New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System

Rachid Cherief; Medjahed Djilali; Khellaf Ould Melha; Sabri T. M. Thabet; Thabet Abdeljawad

doi:10.32323/ujma.1792704

New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System

Abstract

This manuscript investigates a coupled time-fractional Boussinesq-Burgers system (CTFBBS) and presents new analytical solutions for the model. The exact solutions are obtained by employing the Exp$(-\Phi(\eta))$-expansion method combined with the fractional complex transformation. The obtained solution set includes different functional forms, such as hyperbolic and trigonometric solutions, demonstrating the effectiveness and versatility of the proposed analytical technique for nonlinear fractional partial differential equations.To support the analytical findings, numerical simulations based on a finite difference scheme are performed in order to validate the accuracy of the exact solutions. Furthermore, the stability characteristics of the considered system are investigated through linear stability analysis. Finally, the symbolic computation software \textit{Mathematica 11} is utilized to solve the resulting nonlinear algebraic system and to generate surface and contour plots illustrating some representative solutions of the model.

Keywords

Caputo derivative, Exp$(-\Phi(\eta))$-expansion technique, Exact solutions, Fractional traveling wave transformation

Thanks

The author T. Abdeljawad would like to thank Prince Sultan University for paying the APC and for the support through TAS research lab.

References

[1] I. E. Mhlanga, C. M. Khalique, Exact solutions of generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations, J. Appl. Math., 2012(1), Article ID 389017. https://doi.org/10.1155/2012/389017
[2] M. Djilali, A. Hakem, Solving some important nonlinear time-fractional evolution equations by using the ( G′ G )-Expansion Method, J. Sci. Arts., 4(53) (2020), 815-832. https://www.josa.ro/docs/josa 2020 4/a 04 Djilali 815-832 18p.pdf
[3] S. S. Mahmood, M. A. S. Murad, T. Radwan, et al., Analysis of time-fractional soliton solutions for Kudryashov’s law with dual nonlocal nonlinearity and refractive index in optical fibers, AIMS Mathematics, 11(1) (2026), 2384-2405. https://doi.org/10.3934/math.2026097
[4] A. Hossain, N. Alam, F. Hossain, et al. Exploring dynamical patterns and optical solutions of space–time fractional-order double-chain deoxyribonucleic acid model with Atangana’s conformable derivative, Theory Biosci., 145 (2026), Article ID 2. https://doi.org/10.1007/s12064-025-00451-w
[5] Y. F. Patel, M. Izadi, Numerical simulation of foam drainage with fractional reduced differential transform method, Bol. Soc. Mat. Mex., 32 (2026), Article ID 40. https://doi.org/10.1007/s40590-026-00870-9
[6] L. C. Fefferman, Partial Differential Equations in Fluid Mechanics, Cambridge University Press, United Kingdom, 2018.
[7] J. M. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge University Press, New York, 2011.
[8] P. G. Drazin, R. S. Johnson, Solitons: An Introduction, Cambridge University Press, USA, 1989.
[9] A. M. Wazwaz, A Variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equation, Filomat., 31(3) (2017), 831–840. DOI 10.2298/FIL1703831W. http://www.jstor.org/stable/24902181
[10] B. A. Mahmood, M. A. Yousif, A residual power series technique for solving Boussinesq-Burgers equations, Cogent Math., 4(1) (2017), Article ID 1279398. https://doi.org/10.1080/23311835.2017.1279398

[11] A. Esen, O. Tas¸bozan, S. Kutluay, Approximate analytical solutions of the damped Burgers and Boussinesq-Burgers equations, C¸ ankaya Univ. J. Sci. Eng., 11(1) (2014), 65–76. https://dergipark.org.tr/en/download/article-file/386829
[12] M. Khalfallah, Exact traveling wave solutions of the Boussinesq-Burgers equation, Math. Comput. Model., 49(3-4) (2009), 666-671. https://doi.org/10.1016/j.mcm.2008.08.004
[13] X. Li, B. Li, J. Chen, et al., Exact solutions to the Boussinesq-Burgers equations, J. Appl. Math. Phys., 5(9) (2017), 1720-1724. https://doi.org/10.4236/jamp.2017.59145
[14] A. Chen, X. Li, Darboux transformation and soliton solutions for Boussinesq-Burgers equation, Chaos Soliton Fract., 27(1) (2006) 43-49. https://doi.org/10.1016/j.chaos.2004.09.116
[15] W. Zheng-Yan, C. Ai-Hua, Explicit solutions of Boussinesq-Burgers equation, Chin. Phys., 16(5) (2007), Article ID 1233. https://doi.org/10.1088/1009-1963/16/5/011.
[16] L. K. Ravi, S. Saha Ray, S. Sahoo, New exact solutions of coupled Boussinesq-Burgers equations by exp-function method, J. Ocean Eng. Sci., 2(1) (2017), 34-46. https://doi.org/10.1016/j.joes.2016.09.001
[17] P. Wang, B. Tian, W.J. Liu, et al., Lax Pair, Backlund transformation and multisoliton solutions for the Boussinesq-Burgers equations from shallow water waves, Appl. Math. Comput., 218(5) (2011), 1726-1734. https://doi.org/10.1016/j.amc.2011.06.053
[18] A. S. Abdel Rady, E. S. Osman, M. Khalfallah, Multi-soliton solution, rational solution of the Boussinesq-Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 15(5) (2010), 1172-1176. https://doi.org/10.1016/j.cnsns.2009.05.053
[19] A.S.A. Rady, M. Khalfallah, On soliton solutions for Boussinesq-Burgers equations, Commun. Nonlinear Sci. Numer. Simul., 15(4) (2010), 886-894. https://doi.org/10.1016/j.cnsns.2009.05.039
[20] M. O. Al-Amr, Solution of the coupled Boussinesq-Burger’s equations by reduced differential transform method, The 15th International Conference for Informatics and Information Technology (CIIT 2018), (2018).
[21] A. K. Gupta, R. S. Saha, Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq-Burgers equations, Comput. Fluids, 103 (2014), 34-41. https://doi.org/10.1016/j.compfluid.2014.07.008
[22] N. Rahman, S. Akter, H. Or-Roshid, et al., Traveling wave solutions of the (1+1)-dimensional compound KdVB equation by exp(−Φ(η))-expansion method, Global J. Sci. Front. Res. A, Phys. Space Sci., 13(8) (2013), 7-13.
[23] M. H. Bashar, T. Tahseen, N. H. M Shahen, Application of the advanced exp (−Φ(ξ ))-expansion method to the nonlinear conformable time-fractional partial differential equations, Turk. J. Math. Comput. Sci., 13(1)(2021), 68-80. https://izlik.org/JA76MW26SM5
[24] M. M. A. Khater, E. H. M. Zahran, Soliton soltuions of nonlinear evolutions equation by using the extended exp(−ϕ(ξ ))-expansion method, Int. J. Comput. Appl., 145(3) (2016), 1-5.
[25] M. A. E. Abdelrahman, E. H. M. Zahran, M. M. A. Khater, The exp(−ϕ(ξ ))-expansion method and its application for solving nonlinear evolution equations, Int. J. Mod. Nonlinear Theory Appl., 4(1) (2015), 37-47. https://doi.org/10.1007/s12190-008-0159-8
[26] M. M. A. Khater, Extended exp(−ϕ(ξ ))-expansion method for solving the generalized Hirota-Satsuma coupled KdV system, Glob. J. Sci. Front. Res. A, Phys. Space Sci., 15(F7) (2015), 13-21.
[27] X. Tang, H. Liu, Exact traveling wave solutions of equalwidth equation, J. Appl. Math. Phys., 7(10) (2019), 2315-2323. https://doi.org/10.4236/jamp.2019.710157
[28] A. R. Seadawy, D. Lu, M. M. A. Khater, Solitary wave solutions for the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation, J. Ocean Eng. Sci., 2(2) (2017), 137-142. https://doi.org/10.1016/j.joes.2017.05.002
[29] A. Ali, M. A. Iqbal, Q. M. UL-Hassan, et al., An efficient technique for higher order fractional differential equation, SpringerPlus. 5 (2016), Article ID 281. https://doi.org/10.1186/s40064-016-1905-2
[30] M. Kaplan, A. Bekir, A novel analytical method for time-fractional differential equations, Optik, 127(20) (2016), 8209-8214 http://dx.doi.org/10.1016/j.ijleo.2016.05.152.
[31] S. Javeed, S. Saif, A. Waheed, et al., Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers, Result Phys. 9 (2018), 1275-1281 https://doi.org/10.1016/j.rinp.2018.04.026.
[32] S. Kumar, A. Kumar, D. Baleanu, Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger’s equations arise in propagation of shallow water waves, Nonlinear Dyn., 85 (2016), 699-715 https://doi.org/10.1007/s11071-016-2716-2.
[33] L. Meenatchi, M. Kaliyappan, Solving time-fractional nonlinear coupled Boussinesq-Burgers equations arise in propagation of shallow water waves using Adomian decomposition method, AIP Conference Proceedings, 2095(1) (2019), Article ID 030015. https://doi.org/10.1063/1.5097526
[34] M. M. A. Khater, D. Kumar, New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water, J. Ocean Eng. Sci., 2(3) (2017), 223-228. https://doi.org/10.1016/j.joes.2017.07.001
[35] D. Shi, Y. Zhang, W. Liu, et al., Some exact solutions and conservation laws of the coupled time-fractional Boussinesq-Burgers System, Symmetry, 11(1) (2019), Article ID 77. https://doi.org/10.3390/sym11010077.
[36] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, The Netherlands, 2006.
[37] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[38] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[39] T. T. Shone, A. Patra, Solution for non-linear fractional partial differential equations using fractional complex transform, Int. J. Appl. Comput. Math., 5 (2019), Article ID 90. https://doi.org/10.1007/s40819-019-0673-4.
[40] Z. B. Li, J. H. He, Fractional complex transform for fractional differential equations, Math. Comput. Appl.,15(5) (2010), 970-973. https://doi.org/10.3390/mca15050970
[41] Z. B. Li, J. H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A, 2(3) (2011), 121-126.
[42] M. Inc, A. Yusuf, A. I. Aliyu, et al., Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics, Opt. Quantum Electron., 50 (2018), Article ID 190. https://doi.org/10.1007/s11082-018-1459-3.
[43] H. F. Ismael, H. Bulut, H. M. Baskonus, et al., Newly modified method and its application to the coupled Boussinesq equation in ocean engineering with its linear stability analysis, Commun. Theor. Phys., 72(11) (2020), Article ID 115002. https://doi.org/10.1088/1572-9494/aba25f.

Details

Primary Language

English

Subjects

Numerical and Computational Mathematics (Other), Partial Differential Equations

Journal Section

Research Article

Authors

Rachid Cherief
0009-0000-1845-6661
Algeria

Medjahed Djilali
0000-0002-4390-9404
Algeria

Khellaf Ould Melha
0009-0004-3759-2211
Algeria

Sabri T. M. Thabet ^*
0000-0002-4568-9732
India

Thabet Abdeljawad
0000-0002-8889-3768
Saudi Arabia

Early Pub Date

April 11, 2026

Publication Date

-

Submission Date

September 30, 2025

Acceptance Date

March 28, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.32323/ujma.1792704

IZ

https://izlik.org/JA33PM29TW

APA

Cherief, R., Djilali, M., Ould Melha, K., Thabet, S. T. M., & Abdeljawad, T. (2026). New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System. Universal Journal of Mathematics and Applications, Advanced Online Publication. https://doi.org/10.32323/ujma.1792704

AMA

1.Cherief R, Djilali M, Ould Melha K, Thabet STM, Abdeljawad T. New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System. Univ. J. Math. Appl. 2026;(Advanced Online Publication). doi:10.32323/ujma.1792704

Chicago

Cherief, Rachid, Medjahed Djilali, Khellaf Ould Melha, Sabri T. M. Thabet, and Thabet Abdeljawad. 2026. “New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication. https://doi.org/10.32323/ujma.1792704.

EndNote

Cherief R, Djilali M, Ould Melha K, Thabet STM, Abdeljawad T (April 1, 2026) New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System. Universal Journal of Mathematics and Applications Advanced Online Publication

IEEE

[1]R. Cherief, M. Djilali, K. Ould Melha, S. T. M. Thabet, and T. Abdeljawad, “New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System”, Univ. J. Math. Appl., no. Advanced Online Publication, Apr. 2026, doi: 10.32323/ujma.1792704.

ISNAD

Cherief, Rachid - Djilali, Medjahed - Ould Melha, Khellaf - Thabet, Sabri T. M. - Abdeljawad, Thabet. “New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System”. Universal Journal of Mathematics and Applications. Advanced Online Publication (April 1, 2026). https://doi.org/10.32323/ujma.1792704.

JAMA

1.Cherief R, Djilali M, Ould Melha K, Thabet STM, Abdeljawad T. New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System. Univ. J. Math. Appl. 2026. doi:10.32323/ujma.1792704.

MLA

Cherief, Rachid, et al. “New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System”. Universal Journal of Mathematics and Applications, no. Advanced Online Publication, Apr. 2026, doi:10.32323/ujma.1792704.

Vancouver

1.Rachid Cherief, Medjahed Djilali, Khellaf Ould Melha, Sabri T. M. Thabet, Thabet Abdeljawad. New Analytical Solutions and Stability Analysis of Coupled Time-Fractional Boussinesq-Burgers System. Univ. J. Math. Appl. 2026 Apr. 1;(Advanced Online Publication). doi:10.32323/ujma.1792704