Solitary and periodic wave solutions of the loaded Boussinesq and the loaded modified Boussinesq equation

Volume 30, Issue 1, pp 67--74 https://doi.org/10.22436/jmcs.030.01.07

Publication Date: December 01, 2022 Submission Date: April 30, 2022 Revision Date: September 22, 2022 Accteptance Date: October 07, 2022

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 525 Downloads
• 1371 Views

Authors

B. Babajanov - Department of Applied Mathematics and Mathematical Physics, Urgench State University, Urgench, Uzbekistan. F. Abdikarimov - Khorezm Mamun Academy, Khiva, Uzbekistan.

Abstract

In this article, we establish new traveling wave solutions for the loaded Boussinesq equation and the loaded modified Boussinesq equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions and periodic wave solutions of the loaded Boussinesq equation and its modifications. The traveling wave solutions obtained via this method are expressed by hyperbolic functions and the trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions and solitary wave solutions of kink type. This method presents a wider applicability for handling nonlinear wave equations.

Share and Cite

Keywords

• Loaded modified Boussinesq equation
• hyperbolic functions
• trigonometric functions
• periodic wave solutions
• soliton solutions
• functional variable method

MSC

•  34A34
•  34B15
•  35Q51
•  35J60
•  35J66

References

• [1] A. S. Abdel Rady, E. S. Osman, M. Khalfallah, On soliton solutions of the (2 + 1) dimensional Boussinesq equation, Appl. Math. Comput., 219 (2012), 3414–3419
  • View Article
  • MathSciNet
  • Google Scholar


• [2] S. Ahmad, A. Ullah, A. Akg¨ ul, F. Jarad, A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: application to KdV and Fornberg-Witham equations, AIMS Math., 7 (2022), 9389–9404
  • View Article
  • MathSciNet
  • Google Scholar


• [3] M. N. Alam, M. G. Hafez, M.A. Akbar, H.-O.-Roshid, Exact Solutions to the (2+1)-dimensional Boussinesq Equation via exp(-'())-Expansion Method, J. Sci. Res., 7 (2015), 1–10
  • View Article
  • Google Scholar


• [4] B. Babajanov, F. Abdikarimov, Soliton Solutions of the Loaded Modified Calogero-Degasperis Equation, Int. J. Appl. Math., 32 (2022), 381–392
  • View Article
  • Google Scholar


• [5] B. Babajanov, F. Abdikarimov, Expansion Method for the Loaded Modified Zakharov-Kuznetsov Equation, Adv. Math. Models Appl., 7 (2022), 168–177
  • View Article


• [6] B. Babajanov, F. Abdikarimov, Solitary and periodic wave solutions of the loaded modified Benjamin-Bona-Mahony equation via the functional variable method, Res. Math., 30 (2022), 10–20
  • View Article
  • Google Scholar
  • MATH


• [7] B. Babajanov, F. Abdikarimov, Exact Solutions of the Nonlinear Loaded Benjamin-Ono Equation, WSEAS Trans. Math., 21 (2022), 666–670
  • View Article
  • Google Scholar


• [8] B. Babajanov, F. Abdikarimov, The Application of the Functional Variable Method for Solving the Loaded Non-linear Evaluation Equations, Front. Appl. Math. Stat, 8 (2022), 9 pages
  • View Article
  • Google Scholar


• [9] U. I. Baltaeva, On some boundary value problems for a third order loaded integro-differential equation with real parameters, Vestn. Udmurt. Univ., Mat. Mekh. Komp. Nauki, 3 (2012), 3–12
  • View Article
  • Google Scholar
  • MATH


• [10] A. Bekir, A. Cevikel, E. H. M. Zahran, New impressive representations for the soliton behaviors arising from the (2+1)- Boussinesq equation, J. Ocean Eng. Sci., (2022), 6 pages
  • View Article
  • Google Scholar


• [11] A. Biswas, D. Milovic, A. Ranasinghe, Solitary waves of Boussinesq equation in a power law media, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 3738–3742
  • View Article
  • Google Scholar


• [12] Z.-D. Dai, D.-Q. Xian, D.-L. Li, Homoclinic Breather-Wave with Convective Effect for the (1+1)-dimensional Boussinesq Equation, Chinese Phys. Lett., 26 (2009), 1–3
  • View Article
  • Google Scholar


• [13] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkh¨auser , Boston (1997)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] G. Forozani, M. Ghorveei Nosrat, Solitary solution of modified bad and good Boussinesq equation by using of tanh and extended tanh methods, Indian J. Pure Appl. Math., 44 (2013), 497–510
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] A. U. Gulalai, S. Ahmad, M. Inc, Fractal fractional analysis of modified KdV equation under three different kernels, J. Ocean Eng. Sci., (2022), 14 pages
  • View Article
  • Google Scholar


