Solitary Wave Solutions of the Generalized (3+1)-Dimensional Shallow Water-Like Equation by Using Modified Kudryashov Method

Year 2021, Volume: 11 Issue: 1, 157 - 165, 30.06.2021

Asıf Yokuş

https://doi.org/10.37094/adyujsci.883428

Abstract

In this study, the generalized (3+1)-dimensional Shallow Water-Like (SWL) equation, which is one of the evolution equations, is taken into consideration. With the help of this evolution equation discussed, the modified Kudryashov method, traveling wave solutions are successfully obtained. In these solutions, graphs of solitary waves to be obtained by giving special values to arbitrary parameters are presented. At the same time, the effect of change of velocity parameter on the behavior on the solitary wave is examined in the solution obtained. The breaking of the wave is discussed. In this study, complex operations and graphic presentations are presented with the use of a ready-made package program.

Keywords

Generalized (3+1)-Dimensional Shallow Water-Like Equation, modified Kudryashov method, traveling wave solution

References

• [1] Yavuz, M., Sene, N., Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method, Axioms, 9(4), 123, 2020.
• [2] Dungey, J.W., Hydromagnetic Waves. In Physics of the Magnetosphere, Based upon the Proceedings of the Conference Held at Boston College, Springer Science & Business Media, 10, 218, 2012.
• [3] Rezazadeh, H., Mirhosseini-Alizamini, S.M., Eslami, M., Rezazadeh, M., Mirzazadeh, M., Abbagari, S., New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation, Optik, 172, 545-553, 2018.
• [4] Alam, M.N., Akbar, M.A., Traveling wave solutions for the mKdV equation and the Gardner equations by new approach of the generalized (G′/G)-expansion method, Journal of the Egyptian Mathematical Society, 22(3), 402-406, 2014.
• [5] Duran, S., Askin, M., Sulaiman, T.A., New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 7(3), 240-247, 2017.
• [6] Duran, S., Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G'/G, 1/G)-Expansion Method, Adıyaman Üniversitesi Fen Bilimleri Dergisi, 10(2), 585-594, 2020.
• [7] Durur, H., Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method, Modern Physics Letters B, 34(03), 2020.
• [8] Saleem, S., Hussain, M.Z., Aziz, I., A reliable algorithm to compute the approximate solution of KdV-type partial differential equations of order seven. Plos one, 16(1), e0244027, 2021.
• [9] Duran, S., Exact Solutions for Time-Fractional Ramani and Jimbo—Miwa Equations by Direct Algebraic Method, Advanced Science, Engineering and Medicine, 12(7), 982-988, 2020.
• [10] Yokus, A., Durur, H., Ahmad, H., Yao, S.W., Construction of different types analytic solutions for the Zhiber-Shabat equation, Mathematics, 8(6), 908, 2020.
• [11] Yokuş, A., Durur, H., Abro, K.A., Kaya, D., Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis, The European Physical Journal Plus, 135(8), 1-19, 2020.
• [12] Duran, S., Kaya, D., Applications of a new expansion method for finding wave solutions of nonlinear differential equations, World Applied Sciences Journal, 18(11), 1582-1592, 2012.
• [13] Sulaiman, T.A., Bulut, H., Yokus, A., Baskonus, H.M., On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering, Indian Journal of Physics, 93(5), 647-656, 2019.
• [14] Durur, H., Yokuş, A., Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(2), 628-636, 2020.
• [15] Zhang, J., Wei, X., Lu, Y., A generalized (G′ G)-expansion method and its applications, Physics Letters A, 372(20), 3653-3658, 2008.
• [16] Rehman, S.U., Yusuf, A., Bilal, M., Younas, U., Younis, M., Sulaiman, T.A., Application of (G'/G^ 2)-expansion method to microstructured solids, magneto-electro-elastic circular rod and (2+ 1)-dimensional nonlinear electrical lines, Journal| MESA, 11(4), 789-803, 2020.
• [17] Yokuş, A., Durur, H., Ahmad, H., Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system, Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531, 2020.
• [18] Durur, H., Yokuş, A., Vakhnenko-Parkes denkleminin hiperbolik tipte yürüyen dalga çözümü, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 13(2), 550-556, 2020.
• [19] Yokuş, A., Durur, H., Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G') expansion method for nonlinear dynamic theory, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21(2), 590-599, 2019.
• [20] Ismael, H.F., Bulut, H., Baskonus, H.M., Optical soliton solutions to the Fokas–Lenells equation via sine-Gordon expansion method and (m+ ({G'}/{G}))-expansion method, Pramana, 94(1), 35, 2020.
• [21] Durur, H., Ilhan, E., Bulut, H., Novel complex wave solutions of the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Fractal and Fractional, 4(3), 41, 2020.
• [22] Yokus, A., Durur, H., Ahmad, H., Thounthong, P., Zhang, Y.F., Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques, Results in Physics, 19, 103409, 2020.
• [23] Duran, S., Doǧan, K., New wave solutions for nonlinear differential equations using an extended Bernoulli equation as a new expansion method, In ITM Web of Conferences (Vol. 22, p. 01035). EDP Sciences (2018).
• [24] Dusunceli, F., Exact solutions for generalized (3+ 1)-dimensional Shallow Water-Like (SWL) equation, In Conference Proceedings of Science and Technology, 2(1), 55-57, 2019.
• [25] Zayed, E.M.E., Traveling wave solutions for higher dimensional nonlinear evolution equations using the G’/G-expansion method, Journal of Applied Mathematics & Informatics, 28(1_2), 383-395, 2010.
• [26] Zhang, Y., Dong, H., Zhang, X., Yang, H., Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-Like equation, Computers & Mathematics with Applications, 73(2), 246-252 2017.
• [27] Baskonus, H.M., Eskitascioglu, E.I., Complex wave surfaces to the extended shallow water wave model with (2+1)-dimensional, Computational Methods for Differential Equations, 8(3), 585-596, 2020.
• [28] Kumar, D., Seadawy, A.R., Joardar, A.K., Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chinese journal of physics, 56(1), 75-85, 2018.
• [29] Yokus, A., Tuz, M., Güngöz, U., On the exact and numerical complex travelling wave solution to the nonlinear Schrödinger equation, Journal of Difference Equations and Applications, 1-12, 2021.