Elzaki, The new integral transform Elzaki transform, Global Journal of Pure and Applied Mathematics, 7(1)
(2011), 57-64. https://www.researchgate.net/publication/289123241
[8] T. M. Elzaki, Application of new transform “Elzaki Transform” to partial differential equations, Global Journal of
Pure and Applied Mathematics, 7(1) (2011), 65-70. https://www.researchgate.net/publication/236230758
[9] T. M. Elzaki, S. M. Elzaki, E. A. Elnour, On the new integral transform “Elzaki Transform” fundamental properties
investigations and applications, Global Journal of Mathematical Sciences: Theory and Practical, 4(1) (2012), 1-13.
https://www.researchgate.net/publication/258316406
[10] X. Y. Gao, Y. J. Guo, W. R. Shan, Beholding the shallow water waves near an ocean beach or in a lake via a
boussinesq-burgers system, Chaos, Solitons and Fractals, 147 (2021), 110875. https://doi.org/10.1016/j.chaos.
2021.110875
[11] X. Y. Gao, Y. J. Guo, W. R. Shan, On the oceanic/laky shallow-water dynamics through a boussinesqburgers
system, Qualitative Theory of Dynamical Systems, 23(2) (2024), 57. https://doi.org/10.1007/
s12346-023-00905-w
[12] X. T. Gao, B. Tian, T. Y. Zhou, Y. Shen, C. H. Feng, For the shallow water waves: Bilinear-form and similarityreduction
studies on a boussinesq-burgers system, International Journal of Theoretical Physics, 63(7) (2024), 179.
https://doi.org/10.1007/s10773-024-05715-7
[13] P. Guo, X. Wu, L. Wang, New multiple-soliton (kink) solutions for the high-order boussinesq-burgers equation,
Waves in Random and Complex Media, 26(3) (2016), 383-396. https://doi.org/10.1080/17455030.2016.
1158885
[14] A. Gupta, S. S. Ray, Comparison between homotopy perturbation method and optimal homotopy asymptotic method
for the soliton solutions of boussinesq-burger equations, Computers and Fluids, 103 (2014), 34-41. https://doi.
org/10.1016/j.compfluid.2014.07.008q
[15] O. E. Ige, R. A. Oderinu, T. M. Elzaki, Adomian polynomial and elzaki transform method for solving klein gordon
equations, International Journal of Applied Mathematics, 32(3) (2019), 451. https://www.researchgate.net/
publication/335276371
[16] Y. L. Jiang, C. Chen, Lie group analysis and dynamical behavior for classical boussinesq-burgers system, Nonlinear
Analysis: Real World Applications, 47 (2019), 385-397. https://doi.org/10.1016/j.nonrwa.2018.11.010
[17] P. Karunakar, S. Chakraverty, Effect of coriolis constant on geophysical korteweg-de vries equation, Journal of
Ocean Engineering and Science, 4(2) (2019), 113-121. https://doi.org/10.1016/j.joes.2019.02.002
[18] P. Karunakar, S. Chakraverty, Homotopy perturbation method for predicting tsunami wave propagation with crisp
and uncertain parameters, International Journal of Numerical Methods for Heat and Fluid Flow, 31(1) (2020),
92-105. https://doi.org/10.1108/HFF-11-2019-0861
[19] P. Karunakar, S. Chakraverty, T. Rao, K. Ramesh, A. Hussein, Fuzzy fractional shallow water wave equations:
Analysis, convergence of solutions, and comparative study with depth as triangular fuzzy number, Physica Scripta,
99(12) (2024), 125216. https://doi.org/10.1088/1402-4896/ad8afd
[20] M. Khalfallah, Exact traveling wave solutions of the boussinesq-burgers equation, Mathematical and Computer
Modelling, 49(3-4) (2009), 666-671. https://doi.org/10.1016/j.mcm.2008.08.004
[21] R. Kumar, K. S. Pandey, A. Kumar, Dynamical behavior of the solutions of coupled boussinesq-burgers equations
occurring at the seaside beaches, Brazilian Journal of Physics, 52(6) (2022), 201. https://doi.org/10.1007/
s13538-022-01195-4
[22] S. Kumar, S. Rani, Symmetries of optimal system, various closed-form solutions, and propagation of different wave
profiles for the boussinesq-burgers system in ocean waves, Physics of Fluids, 34(3) (2022). https://doi.org/10.