• [16] A. Harir, S. Melliani, L. S. Chadli, Solving fuzzy Burgers equation by variational iteration method, J. Math. Comput. Sci., 21 (2020), 136–149
  • View Article
  • Google Scholar


• [17] R. Hirota, Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices, J. Math. Phys., 14 (1973), 810–814
  • View Article
  • MathSciNet
  • Google Scholar


• [18] M. Hosseini, H. Abdollahzadeh, M. Abdollahzadeh, Exact Travelling Solutions for the Sixth-order Boussinesq Equation, J. Math. Comput. Sci, 2 (2011), 376–387
  • View Article
  • Google Scholar


• [19] M. S. Ismail, A. G. Bratsos, A predictor-corrector scheme for the numerical solution of the Boussinesq equation, J. Appl. Math. Comput., 13 (2003), 11–27
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] M. Javidi, Y. Jalilian, Exact solitary wave solution of Boussinesq equation by VIM, Chaos Solitons Fractals, 36 (2008), 1256–1260
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] A. B. Khasanov, U. A. Hoitmetov, On integration of the loaded mKdV equation in the class of rapidly decreasing functions, Izv. Irkutsk. Gos. Univ. Ser. Mat., 38 (2021), 19–35
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] A. Kneser, Belastete integralgleichungen, Rend. Circ. Matem. Palermo, 37 (1914), 169–197
  • View Article
  • Google Scholar
  • MATH


• [23] A. I. Kozhanov, A nonlinear loaded parabolic equation and a related inverse problem, Math. Notes, 76 (2004), 784–795
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] C. Liu, D. Dai, Exact periodic solitary wave solutions for the (2 + 1)-dimensional Boussinesq equation, J. Math. Anal. Appl., 367 (2010), 444–450
  • View Article
  • MathSciNet
  • Google Scholar


• [25] L. Lichtenstein, Vorlesungen ¨uber einige Klassen nichtlinearer Integralgleichungen und Integro-Differential-Gleichungen nebst Anwendungen, Springer-Verlag, Berlin (1931)
  • View Article
  • Google Scholar
  • MATH


• [26] V.G. Makhankov, Dynamics of classical solitons (in nonintegrable systems), Phys. Rep., 35 (1978), 1–128
  • View Article
  • MathSciNet
  • Google Scholar


• [27] A. M. Nahuˇsev, Boundary value problems for loaded integro-differential equations of hyperbolic type and some of their applications to the prediction of ground moisture, Differ. Uravn., 15 (1979), 96–105
  • View Article
  • Google Scholar


• [28] A. M. Nakhushev, Loaded equations and their applications, Differ. Uravn., 19 (1983), 86–94
  • View Article
  • Google Scholar
  • MATH


• [29] T. Nazir, M. Abbas, M. K. Iqbal, A new quintic B-spline approximation for numerical treatment of Boussinesq equation, J. Math. Comput. Sci., 20 (2020), 30–42
  • View Article
  • Google Scholar


• [30] L. T. K. Nguyen, Soliton solution of good Boussinesq equation, Vietnam J. Math., 44 (2016), 375–385
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] G. U. Urazboev, I. I. Baltaeva, I. D. Rakhimov, Generalized (G’/G)-Expansion method for loaded Korteweg-de Vries equation, Sib. Zh. Ind. Mat., 24 (2021), 139–147
  • View Article
  • MathSciNet
  • MATH


• [32] A.-M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Appl. Math. Comput., 192 (2007), 479–486
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [33] A.-M. Wazwaz, New travelling wave solutions to the Boussinesq and the Klein-Gordon equations, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 889–901
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] A. B. Yakhshimuratov, B. A. Babajanov, Integration of equations of Kaup system kind with self-consistent source in class of periodic functions, Ufa Math. J., 12 (2020), 104-–114
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] A. Yildirim, S.T. Mohyud-Din, A variational approach for soliton solutions of good Boussinesq equation, J. King Saud Univ. Sci., 22 (2010), 205–208
  • View Article
  • Google Scholar


• [36] X.-L. Yin, S.-X. Kong, Y.-Q. Liu, Zheng X.-T. Zheng, New scheme for nonlinear Schr¨odinger equations with variable coefficients, J. Math. Comput. Sci., 19 (2019), 151–157
  • View Article
  • Google Scholar


• [37] A. Zerarka, S. Ouamane, Application of the functional variable method to a class of nonlinear wave equations, World J. Model. Simul., 6 (2010), 150–160
  • View Article
  • Google Scholar


• [38] A. Zerarka, S. Ouamane, A. Attaf, On the functional variable method for finding exact solutions to a class of wave equations, Appl. Math. Comput., 217 (2010), 2897–2904
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH