Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

Vina Apriliani, Ikhsan Maulidi, Budi Azhari

Abstract


One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations.

 


Keywords


Extended F-Expansion Method; Modifiead Korteweg-de Vries (mKdV) Equation.

Full Text:

PDF

References


Abassy, T. A., El-Tawil, M. A., & Saleh, H. K. (2004). The solution of KdV and mKdV equations using Adomian Pade approximation. International Journal of Nonlinear Sciences and Numerical Simulation, 5(4), 327–339.

Ak, T., Karakoc, S. B. G., & Biswas, A. (2017). A new approach for numerical solution of modified Korteweg-de Vries equation. Iranian Journal of Science and Technology, Transactions A: Science, 41(4), 1109–1121.

Akbar, M. A., & Ali, N. H. M. (2017). The improved F-expansion method with Riccati equation and its applications in mathematical physics. Cogent Mathematics, 4(1), 1–19.

Al-Fhaid, A. S. (2012). New exact solutions for the modified KdV-KP equation using the extended F-expansion method. Applied Mathematical Sciences, 6(107), 5315–5332.

Apriliani, V. (2015). Modifikasi Metode Ekspansi-F untuk Menyelesaikan Persamaan Boussinesq Orde Empat. Skripsi: Institut Pertanian Bogor.

Başhan, A., Uçar, Y., Yağmurlu, N. M., & Esen, A. (2016). Numerical solution of the complex modified Korteweg-de Vries equation by DQM. Journal of Physics: Conference Series, 766(1), 12–28.

Bashir, M. A., & Alhakim, L. A. (2013). New F expansion method and its applications to modified KdV equation. Journal of Mathematics Research, 5(4), 83–96.

Chai, Y., Jia, T., Hao, H., & Zhang, J. (2014). Exp-function method for a generalized mKdV equation. Discrete Dynamics in Nature and Society, 2014(1), 1–8. https://doi.org/10.1155/2014/153974

Gao, F., Yang, X. J., & Ju, Y. (2019). Exact traveling-wave solutions for one-dimensional modified korteweg–de vries equation defined on cantor sets. Fractals, 27(1), 1–9.

He, Y., Zhao, Y.-M., & Long, Y. (2013). New exact solutions for a higher-order wave equation of KdV type using extended F-expansion method. Mathematical Problems in Engineering, 2013(1), 1–8. https://doi.org/10.1155/2014/848069

Islam, M. S., Akbar, M. A., & Khan, K. (2017). The improved F-expansion method and its application to the MEE circular rod equation and the ZKBBM equation. Cogent Mathematics, 4(1), 1–14.

Islam, M. S., Akbar, M. A., & Khan, K. (2018). Analytical solutions of nonlinear Klein–Gordon equation using the improved F-expansion method. Optical and Quantum Electronics, 50(224), 1–11.

Islam, M. S., Khan, K., & Akbar, M. A. (2015). An analytical method for finding exact solutions of modified Korteweg–de Vries equation. Results in Physics, 5, 131–135.

Islam, M. S., Khan, K., & Akbar, M. A. (2017). Application of the improved F-expansion method with Riccati equation to find the exact solution of the nonlinear evolution equations. Journal of the Egyptian Mathematical Society, 25(1), 13–18.

Ji, J. L., & Zhu, Z. N. (2017). Soliton solutions of an integrable nonlocal modified Korteweg–de Vries equation through inverse scattering transform. Journal of Mathematical Analysis and Applications, 453(2), 973–984.

Munk, W. H. (1949). The solitary wave theory and its application to surf problems. Annals of the New York Academy of Sciences, 51(3), 376–424.

Nuruddeen, R. I. (2018). Multiple soliton solutions for the (3+1) conformable space–time fractional modified korteweg–de-vries equations. Journal of Ocean Engineering and Science, 3(1), 11–18.

Pandir, Y., & Duzgun, H. H. (2019). New exact solutions of the space-time fractional cubic Schrodinger equation using the new type F-expansion method. Waves in Random and Complex Media, 29(3), 425–434.

Trogdon, T., Olver, S., & Deconinck, B. (2012). Numerical inverse scattering for the Korteweg–de Vries and modified Korteweg–de Vries equations. Physica D: Nonlinear Phenomena, 241(11), 1003–1025.

Zhao, Y.-M. (2013). F-expansion method and its application for finding new exact solutions to the kudryashov-sinelshchikov equation. Journal of Applied Mathematics, 2013(1), 1–7. https://doi.org/10.1155/2013/89576




DOI: http://dx.doi.org/10.24042/ajpm.v11i1.5153

Refbacks

  • There are currently no refbacks.


 

Creative Commons License
Al-Jabar : Jurnal Pendidikan Matematika is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.