1063/5.0085927
[23] X. Li, A. Chen, Darboux transformation and multi-soliton solutions of boussinesq-burgers equation, Physics Letters
A, 342(5-6) (2005), 413-420. https://doi.org/10.1016/j.physleta.2005.05.083
[24] M. Li, W. Hu, C.Wu, Rational solutions of the classical boussinesq-burgers system, Nonlinear Dynamics, 94 (2018),
1291-1302. https://doi.org/10.1007/s11071-018-4424-6
[25] Z. Liang, Z. L. Feng, L. C. Yin, Some new exact solutions of Jacobian elliptic function about the generalized
boussinesq equation and boussinesq-burgers equation, Chinese Physics B, 17(2) (2008), 403. https://doi.org/10.
1088/1674-1056/17/2/009
[26] F. Y. Liu, Y. T. Gao, Lie group analysis for a higher-order boussinesq-burgers system, Applied Mathematics Letters,
132 (2022), 108094. https://doi.org/10.1016/j.aml.2022.108094
[27] M. Liu, H. Xu, Z.Wang, Exact solutions and bifurcations of the time-fractional coupled boussinesq-burgers equation,
Physica Scripta, 98(11) (2023), 115040. https://doi.org/10.1088/1402-4896/ad03c4
[28] J. Lu, Y. Sun, Numerical approaches to time fractional boussinesq-burgers equations, Fractals, 29(08) (2021),
2150244. https://doi.org/10.1142/S0218348X21502443
[29] B. A. Mahmood, M. A. Yousif, A residual power series technique for solving boussinesq-burgers equations, Cogent
Mathematics, 4(1) (2017), 1279398. https://doi.org/10.1080/23311835.2017.1279398
[30] A. M. Mubaraki, R. Nuruddeen, K. K. Ali, J. G´omez-Aguilar, Additional solitonic and other analytical solutions
for the higher-order boussinesq-burgers equation, Optical and Quantum Electronics, 56(2) (2024), 165. https:
//doi.org/10.1007/s11082-023-05744-2
[31] Z. M. Odibat, A study on the convergence of homotopy analysis method, Applied Mathematics and Computation,
217(2) (2010), 782-789. https://doi.org/10.1016/j.amc.2010.06.017
[32] A. Podder, M. A. Arefin, K. A. Gepreel, M. H. Uddin, M. A. Akbar, Diverse soliton wave profile assessment to
the fractional order nonlinear landau-ginzburg-higgs and coupled boussinesq-burger equations, Results in Physics, 65
(2024), 107994. https://doi.org/10.1016/j.rinp.2024.107994
[33] A. A. Rady, M. Khalfallah, On soliton solutions for boussinesq-burgers equations, Communications in Nonlinear
Science and Numerical Simulation, 15(4) (2010), 886-894. https://doi.org/10.1016/j.cnsns.2009.05.039
[34] A. A. Rady, E. Osman, M. Khalfallah, Multi-soliton solution, rational solution of the boussinesq-burgers equations,
Communications in Nonlinear Science and Numerical Simulation, 15(5) (2010), 1172-1176. https://doi.org/10.
1016/j.cnsns.2009.05.053
[35] P. Rao, D. Mohapatra, S. Chakraverty, D. Roy, Vibration analysis of non-homogenous single-link flexible manipulator
in uncertain environment, Applied Mathematical Modelling, (2025), 115939. https://doi.org/10.1016/j.
