Traveling wave solutions and numerical solutions for a mBBM equation
Keywords:
mBBM equation; complex method; extended direct algebraic method; ODM
Abstract
In this paper, some exact meromorphic solutions and generalized trigonometric solutions of the space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation are established by a new transformation and reliable methods. Moreover, some numerical solutions are obtained by using the optimal decomposition method (ODM), and their accuracy is shown in tables and images.
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References
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[13] A. Eremenko: Meromorphic solutions of equations of Briot-Bouquet type, Teor. Funkc. Funkc. Anal. IhPrilozh. 38, 48-56 (1982).
[14] A. Eremenko, L.W. Liao, T.W. Ng: Meromorphic solutions of higher order Briot–Bouquet differential equations, Math. Proc. Cambridge Philos. Soc. 146, 197-206 (2009).
[15] Y.Y. Gu, C.F. Wu, X. Yao, W.J. Yuan: Characterizations of all real solutions for the KdV equation and WR, Appl. Math. Lett. 107, 106446 (2020).
[16] W.J. Yuan, Y.D. Shang, Y. Huang, H. Wang: The representation of meromorphic solutions of certain ordinary differential equations and its applications, Sci. Sinica Math. 43, 563–575 (2013).
[17] S. Lang: Elliptic functions, Springer, New York 1987.
[18] W. Yuan, F. Meng, Y. Huang, Y. Wu: All traveling wave exact solutions of the variant Boussinesq equations, Appl. Math. Comput. 268, 865-872 (2015).
[19] Y. Gu, W. Yuan, N. Aminakbari, Q. Jiang: Exact solutions of the Vakhnenko-Parkes equation with complex method, J. Funct. Spaces. 2017, 6 (2017).
[20] H. Rezazadeh, H.Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou: New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chin. J. Phys. 56, 2805-2816 (2018).
[21] M.A. Bashir, A.A. Moussa: The cotha(ξ) Expansion Method and its Application to the Davey-Stewartson Equation, Appl. Math. Sci. 8, 3851-3868 (2014).
[22] Z. Odibat: An optimized decomposition method for nonlinear ordinary and partial differential equations, Phys. A. 541, 123323 (2020).
[2] V. Varlamov, Y. Liu: Cauchy problem for the Ostrovsky equation, Discrete Continuous Dynam. Syst. 10, 731 (2004) .
[3] J.C. Saut, N. Tzvetkov: Global well-posedness for the KP-BBM equations, Appl. Math. Res. eXpress. 2004, 1-16 (2004).
[4] E. Yusufoğlu: New solitonary solutions for the MBBM equations using Exp-function method, Phys. Lett. A. 372, 442-446 (2008).
[5] S. Abbasbandy, A. Shirzadi: The first integral method for modified Benjamin–Bona–Mahony equation, Commun. Nonlinear Sci. Numer. Simul. 15, 1759-1764 (2010).
[6] E.M.E. Zayed, S. Al-Joudi: Applications of an extended (G′/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Math. Prob. Engr. 2010, 19 (2010).
[7] J.F. Alzaidy: Fractional sub-equation method and its applications to the space–time fractional differential equations in mathematical physics, Br. J. Math. Comput. Sci. 3, 153-163 (2013).
[8] A. Bekir, Ö. Güner, Ö. Ünsal: The first integral method for exact solutions of nonlinear fractional differential equations, J. Comput. Nonlinear Dyn. 10, 021020 (2015).
[9] A.H. Arnous: Solitary wave solutions of space-time FDEs using the generalized Kudryashov method, Acta univ. apulensis. 42, 41-51 (2015).
[10] S. Javeed, S. Saif, A. Waheed, D. Baleanu: Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers, Results phys. 9, 1275-1281 (2018).
[11] M.T. Islam, M.A. Akbar, M.A.K. Azad: The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation, J. Mech. Cont. Math. Sci. 13, 56-71 (2018).
[12] J.V.D.C. Sousa, E.C. de Oliveira: A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl. 16, 83-96 (2017).
[13] A. Eremenko: Meromorphic solutions of equations of Briot-Bouquet type, Teor. Funkc. Funkc. Anal. IhPrilozh. 38, 48-56 (1982).
[14] A. Eremenko, L.W. Liao, T.W. Ng: Meromorphic solutions of higher order Briot–Bouquet differential equations, Math. Proc. Cambridge Philos. Soc. 146, 197-206 (2009).
[15] Y.Y. Gu, C.F. Wu, X. Yao, W.J. Yuan: Characterizations of all real solutions for the KdV equation and WR, Appl. Math. Lett. 107, 106446 (2020).
[16] W.J. Yuan, Y.D. Shang, Y. Huang, H. Wang: The representation of meromorphic solutions of certain ordinary differential equations and its applications, Sci. Sinica Math. 43, 563–575 (2013).
[17] S. Lang: Elliptic functions, Springer, New York 1987.
[18] W. Yuan, F. Meng, Y. Huang, Y. Wu: All traveling wave exact solutions of the variant Boussinesq equations, Appl. Math. Comput. 268, 865-872 (2015).
[19] Y. Gu, W. Yuan, N. Aminakbari, Q. Jiang: Exact solutions of the Vakhnenko-Parkes equation with complex method, J. Funct. Spaces. 2017, 6 (2017).
[20] H. Rezazadeh, H.Tariq, M. Eslami, M. Mirzazadeh, Q. Zhou: New exact solutions of nonlinear conformable time-fractional Phi-4 equation, Chin. J. Phys. 56, 2805-2816 (2018).
[21] M.A. Bashir, A.A. Moussa: The cotha(ξ) Expansion Method and its Application to the Davey-Stewartson Equation, Appl. Math. Sci. 8, 3851-3868 (2014).
[22] Z. Odibat: An optimized decomposition method for nonlinear ordinary and partial differential equations, Phys. A. 541, 123323 (2020).
Published
2022-07-04
How to Cite
Ni, W., & Li, Y. (2022). Traveling wave solutions and numerical solutions for a mBBM equation. Journal of Progressive Research in Mathematics, 19(2), 1-14. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2145
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