Effect of diffusion parameters on traveling wave solutions of singular perturbed Boussinesq equation

Abstract

This paper focuses on the nature of the traveling wave solutions of the singular perturbed (sixth-order) Boussinesq equation, which is important for the physical relationships of water waves and shows strong interactions. Most studies of this equation have been analyzed by numerical methods and it has been observed that analytical solutions are limited. This has formed the main motivation for an analytical method, the Kudryashov method, for the generation of traveling wave solutions. One of the important goals of this work is to analyze the wave propagation phenomena in detail. For this purpose, the model has been extended by adding parameters to the diffusion term, which can illuminate shallow fluid layers and nonlinear atomic phenomena. This model aims to provide a deeper understanding of the motion of waves and the behavior of fluids, as well as its solution by the analytical method. To deepen the physical discussion, the responses of the solution are simulated for different values of the diffusion coefficient and the parameter associated with the wave velocity.

Keywords

The Singular Perturbed (sixth-order) Boussinesq Equation / The Kudryashov method / Traveling wave solution / Nonlinear wave propagation

Funding

None.

Declaration of competing interest

The authors declare that they have no conflict of interest regarding the publication of this article.

References

[1] Boussinesq J. Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J Math Pures Appl. 1872;17:55-108.

[2] Helal MA, Seadawy AR, Zekry M. Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications. Chin J Phys. 2017; 55(2):378-385.

[3] Peregrine DH. Long waves on a beach. J Fluid Mech. 1967; 27(4):815-827.

[4] McCowan AD. The range of application of Boussinesq type numerical short wave models.

[5] Dash RK, Daripa P. Analytical and numerical studies of a singularly perturbed Boussinesq equation. Appl Math Comput. 2002; 126(1):1-30.

[6] Rahimian M, Toomanian M, Nadjafikhah M. Approximate symmetry and exact solutions of the singularly perturbed Boussinesq equation. Commun Nonlinear Sci Numer Simul. 2017;53:1-9.

[7] Daripa P, Hua W. A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: filtering and regularization techniques. Appl Math Comput. 1999; 101(2-3):159-207.

[8] Zou ZL. Higher order Boussinesq equations. Ocean Eng. 1999; 26(8):767-792.

[9] Esfahani A, Farah LG. Local well-posedness for the sixth-order Boussinesq equation. J Math Anal Appl. 2012; 385(1):230-242.

[10] Wazwaz AM. Multiple-soliton solutions for the ninth-order KdV equation and sixth-order Boussinesq equation. Appl Math Comput. 2008; 203(1):277-283.

[11] Grilli ST, Skourup J, Svendsen IA. An efficient boundary element method for nonlinear water waves. Eng Anal Bound Elem. 1989; 6(2):97-107.

[12] Ershkov S, Burmasheva N, Leshchenko DD, Prosviryakov EY. Exact Solutions of the Oberbeck-Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows. Symmetry 2023; 15(9):1730.

[13] Barletta A, Celli M, Rees DAS. On the use and misuse of the Oberbeck-Boussinesq approximation. Physics 2023; 5(1):298-309.

[14] Zeytounian RK. Asymptotic modeling of atmospheric flows. Springer Science & Business Media. 2012.

[15] Madsen PA, Sørensen OR. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coast Eng. 1992; 18(3-4):183-204.

[16] Khaliq S, Ullah A, Ahmad S, Akgül A, Yusuf A, Sulaiman TA. Some novel analytical solutions of a new extented (2+1)-dimensional Boussinesq equation using a novel method. J Ocean Eng Sci. 2022.

[17] Zhao B, Wang J, Dong H, Fu L. Periodic solution and asymptotic behavior of the three-dimensional sixth-order Boussinesq equation in shallow water waves. Nonlinear Dyn. 2024; 112(1):643-659.

[18] Hirota R. The direct method in soliton theory (No. 155). Cambridge University Press. 2004.

[19] Wang Z, Qin Y, Zou L. Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation. Chin Phys B. 2017; 26(5):050504.

[20] Yang X, Zhang Z, Zhang N. Properties and stability analysis of the sixth-order Boussinesq equations for Rossby waves. Chin J Phys. 2023.

[21] Charlier C, Lenells J, Wang DS. The “good” Boussinesq equation: long-time asymptotics. Anal PDE. 2023; 16(6):1351-1388.

[22] Gao Y, Li Y, Su C. Well-posedness for good Boussinesq equations subject to quasi-periodic initial data. 2020. arXiv preprint arXiv:2006.07894.

[23] N’Gbo NG, Xia Y. Traveling wave solution of bad and good modified Boussinesq equations with conformable fractional-order derivative. Qual Theory Dyn Syst. 2022;21:1-21.

[24] Yang Z, Wang X. Blowup of solutions for the “bad” Boussinesq-type equation. J Math Anal Appl. 2003; 285(1):282-298.

[25] Christov CI, Maugin GA, Velarde MG. Well-posed Boussinesq paradigm with purely spatial higher-order derivatives. Phys Rev E. 1996; 54(4):3621.

[26] Yavuz M, Özdemir N. Analysis of an epidemic spreading model with exponential decay law. Math Sci Appl E-Notes. 2020; 8(1):142-154.

[27] El-Gamel M, Mohamed N, Adel W. Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis. Math Model Numer Simul Appl. 2023; 3(4):351-375.

[28] Yavuz M, Özdemir N. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler Kernel. Discret Contin Dyn Syst S. 2020; 13(3): 995-1006.

[29] Khan A, Khan A, Sinan M. Ion temperature gradient modes driven soliton and shock by reduction perturbation method for electron-ion magneto-plasma. Math Model Numer Simul Appl. 2022; 2(1):1-12.

[30] Yavuz M, Ozdemir N. An integral transform solution for fractional advection-diffusion problem. Math Stud Appl. 2018;442-446.

[31] Jena SR, Sahu I. A reliable method for voltage of telegraph equation in one and two space variables in electrical transmission: approximate and analytical approach. Phys Scr. 2023; 98(10):105216.

[32] Sahu I, Jena SR. SDIQR mathematical modelling for COVID-19 of Odisha associated with influx of migrants based on Laplace Adomian decomposition technique. Model Earth Syst Environ. 2023; 9(4):4031-4040.

[33] Jena SR, Paul AK. Fifth step block method and shooting constant for third order nonlinear dynamical systems. Int J Syst Assur Eng Manag. 2024; 15(6):2218-2229.

[34] Jena SR, Senapati A. One-dimensional heat and advection-diffusion equation based on improvised cubic B-spline collocation, finite element method and Crank-Nicolson technique. Int Commun Heat Mass. 2023;147:106958.

[35] Kumar A, Kumar M, Goswami P. Numerical solution of coupled system of Emden-Fowler equations using artificial neural network technique. Int J Optim Control: Theor Appl. 2024; 14(1):62-73.

[36] Jena SR, Senapati A. Explicit and implicit numerical investigations of one-dimensional heat equation based on spline collocation and Thomas algorithm. Soft Comput. 2024; 28(20):12227-12248.

[37] Arslan D, Çelik E. An approximate solution of singularly perturbed problem on uniform mesh. Int J Optim Control Theor Appl. 2024; 14(1):74-80.

[38] Durur H, Yokuş A, Duran S. Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions. Int J Mod Phys B. 2024; 38(04):2450054.

[39] Vucheva V, Kolkovska N. A symplectic high-order accurate numerical method for the sixth order Boussinesq equation. In: AIP Conference Pro-ceedings (Vol. 2953, No. 1). AIP Publishing. 2023.

[40] Huntul MJ, Abbas M, Iqbal MK. An inverse problem for investigating the time-dependent coefficient in a higher-order equation. Comput Appl Math. 2022; 41(3):120.

[41] Arslan D. Approximate solutions of singularly perturbed nonlinear ill-posed and sixth-order boussinesq equations with hybrid method. Bitlis Eren Univ Fen Bilim Derg. 2019; 8(2):451-458. https://doi.org/10.17798/bitlisfen.491847

[42] Song C, Li J, Gao R. Nonexistence of global solutions to the initial boundary value problem for the singularly perturbed sixth-order boussinesq-type equation. J Appl Math. 2014;2014.

[43] Daripa P. Higher-order Boussinesq equations for two-way propagation of shallow water waves. Eur J Mech B Fluids. 2006; 25(6):1008-1021.

[44] Yokuş A, Durur H, Duran S. Simulation and refraction event of complex hyperbolic type solitary wave in plasma and optical fiber for the perturbed Chen-Lee-Liu equation. Opt Quantum Electron. 2021;53:1-17.

[45] Triki H, Benlalli A, Wazwaz AM. Exact solutions of the generalized Pochhammer-Chree equation with sixth-order dispersion. Rom J Phys. 2015;60:935-951.

[46] Jiao X, Yao R, Lou SY. Approximate similarity reduction for singularly perturbed Boussinesq equation via symmetry perturbation and direct method. J Math Phys. 2008; 49(9):093505.

[47] Özgeç AR. Nonlineer iletim hatlarında dalgaların Wavelet ve Fourier transform yöntemiyle incelenmesi (Master’s thesis, Adnan Menderes Üniversitesi, Fen Bilimleri Enstitüsü). 2010.

[48] Greiner W. Quantum mechanics: an introduction. Springer Science & Business Media; 2011.

[49] Uthayakumar T, Al Sakkaf L, Al Khawaja U. Peregrine solitons of the higher-order, inhomogeneous, coupled, discrete, and nonlocal nonlinear Schrödinger equations. Front Phys. 2020;8:596886.