apm.2025.115939
[36] X. Rui, Darboux transformations and soliton solutions for classical boussinesq-burgers equation, Communications
in Theoretical Physics, 50(3) (2008), 579. https://doi.org/10.1088/0253-6102/50/3/08
[37] M. Sahoo, S. Chakraverty, Dynamics of tsunami wave propagation in uncertain environment, Computational and
Applied Mathematics, 43(5) (2024), 266. https://doi.org/10.1007/s40314-024-02776-6
[38] D. Shi, Y. Zhang, W. Liu, J. Liu, Some exact solutions and conservation laws of the coupled time-fractional
boussinesq-burgers system, Symmetry, 11(1) (2019), 77. https://doi.org/10.3390/sym11010077
[39] R. Vana, P. Karunakar, A fuzzy semi-analytical approach for modeling uncertainties in solitary wave solution
of coupled nonlinear boussinesq equations, Physica Scripta, 99(10) (2024), 105218. https://doi.org/10.1088/
1402-4896/ad72aa
[40] R. Vana, P. Karunakar, Computational approach and convergence analysis for intervalbased solution of the
benjamin-bona-mahony equation with imprecise parameters, Engineering Computations: International Journal for
Computer-Aided Engineering, 41(4) (2024), 1067-1085. https://doi.org/10.1108/EC-12-2023-0905
[41] R. Vana, P. Karunakar, Fuzzy uncertainty modeling of generalized hirota-satsuma coupled korteweg-de vries equation,
Physics of Fluids, 36(9) (2024), 096609. https://doi.org/10.1063/5.0226445
[42] R. Vana, P. Karunakar, Uncertainties in regularized long-wave equation and its modified form: A triangular fuzzybased
approach, Physics of Fluids, 36(4) (2024), 046610. https://doi.org/10.1063/5.0206452
[43] R. Vana, K. Perumandla, S. Chakraverty, Semi-analytical and numerical computations for the solution of uncertain
fractional benjamin bona mahony equation with triangular fuzzy number, Journal of Computational Applied
Mechanics, 56(1) (2025), 196-221. https://doi.org/10.22059/jcamech.2024.387317.1318
[44] A. C. Varsoliwala, T. R. Singh, An approximate analytical solution of non linear partial differential equation for
water infiltration in unsaturated soils by combined elzaki transform and adomian decomposition method, in: Journal
of Physics: Conference Series, Vol. 1473, IOP Publishing, (2020), 012009. https://doi.org/10.1088/1742-6596/
1473/1/012009
[45] A. C. Varsoliwala, T. R. Singh, Mathematical modeling of tsunami wave propagation at mid ocean and its
amplification and run-up on shore, Journal of Ocean Engineering and Science, 6(4) (2021), 367-375. https:
//doi.org/10.1016/j.joes.2021.03.003
[46] A. C. Varsoliwala, T. R. Singh, Solution of fingering phenomenon arising in porous media in horizontal direction
by combination of elzaki transform and adomian decomposition method, in: Advances in Mathematical Modelling,
Applied Analysis and Computation: Proceedings of ICMMAAC 2021, Springer, (2022), 495-506. https://doi.
org/10.1007/978-981-19-0179-9_29
[47] A. C. Varsoliwala, T. R. Singh, Mathematical modeling of atmospheric internal waves phenomenon and its solution
by elzaki adomian decomposition method, Journal of Ocean Engineering and Science, 7(3) (2022), 203-212. https:
//doi.org/10.1016/j.joes.2021.07.010
[48] Y. H.Wang, CTE method to the interaction solutions of boussinesq-burgers equations, Applied Mathematics Letters,
38 (2014), 100-105. https://doi.org/10.1016/j.aml.2014.07.014
[49] K. J. Wang, G. D. Wang, H. W. Zhu, A new perspective on the study of the fractal coupled boussinesq-burger
equation in shallow water, Fractals, 29(05) (2021), 2150122. https://doi.org/10.1142/S0218348X2150122X
[50] C. C. Zhang, A. H. Chen, Bilinear form and new multi-soliton solutions of the classical boussinesq-burgers system,
Applied Mathematics Letters, 58 (2016), 133-139. https://doi.org/10.1016/j.aml.2016.02.015
[51] Y. Zhang, L. Wang, Application of laplace adomian decomposition method for fractional fokker-planck equation
and time fractional coupled boussinesq-burger equations, Engineering Computations, 41(4) (2024), 793-818. https:
//doi.org/10.1108/EC-06-2023-0275
[52] P. M. Zu, X. M. Zhang, Study of the generalized hypothetical syllogism for some well known families of fuzzy
implications with respect to strict t-norm, Iranian Journal of Fuzzy Systems, 21(2) (2024), 181-200. https://doi.
org/10.22111/ijfs.2024.46361.8164
[53] J. M. Zuo, Y. M. Zhang, The hirota bilinear method for the coupled burgers equation and the high-order boussinesqburgers equation, Chinese Physics B, 20(1) (2011), 010205. https://doi.org/10.1088/1674-1056/20/1/010205
Iranian Journal of Fuzzy Systems
Volume 22, Issue 6
November and December 2025
Pages 21-39

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